Multimedia Image and Video Processing

Multimedia Image and Video Processing
MULTIMEDIA
IMAGE and VIDEO
PROCESSING
© 2001 by CRC Press LLC
IMAGE PROCESSING SERIES
Series Editor: Phillip A. Laplante
Published Titles
Image and Video Compression for Multimedia Engineering
Yun Q. Shi and Huiyang Sun
Forthcoming Titles
Adaptive Image Processing: A Computational Intelligence
Perspective
Ling Guan, Hau-San Wong, and Stuart William Perry
Shape Analysis and Classification: Theory and Practice
Luciano da Fontoura Costa and Roberto Marcondes Cesar, Jr.
© 2001 by CRC Press LLC
MULTIMEDIA
IMAGE and VIDEO
PROCESSING
Edited by
Ling Guan
Sun-Yuan Kung
Jan Larsen
CRC Press
Boca Raton London New York Washington, D.C.
© 2001 by CRC Press LLC
Library of Congress Cataloging-in-Publication Data
Multimedia image and video processing / edited by Ling Guan, Sun-Yuan Kung, Jan Larsen.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-3492-6 (alk.)
1. Multimedia systems. 2. Image processing—Digital techniques. I. Guan, Ling. II.
Kung, S.Y. (Sun Yuan) III. Larsen, Jan.
QA76.575 2000
006.4′2—dc21
00-030341
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Contents
1 Emerging Standards for Multimedia Applications
Tsuhan Chen
1.1 Introduction
1.2 Standards
1.3 Fundamentals of Video Coding
1.3.1 Transform Coding
1.3.2 Motion Compensation
1.3.3 Summary
1.4 Emerging Video and Multimedia Standards
1.4.1 H.263
1.4.2 H.26L
1.4.3 MPEG-4
1.4.4 MPEG-7
1.5 Standards for Multimedia Communication
1.6 Conclusion
References
2 An Efficient Algorithm and Architecture for Real-Time Perspective Image
Warping
Yi Kang and Thomas S. Huang
2.1 Introduction
2.2 A Fast Algorithm for Perspective Transform
2.2.1 Perspective Transform
2.2.2 Existing Approximation Methods
2.2.3 Constant Denominator Method
2.2.4 Simulation Results
2.2.5 Sprite Warping Algorithm
2.3 Architecture for Sprite Warping
2.3.1 Implementation Issues
2.3.2 Memory Bandwidth Reduction
2.3.3 Architecture
2.4 Conclusion
References
©2001 CRC Press LLC
3
Application-Specific Multimedia Processor Architecture
Yu Hen Hu and Surin Kittitornkun
3.1 Introduction
3.1.1 Requirements of Multimedia Signal Processing (MSP) Hardware
3.1.2 Strategies: Matching Micro-Architecture and Algorithm
3.2 Systolic Array Structure Micro-Architecture
3.2.1 Systolic Array Design Methodology
3.2.2 Array Structures for Motion Estimation
3.3 Dedicated Micro-Architecture
3.3.1 Design Methodologies for Dedicated Micro-Architecture
3.3.2 Feed-Forward Direct Synthesis: Fast Discrete Cosine Transform (DCT)
3.3.3 Feedback Direct Synthesis: Huffman Coding
3.4 Concluding Remarks
References
4 Superresolution of Images with Learned Multiple Reconstruction Kernels
Frank M. Candocia and Jose C. Principe
4.1 Introduction
4.2 An Approach to Superresolution
4.2.1 Comments and Observations
4.2.2 Finding Bases for Image Representation
4.2.3 Description of the Methodology
4.3 Image Acquisition Model
4.4 Relating Kernel-Based Approaches
4.4.1 Single Kernel
4.4.2 Family of Kernels
4.5 Description of the Superresolution Architecture
4.5.1 The Training Data
4.5.2 Clustering of Data
4.5.3 Neighborhood Association
4.5.4 Superresolving Images
4.6 Results
4.7 Issues and Notes
4.8 Conclusions
References
5
Image Processing Techniques for Multimedia Processing
N. Herodotou, K.N. Plataniotis, and A.N. Venetsanopoulos
5.1 Introduction
5.2 Color in Multimedia Processing
5.3 Color Image Filtering
5.3.1 Fuzzy Multichannel Filters
5.3.2 The Membership Functions
5.3.3 A Combined Fuzzy Directional and Fuzzy Median Filter
5.3.4 Application to Color Images
5.4 Color Image Segmentation
5.4.1 Histogram Thresholding
5.4.2 Postprocessing and Region Merging
5.4.3 Experimental Results
5.5 Facial Image Segmentation
5.5.1 Extraction of Skin-Tone Regions
©2001 CRC Press LLC
5.5.2 Postprocessing
5.5.3 Shape and Color Analysis
5.5.4 Fuzzy Membership Functions
5.5.5 Meta-Data Features
5.5.6 Experimental Results
5.6 Conclusions
References
6
Intelligent Multimedia Processing
Ling Guan, Sun-Yuan Kung, and Jenq-Neng Hwang
6.1 Introduction
6.1.1 Neural Networks and Multimedia Processing
6.1.2 Focal Technical Issues Addressed in the Chapter
6.1.3 Organization of the Chapter
6.2 Useful Neural Network Approaches to Multimedia Data Representation, Classification, and Fusion
6.2.1 Multimedia Data Representation
6.2.2 Multimedia Data Detection and Classification
6.2.3 Hierarchical Fuzzy Neural Networks as Linear Fusion Networks
6.2.4 Temporal Models for Multimodal Conversion and Synchronization
6.3 Neural Networks for IMP Applications
6.3.1 Image Visualization and Segmentation
6.3.2 Personal Authentication and Recognition
6.3.3 Audio-to-Visual Conversion and Synchronization
6.3.4 Image and Video Retrieval, Browsing, and Content-Based Indexing
6.3.5 Interactive Human–Computer Vision
6.4 Open Issues, Future Research Directions, and Conclusions
References
7
On Independent Component Analysis for Multimedia Signals
Lars Kai Hansen, Jan Larsen, and Thomas Kolenda
7.1 Background
7.2 Principal and Independent Component Analysis
7.3 Likelihood Framework for Independent Component Analysis
7.3.1 Generalization and the Bias-Variance Dilemma
7.3.2 Noisy Mixing of White Sources
7.3.3 Separation Based on Time Correlation
7.3.4 Likelihood
7.4 Separation of Sound Signals
7.4.1 Sound Separation using PCA
7.4.2 Sound Separation using Molgedey–Schuster ICA
7.4.3 Sound Separation using Bell–Sejnowski ICA
7.4.4 Comparison
7.5 Separation of Image Mixtures
7.5.1 Image Segmentation using PCA
7.5.2 Image Segmentation using Molgedey–Schuster ICA
7.5.3 Discussion
7.6 ICA for Text Representation
7.6.1 Text Analysis
7.6.2 Latent Semantic Analysis — PCA
7.6.3 Latent Semantic Analysis — ICA
©2001 CRC Press LLC
7.7 Conclusion
Acknowledgment
Appendix A
References
8 Image Analysis and Graphics for Multimedia Presentation
Tülay Adali and Yue Wang
8.1 Introduction
8.2 Image Analysis
8.2.1 Pixel Modeling
8.2.2 Model Identification
8.2.3 Context Modeling
8.2.4 Applications
8.3 Graphics Modeling
8.3.1 Surface Reconstruction
8.3.2 Physical Deformable Models
8.3.3 Deformable Surface–Spine Models
8.3.4 Numerical Implementation
8.3.5 Applications
References
9
Combined Motion Estimation and Transform Coding in Compressed Domain
Ut-Va Koc and K.J. Ray Liu
9.1 Introduction
9.2 Fully DCT-Based Motion-Compensated Video Coder Structure
9.3 DCT Pseudo-Phase Techniques
9.4 DCT-Based Motion Estimation
9.4.1 The DXT-ME Algorithm
9.4.2 Computational Issues and Complexity
9.4.3 Preprocessing
9.4.4 Adaptive Overlapping Approach
9.4.5 Simulation Results
9.5 Subpixel DCT Pseudo-Phase Techniques
9.5.1 Subpel Sinusoidal Orthogonality Principles
9.6 DCT-Based Subpixel Motion Estimation
9.6.1 DCT-Based Half-Pel Motion Estimation Algorithm (HDXT-ME)
9.6.2 DCT-Based Quarter-Pel Motion Estimation Algorithm (QDXT-ME
and Q4DXT-ME)
9.6.3 Simulation Results
9.7 DCT-Based Motion Compensation
9.7.1 Integer-Pel DCT-Based Motion Compensation
9.7.2 Subpixel DCT-Based Motion Compensation
9.7.3 Simulation
9.8 Conclusion
References
10 Object-Based Analysis–Synthesis Coding Based on Moving 3D Objects
Jörn Ostermann
10.1 Introduction
10.2 Object-Based Analysis–Synthesis Coding
10.3 Source Models for OBASC
©2001 CRC Press LLC
10.3.1 Camera Model
10.3.2 Scene Model
10.3.3 Illumination Model
10.3.4 Object Model
10.4 Image Analysis for 3D Object Models
10.4.1 Overview
10.4.2 Motion Estimation for R3D
10.4.3 MF Objects
10.5 Optimization of Parameter Coding for R3D and F3D
10.5.1 Motion Parameter Coding
10.5.2 2D Shape Parameter Coding
10.5.3 Coding of Component Separation
10.5.4 Flexible Shape Parameter Coding
10.5.5 Color Parameters
10.5.6 Control of Parameter Coding
10.6 Experimental Results
10.7 Conclusions
References
11 Rate-Distortion Techniques in Image and Video Coding
Aggelos K. Katsaggelos and Gerry Melnikov
11.1 The Multimedia Transmission Problem
11.2 The Operational Rate-Distortion Function
11.3 Problem Formulation
11.4 Mathematical Tools in RD Optimization
11.4.1 Lagrangian Optimization
11.4.2 Dynamic Programming
11.5 Applications of RD Methods
11.5.1 QT-Based Motion Estimation and Motion-Compensated Interpolation
11.5.2 QT-Based Video Encoding
11.5.3 Hybrid Fractal/DCT Image Compression
11.5.4 Shape Coding
11.6 Conclusions
References
12 Transform Domain Techniques for Multimedia Image and Video Coding
S. Suthaharan, S.W. Kim, H.R. Wu, and K.R. Rao
12.1 Coding Artifacts Reduction
12.1.1 Introduction
12.1.2 Methodology
12.1.3 Experimental Results
12.1.4 More Comparison
12.2 Image and Edge Detail Detection
12.2.1 Introduction
12.2.2 Methodology
12.2.3 Experimental Results
12.3 Summary
References
©2001 CRC Press LLC
13 Video Modeling and Retrieval
Yi Zhang and Tat-Seng Chua
13.1 Introduction
13.2 Modeling and Representation of Video: Segmentation vs.
Stratification
13.2.1 Practical Considerations
13.3 Design of a Video Retrieval System
13.3.1 Video Segmentation
13.3.2 Logging of Shots
13.3.3 Modeling the Context between Video Shots
13.4 Retrieval and Virtual Editing of Video
13.4.1 Video Shot Retrieval
13.4.2 Scene Association Retrieval
13.4.3 Virtual Editing
13.5 Implementation
13.6 Testing and Results
13.7 Conclusion
References
14 Image Retrieval in Frequency Domain Using DCT Coefficient Histograms
Jose A. Lay and Ling Guan
14.1 Introduction
14.1.1 Multimedia Data Compression
14.1.2 Multimedia Data Retrieval
14.1.3 About This Chapter
14.2 The DCT Coefficient Domain
14.2.1 A Matrix Description of the DCT
14.2.2 The DCT Coefficients in JPEG and MPEG Media
14.2.3 Energy Histograms of the DCT Coefficients
14.3 Frequency Domain Image/Video Retrieval Using DCT Coefficients
14.3.1 Content-Based Retrieval Model
14.3.2 Content-Based Search Processing Model
14.3.3 Perceiving the MPEG-7 Search Engine
14.3.4 Image Manipulation in the DCT Domain
14.3.5 The Energy Histogram Features
14.3.6 Proximity Evaluation
14.3.7 Experimental Results
14.4 Conclusions
References
15 Rapid Similarity Retrieval from Image and Video
Kim Shearer, Svetha Venkatesh, and Horst Bunke
15.1 Introduction
15.1.1 Definitions
15.2 Image Indexing and Retrieval
15.3 Encoding Video Indices
15.4 Decision Tree Algorithms
15.4.1 Decision Tree-Based LCSG Algorithm
15.5 Decomposition Network Algorithm
15.5.1 Decomposition-Based LCSG Algorithm
15.6 Results of Tests Over a Video Database
©2001 CRC Press LLC
15.6.1 Decomposition Network Algorithm
15.6.2 Inexact Decomposition Algorithm
15.6.3 Decision Tree
15.6.4 Results of the LCSG Algorithms
15.7 Conclusion
References
16 Video Transcoding
Tzong-Der Wu, Jenq-Neng Hwang, and Ming-Ting Sun
16.1 Introduction
16.2 Pixel-Domain Transcoders
16.2.1 Introduction
16.2.2 Cascaded Video Transcoder
16.2.3 Removal of Frame Buffer and Motion Compensation Modules
16.2.4 Removal of IDCT Module
16.3 DCT Domain Transcoder
16.3.1 Introduction
16.3.2 Architecture of DCT Domain Transcoder
16.3.3 Full-Pixel Interpolation
16.3.4 Half-Pixel Interpolation
16.4 Frame-Skipping in Video Transcoding
16.4.1 Introduction
16.4.2 Interpolation of Motion Vectors
16.4.3 Search Range Adjustment
16.4.4 Dynamic Frame-Skipping
16.4.5 Simulation and Discussion
16.5 Multipoint Video Bridging
16.5.1 Introduction
16.5.2 Video Characteristics in Multipoint Video Conferencing
16.5.3 Results of Using the Coded Domain and Transcoding Approaches
16.6 Summary
References
17 Multimedia Distance Learning
Sachin G. Deshpande, Jenq-Neng Hwang, and Ming-Ting Sun
17.1 Introduction
17.2 Interactive Virtual Classroom Distance Learning Environment
17.2.1 Handling the Electronic Slide Presentation
17.2.2 Handling Handwritten Text
17.3 Multimedia Features for On-Demand Distance Learning Environment
17.3.1 Hypervideo Editor Tool
17.3.2 Automating the Multimedia Features Creation for On-Demand System
17.4 Issues in the Development of Multimedia Distance Learning
17.4.1 Error Recovery, Synchronization, and Delay Handling
17.4.2 Fast Encoding and Rate Control
17.4.3 Multicasting
17.4.4 Human Factors
17.5 Summary and Conclusion
References
©2001 CRC Press LLC
18 A New Watermarking Technique for Multimedia Protection
Chun-Shien Lu, Shih-Kun Huang, Chwen-Jye Sze, and Hong-Yuan Mark Liao
18.1 Introduction
18.1.1 Watermarking
18.1.2 Overview
18.2 Human Visual System-Based Modulation
18.3 Proposed Watermarking Algorithms
18.3.1 Watermark Structures
18.3.2 The Hiding Process
18.3.3 Semipublic Authentication
18.4 Watermark Detection/Extraction
18.4.1 Gray-Scale Watermark Extraction
18.4.2 Binary Watermark Extraction
18.4.3 Dealing with Attacks Including Geometric Distortion
18.5 Analysis of Attacks Designed to Defeat HVS-Based Watermarking
18.6 Experimental Results
18.6.1 Results of Hiding a Gray-Scale Watermark
18.6.2 Results of Hiding a Binary Watermark
18.7 Conclusion
References
19 Telemedicine: A Multimedia Communication Perspective
Chang Wen Chen and Li Fan
19.1 Introduction
19.2 Telemedicine: Need for Multimedia Communication
19.3 Telemedicine over Various Multimedia Communication Links
19.3.1 Telemedicine via ISDN
19.3.2 Medical Image Transmission via ATM
19.3.3 Telemedicine via the Internet
19.3.4 Telemedicine via Mobile Wireless Communication
19.4 Conclusion
References
©2001 CRC Press LLC
Preface
Multimedia is one of the most important aspects of the information era. Although there are
books dealing with various aspects of multimedia, a book comprehensively covering system,
processing, and application aspects of image and video data in a multimedia environment is
urgently needed. Contributed by experts in the field, this book serves this purpose.
Our goal is to provide in a single volume an introduction to a variety of topics in image and
video processing for multimedia. An edited compilation is an ideal format for treating a broad
spectrum of topics because it provides the opportunity for each topic to be written by an expert
in that field.
The topic of the book is processing images and videos in a multimedia environment. It covers
the following subjects arranged in two parts: (1) standards and fundamentals: standards, multimedia architecture for image processing, multimedia-related image processing techniques,
and intelligent multimedia processing; (2) methodologies, techniques, and applications: image and video coding, image and video storage and retrieval, digital video transmission, video
conferencing, watermarking, distance education, video on demand, and telemedicine.
The book begins with the existing standards for multimedia, discussing their impacts to
multimedia image and video processing, and pointing out possible directions for new standards.
The design of multimedia architectures is based on the standards. It deals with the way
visual data is being processed and transmitted at a more practical level. Current and new
architectures, and their pros and cons, are presented and discussed in Chapters 2 to 4.
Chapters 5 to 8 focus on conventional and intelligent image processing techniques relevant to
multimedia, including preprocessing, segmentation, and feature extraction techniques utilized
in coding, storage, and retrieval and transmission, media fusion, and graphical interface.
Compression and coding of video and images are among the focusing issues in multimedia.
New developments in transform- and motion-based algorithms in the compressed domain,
content- and object-based algorithms, and rate–distortion-based encoding are presented in
Chapters 9 to 12.
Chapters 13 to 15 tackle content-based image and video retrieval. They cover video modeling
and retrieval, retrieval in the transform domain, indexing, parsing, and real-time aspects of
retrieval.
The last chapters of the book (Chapters 16 to 19) present new results in multimedia application areas, including transcoding for multipoint video conferencing, distance education,
watermarking techniques for multimedia processing, and telemedicine.
Each chapter has been organized so that it can be covered in 1 to 2 weeks when this book is
used as a principal reference or text in a senior or graduate course at a university.
It is generally assumed that the reader has prior exposure to the fundamentals of image and
video processing. The chapters have been written with an emphasis on a tutorial presentation
so that the reader interested in pursuing a particular topic further will be able to obtain a solid
introduction to the topic through the appropriate chapter in this book. While the topics covered
are related, each chapter can be read and used independently of the others.
©2001 CRC Press LLC
This book is primarily a result of the collective efforts of the chapter authors. We are
very grateful for their enthusiastic support, timely response, and willingness to incorporate
suggestions from us, from other contributing authors, and from a number of our colleagues
who served as reviewers.
Ling Guan
Sun-Yuan Kung
Jan Larsen
©2001 CRC Press LLC
Contributors
Tülay Adali University of Maryland, Baltimore, Maryland
Horst Bunke Institute für Informatik und Angewandte Mathematik, Universität Bern,
Switzerland
Frank M. Candocia University of Florida, Gainesville, Florida
Chang Wen Chen University of Missouri, Columbia, Missouri
Tsuhan Chen Carnegie Mellon University, Pittsburgh, Pennsylvania
Tat-Seng Chua National University of Singapore, Kentridge, Singapore
Sachin G. Deshpande University of Washington, Seattle, Washington
Li Fan University of Missouri, Columbia, Missouri
Ling Guan University of Sydney, Sydney, Australia
Lars Kai Hansen Technical University of Denmark, Lyngby, Denmark
N. Herodotou University of Toronto, Toronto, Ontario, Canada
Yu Hen Hu University of Wisconsin-Madison, Madison, Wisconsin
Shih-Kun Huang Institute of Information Science, Academia Sinica, Taiwan, China
Thomas S. Huang Beckman Institute, University of Illinois at Urbana-Champaign,
Urbana, Illinois
Jenq-Neng Hwang University of Washington, Seattle, Washington
Yi Kang Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, Illinois
Aggelos K. Katsaggelos Northwestern University, Evanston, Illinois
S.W. Kim Korea Advanced Institute of Science and Technology, Taejon, Korea
Surin Kittitornkun University of Wisconsin-Madison, Madison, Wisconsin
Ut-Va Koc Lucent Technologies Bell Labs, Murray Hill, New Jersey
Thomas Kolenda Technical University of Denmark, Lyngby, Denmark
©2001 CRC Press LLC
Sun-Yuan Kung Princeton University, Princeton, New Jersey
Jan Larsen Technical University of Denmark, Lyngby, Denmark
Jose A. Lay University of Sydney, Sydney, Australia
Hong-Yuan Mark Liao Institute of Information Science, Academia Sinica, Taipei, Taiwan
K.J. Ray Liu University of Maryland, College Park, Maryland
Chun-Shien Lu Institute of Information Science, Academia Sinica, Taipei, Taiwan
Gerry Melnikov Northwestern University, Evanston, Illinois
Jörn Ostermann AT&T Labs — Research, Red Bank, New Jersey
K.N. Plataniotis University of Toronto, Toronto, Ontario, Canada
Jose C. Principe University of Florida, Gainesville, Florida
K.R. Rao University of Texas at Arlington, Arlington, Texas
Kim Shearer Curtin University of Technology, Perth, Australia
Ming-Ting Sun University of Washington, Seattle, Washington
S. Suthaharan Tennessee State University, Nashville, Tennessee
Chwen-Jye Sze Institute of Information Science, Academia Sinica, Taiwan, China
A.N. Venetsanopoulos University of Toronto, Toronto, Ontario, Canada
Svetha Venkatesh Curtin University of Technology, Perth, Australia
Yue Wang Catholic University of America, Washington, D.C.
H.R. Wu Monash University, Clayton, Victoria, Australia
Tzong-Der Wu University of Washington, Seattle, Washington
Yi Zhang National University of Singapore, Kent Ridge, Singapore
©2001 CRC Press LLC
Chapter 1
Emerging Standards for Multimedia Applications
Tsuhan Chen
1.1
Introduction
Due to the rapid growth of multimedia communication, multimedia standards have received
much attention during the last decade. This is illustrated by the extremely active development
in several international standards including H.263, H.263 Version 2 (informally known as
H.263+), H.26L, H.323, MPEG-4, and MPEG-7. H.263 Version 2, developed to enhance
an earlier video coding standard H.263 in terms of coding efficiency, error resilience, and
functionalities, was finalized in early 1997. H.26L is an ongoing standard activity searching
for advanced coding techniques that can be fundamentally different from H.263. MPEG-4, with
its emphasis on content-based interactivity, universal access, and compression performance,
was finalized with Version 1 in late 1998 and with Version 2 1 year later. The MPEG-7 activity,
which has begun since the first call for proposals in late 1998, is developing a standardized
description of multimedia materials, including images, video, text, and audio, in order to
facilitate search and retrieval of multimedia content. By examining the development of these
standards in this chapter, we will see the trend of video technologies progressing from pixelbased compression techniques to high-level image understanding. At the end of the chapter,
we will also introduce H.323, an ITU-T standard designed for multimedia communication over
networks that do not guarantee quality of service (QoS), and hence very suitable for Internet
applications.
The chapter is outlined as follows. In Section 1.2, we introduce the basic concepts of
standards activities. In Section 1.3, we review the fundamentals of video coding. In Section 1.4,
we study recent video and multimedia standards, including H.263, H.26L, MPEG-4, and
MPEG-7. In Section 1.5, we briefly introduce standards for multimedia communication,
focusing on ITU-T H.323. We conclude the chapter with a brief discussion on the trend of
multimedia standards (Section 1.6).
1.2
Standards
Standards are essential for communication. Without a common language that both the
transmitter and the receiver understand, communication is impossible. In multimedia communication systems the language is often defined as a standardized bitstream syntax. Adoption of
©2001 CRC Press LLC
standards by equipment manufacturers and service providers increases the customer base and
hence results in higher volume and lower cost. In addition, it offers consumers more freedom
of choice among manufacturers, and therefore is welcomed by the consumers.
For transmission of video or multimedia content, standards play an even more important
role. Not only do the transmitter and the receiver need to speak the same language, but the
language also has to be efficient (i.e., provide high compression of the content), due to the
relatively large amount of bits required to transmit uncompressed video and multimedia data.
Note, however, that standards do not specify the whole communication process. Although
it defines the bitstream syntax and hence the decoding process, a standard usually leaves the
encoding processing open to the vendors. This is the standardize-the-minimum philosophy
widely adopted by most video and multimedia standards. The reason is to leave room for
competition among different vendors on the encoding technologies, and to allow future technologies to be incorporated into the standards, as they become mature. The consequence
is that a standard does not guarantee the quality of a video encoder, but it ensures that any
standard-compliant decoder can properly receive and decode the bitstream produced by any
encoder.
Existing standards may be classified into two groups. The first group comprises those
that are decided upon by a mutual agreement between a small number of companies. These
standards can become very popular in the marketplace, thereby leading other companies to
also accept them. So, they are often referred to as the de facto standards. The second set of
standards is called the voluntary standards. These standards are defined by volunteers in open
committees. These standards are agreed upon based on the consensus of all the committee
members. These standards need to stay ahead of the development of technologies, in order
to avoid any disagreement between those companies that have already developed their own
proprietary techniques.
For multimedia communication, there are several organizations responsible for the definition
of voluntary standards. One is the International Telecommunications Union–Telecommunication Standardization Sector (ITU-T), originally known as the International Telephone and
Telegraph Consultative Committee (CCITT). Another one is the International Standardization
Organization (ISO). Along with the Internet Engineering Task Force (IETF), which defines
multimedia delivery for the Internet, these three organizations form the core of standards
activities for modern multimedia communication.
Both ITU-T and ISO have defined different standards for video coding. These standards are
summarized in Table 1.1. The major differences between these standards lie in the operating bit
rates and the applications for which they are targeted. Note, however, that each standard allows
for operating at a wide range of bit rates; hence each can be used for all the applications in
principle. All these video-related standards follow a similar framework in terms of the coding
algorithms; however, there are differences in the ranges of parameters and some specific coding
modes.
1.3
Fundamentals of Video Coding
In this section, we review the fundamentals of video coding. Figure 1.1 shows the general
data structure of digital video. A video sequence is composed of pictures updated at a certain
rate, sometimes with a number of pictures grouped together (group of pictures [GOP]). Each
picture is composed of several groups of blocks (GOBs), sometimes called the slices. Each
GOB contains a number of macroblocks (MBs), and each MB is composed of four luminance
©2001 CRC Press LLC
Table 1.1 Video Coding Standards Developed by Various Organizations
Organization
ITU-T
ISO
ISO
ITU-T
ISO
Standard
H.261
IS 11172-2
MPEG-1 Video
IS 13818-2
MPEG-2 Videoa
H.263
IS 14496-2
MPEG-4 Video
H.26L
Typical Bit Rate
1.2 Mbits/s
Typical Applications
ISDN Video Phone
CD-ROM
4–80 Mbits/s
SDTV, HDTV
p× 64 kbits/s, p =1 . . . 30
64 kbits/s or below
24–1024 kbits/s
PSTN Video Phone
A variety of
applications
<64 kbits/s
A variety of
ITU-T
applications
a ITU-T also actively participated in the development of MPEG-2 Video. In fact,
ITU-T H.262 refers to the same standard and uses the same text as IS 13818-2.
blocks, 8 × 8 pixels each, which represent the intensity variation, and two chrominance blocks
(CB and CR ), which represent the color information.
FIGURE 1.1
Data structure of digital video.
The coding algorithm widely used in most video coding standards is a combination of the
discrete cosine transform (DCT) and motion compensation. DCT is applied to each block to
transform the pixel values into DCT coefficients in order to remove the spatial redundancy. The
DCT coefficients are then quantized and zigzag scanned to provide a sequence of symbols, with
each symbol representing a number of zero coefficients followed by one nonzero coefficient.
These symbols are then converted into bits by entropy coding (e.g., variable-length coding
[VLC]). On the other hand, temporal redundancy is removed by motion compensation (MC).
The encoder estimates the motion by matching each macroblock in the current picture with
the reference picture (usually the previous picture) to find the motion vector that specifies the
best matching area. The residue is then coded and transmitted with the motion vectors. We
now discuss these techniques in detail.
©2001 CRC Press LLC
1.3.1
Transform Coding
Transform coding has been widely used to remove redundancy between data samples. In
transform coding, a set of data samples is first linearly transformed into a set of transform
coefficients. These coefficients are then quantized and coded. A proper linear transform
should decorrelate the input samples, and hence remove the redundancy. Another way to look
at this is that a properly chosen transform can concentrate the energy of input samples into a
small number of transform coefficients, so that resulting coefficients are easier to code than
the original samples.
The most commonly used transform for video coding is the DCT [1, 2]. In terms of both
objective coding gain and subjective quality, the DCT performs very well for typical image
data. The DCT operation can be expressed in terms of matrix multiplication by:
Z = CT XC
where X represents the original image block and Z represents the resulting DCT coefficients.
The elements of C, for an 8 × 8 image block, are defined as
√
(2m + 1)nπ
1/(2 2) when n = 0
Cmn = kn cos
where kn =
16
1/2
otherwise
After the transform, the DCT coefficients in Z are quantized. Quantization implies loss of
information and is the primary source of actual compression in the system. The quantization
step size depends on the available bit rate and can also depend on the coding modes. Except
for the intra-DC coefficients that are uniformly quantized with a step size of 8, an enlarged
“dead zone” is used to quantize all other coefficients in order to remove noise around zero.
Typical input–output relations for these two cases are shown in Figure 1.2.
FIGURE 1.2
Quantization with and without the “dead zone.”
The quantized 8 × 8 DCT coefficients are then converted into a one-dimensional (1D)
array for entropy coding by an ordered scanning operation. Figure 1.3 shows the zigzag scan
order used in most standards for this conversion. For typical video data, most of the energy
concentrates in the low-frequency coefficients (the first few coefficients in the scan order) and
the high-frequency coefficients are usually very small and often quantized to zero. Therefore,
the scan order in Figure 1.3 can create long runs of zero-valued coefficients, which is important
for efficient entropy coding, as we discuss in the next paragraph.
©2001 CRC Press LLC
FIGURE 1.3
Scan order of the DCT coefficients.
The resulting 1D array is then decomposed into segments, with each segment containing
either a number of consecutive zeros followed by a nonzero coefficient or a nonzero coefficient
without any preceding zeros. Let an event represent the pair (run, level), where “run” represents
the number of zeros and “level” represents the magnitude of the nonzero coefficient. This
coding process is sometimes called “run-length coding.” Then, a table is built to represent
each event by a specific codeword (i.e., a sequence of bits). Events that occur more often
are represented by shorter codewords, and less frequent events are represented by longer
codewords. This entropy coding process is therefore called VLC or Huffman coding. Table 1.2
shows part of a sample VLC table. In this table, the last bit “s” of each codeword denotes the
sign of the level, “0” for positive and “‘1” for negative. It can be seen that more likely events
(i.e., short runs and low levels), are represented with short codewords, and vice versa.
At the decoder, all the above steps are reversed one by one. Note that all the steps can be
exactly reversed except for the quantization step, which is where loss of information arises.
This is known as “lossy” compression.
1.3.2
Motion Compensation
The transform coding described in the previous section removes spatial redundancy within
each frame of video content. It is therefore referred to as intra coding. However, for video
material, inter coding is also very useful. Typical video material contains a large amount of
redundancy along the temporal axis. Video frames that are close in time usually have a large
amount of similarity. Therefore, transmitting the difference between frames is more efficient
than transmitting the original frames. This is similar to the concept of differential coding and
predictive coding. The previous frame is used as an estimate of the current frame, and the
residual, the difference between the estimate and the true value, is coded. When the estimate
is good, it is more efficient to code the residual than the original frame.
Consider the fact that typical video material is a camera’s view of moving objects. Therefore,
it is possible to improve the prediction result by first estimating the motion of each region in
the scene. More specifically, the encoder can estimate the motion (i.e., displacement) of each
block between the previous frame and the current frame. This is often achieved by matching
each block (actually, macroblock) in the current frame with the previous frame to find the best
matching area,1 as illustrated in Figure 1.4. This area is then offset accordingly to form the
estimate of the corresponding block in the current frame. Now, the residue has much less energy
than the original signal and therefore is much easier to code to within a given average error.
©2001 CRC Press LLC
Table 1.2 Part of a Sample
VLC Table
Run
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
...
Level
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
1
2
3
4
5
1
2
3
4
...
Code
11s
0100 s
0010 1s
0000 110s
0010 0110 s
0010 0001 s
0000 0010 10s
0000 0001 1101 s
0000 0001 1000 s
0000 0001 0011 s
0000 0001 0000 s
0000 0000 1101 0s
0000 0000 1100 1s
0000 0000 1100 0s
0000 0000 1011 1s
011s
0001 10s
0010 0101 s
0000 0011 00s
0000 0001 1011 s
0000 0000 1011 0s
0000 0000 1010 1s
0101 s
0000 100s
0000 0010 11s
0000 0001 0100 s
0000 0000 1010 0s
0011 1s
0010 0100 s
0000 0001 1100 s
0000 0000 1001 1s
...
This process is called motion compensation (MC), or more precisely, motion-compensated
prediction [3, 4]. The residue is then coded using the same process as that of intra coding.
Pictures that are coded without any reference to previously coded pictures are called intra
pictures, or simply I pictures (or I frames). Pictures that are coded using a previous picture
as a reference for prediction are called inter or predicted pictures, or simply P pictures (or
P frames). However, note that a P picture may also contain some intra-coded macroblocks.
The reason is as follows. For a certain macroblock, it may be impossible to find a good enough
matching area in the reference picture to be used for prediction. In this case, direct intra coding
of such a macroblock is more efficient. This situation happens often when there is occlusion
or intense motion in the scene.
1 Note, however, that the standard does not specify how motion estimation should be done. Motion estimation can be a
very computationally intensive process and is the source of much of the variation in the quality produced by different
encoders.
©2001 CRC Press LLC
FIGURE 1.4
Motion compensation.
During motion compensation, in addition to bits used for coding the DCT coefficients of the
residue, extra bits are required to carry information about the motion vectors. Efficient coding
of motion vectors is therefore also an important part of video coding. Because motion vectors
of neighboring blocks tend to be similar, differential coding of the horizontal and vertical
components of motion vectors is used. That is, instead of coding motion vectors directly, the
previous motion vector or multiple neighboring motion vectors are used as a prediction for
the current motion vector. The difference, in both the horizontal and vertical components,
is then coded using a VLC table, part of which is shown in Table 1.3. Note two things in
Table 1.3 Part of a
VLC Table for Coding
Motion Vectors
MVD
...
0
1
2 & −30
3 & −29
4 & −28
5 & −27
6 & −26
7 & −25
0000 0111
0000 1001
0000 1011
0000 111
0001 1
0011
011
1
010
0010
0001 0
0000 110
0000 1010
0000 1000
0000 0110
−7 & 25
−6 & 26
−5 & 27
−4 & 28
−3 & 29
−2 & 30
−1
...
©2001 CRC Press LLC
Code
...
...
this table. First, short codewords are used to represent small differences, because these are
more likely events. Second, one codeword can represent up to two possible values for motion
vector difference. Because the allowed range of both the horizontal component and the vertical
component of motion vectors is restricted to the range of −15 to +15, only one will yield a
motion vector with the allowable range. Note that the ±15 range for motion vector values
may not be adequate for high-resolution video with large amounts of motion; some standards
provide a way to extend this range as either a basic or optional feature of their design.
1.3.3
Summary
Video coding can be summarized into the block diagram in Figure 1.5. The left-hand side
of the figure shows the encoder and the right-hand side shows the decoder. At the encoder, the
input picture is compared with the previously decoded frame with motion compensation. The
difference signal is DCT transformed and quantized, and then entropy coded and transmitted.
At the decoder, the decoded DCT coefficients are inverse DCT transformed and then added to
the previously decoded picture with loop-filtered motion compensation.
FIGURE 1.5
Block diagram of video coding.
1.4
Emerging Video and Multimedia Standards
Most early video coding standards, including H.261, MPEG-1, and MPEG-2, use the same
hybrid DCT-MC framework as described in the previous sections, and they have very specific
©2001 CRC Press LLC
functionalities and targeted applications. The new generation of video coding standards,
however, contains many optional modes and supports a larger variety of functionalities. We
now introduce the new functionalities provided in these new standards, including H.263, H.26L,
MPEG-4, and MPEG-7.
1.4.1
H.263
The H.263 design project started in 1993, and the standard was approved at a meeting of
ITU-T SG 15 in November 1995 (and published in March 1996) [5]. Although the original
goal of this endeavor was to design a video coding standard suitable for applications with bit
rates around 20 kbits/s (the so-called very-low-bit-rate applications), it became apparent that
H.263 could provide a significant improvement over H.261 at any bit rate. In essence, H.263
combines the features of H.261 with several new methods, including the half-pixel motion
compensation first found in MPEG-1 and other techniques. Compared to an earlier standard
H.261, H.263 can provide 50% or more savings in the bit rate needed to represent video at a
given level of perceptual quality at very low bit rates. In terms of signal-to-noise ratio (SNR),
H.263 can provide about a 3-dB gain over H.261 at these very low rates. In fact, H.263 provides
superior coding efficiency to that of H.261 at all bit rates (although not nearly as dramatic an
improvement when operating above 64 kbits/s). H.263 can also provide a significant bit rate
savings when compared to MPEG-1 at higher rates (perhaps 30% at around 1 Mbit/s).
H.263 represents today’s state of the art for standardized video coding. Essentially any bit
rate, picture resolution, and frame rate for progressive-scanned video content can be efficiently
coded with H.263. H.263 is structured around a “baseline” mode of operation, which defines
the fundamental features supported by all decoders, plus a number of optional enhanced modes
of operation for use in customized or higher performance applications. Because of its high
performance, H.263 was chosen as the basis of the MPEG-4 video design, and its baseline
mode is supported in MPEG-4 without alteration. Many of its optional features are now also
found in some form in MPEG-4.
In addition to the baseline mode, H.263 includes a number of optional enhancement features
to serve a variety of applications. The original version of H.263 had about four such optional
modes. The latest version of H.263, known informally as H.263+ or H.263 Version 2, extends
the number of negotiable options to 16 [5]. These enhancements provide either improved
quality or additional capabilities to broaden the range of applications. Among the new negotiable coding options specified by H.263 Version 2, five of them are intended to improve
the coding efficiency. These are the advanced intra coding mode, alternate inter VLC mode,
modified quantization mode, deblocking filter mode, and improved PB-frame mode. Three
optional modes are especially designed to address the needs of mobile video and other unreliable transport environments. They are the slice structured mode, reference picture selection
mode, and independent segment decoding mode. The temporal, SNR, and spatial scalability
modes support layered bitstream scalability, similar to those provided by MPEG-2.
There are two other enhancement modes in H.263 Version 2: the reference picture resampling mode and reduced-resolution update mode. The former allows a previously coded picture
to be resampled, or warped, before it is used as a reference picture.
Another feature of H.263 Version 2 is the use of supplemental information, which may
be included in the bitstream to signal enhanced display capabilities or to provide tagging
information for external use. One use of the supplemental enhancement information is to
specify the chroma key for representing transparent and semitransparent pixels [6].
Each optional mode is useful in some applications, but few manufacturers would want to
implement all of the options. Therefore, H.263 Version 2 contains an informative specification
of three levels of preferred mode combinations to be supported. Each level contains a number
©2001 CRC Press LLC
of options to be supported by an equipment manufacturer. Such information is not a normative
part of the standard. It is intended only to provide manufacturers some guidelines as to which
modes are more likely to be widely adopted across a full spectrum of terminals and networks.
Three levels of preferred modes are described in H.263 Version 2, and each level supports
the optional modes specified in lower levels. In addition to the level structure is a discussion
indicating that because the advanced prediction mode was the most beneficial of the original H.263 modes, its implementation is encouraged not only for its performance but for its
backward compatibility with the original H.263.
The first level is composed of
• The advanced intra coding mode
• The deblocking filter mode
• Full-frame freeze by supplementary enhancement information
• The modified quantization mode
Level 2 supports, in addition to modes supported in Level 1
• The unrestricted motion vector mode
• The slice structured mode
• The simplest resolution-switching form of the reference picture resampling mode
In addition to these modes, Level 3 further supports
• The advanced prediction mode
• The improved PB-frames mode
• The independent segment decoding mode
• The alternative inter VLC mode
1.4.2
H.26L
H.26L is an effort to seek efficient video coding algorithms that can be fundamentally different from the MC-DCT framework used in H.261 and H.263. When finalized, it will be
a video coding standard that provides better quality and more functionalities than existing
standards. The first call for proposals for H.26L was issued in January 1998. According to the
call for proposals, H.26L is aimed at very-low-bit-rate, real-time, low-end-to-end delay coding
for a variety of source materials. It is expected to have low complexity, permitting software
implementation, enhanced error robustness (especially for mobile networks), and adaptable
rate control mechanisms. The applications targeted by H.26L include real-time conversational
services, Internet video applications, sign language and lip-reading communication, video storage and retrieval services (e.g., VOD), video store and forward services (e.g., video mail), and
multipoint communication over heterogeneous networks. The schedule for H.26L activities is
shown in Table 1.4.
©2001 CRC Press LLC
Table 1.4 Schedule for H.26L
Jan 1998
Nov 1998
Jan 1999
Nov 1999
Aug 2001
May 2002
1.4.3
Call for proposals
Evaluation of the proposals
1st test model of H.26L (TML1)
Final major feature adoptions
Determination
Decision
MPEG-4
MPEG-4 [7] was originally created as a standard for very low bit rate coding of limitedcomplexity audiovisual material. The scope was later extended to supporting new functionalities such as content-based interactivity, universal access, and high-compression coding of
general material for a wide bit-rate range. It also emphasizes flexibility and extensibility. The
concept of content-based coding of MPEG-4 is shown inz Figure 1.6. Each input pictureis
decomposed into a number of arbitrarily shaped regions called video object planes (VOPs).
Each VOP is then coded with a coding algorithm that is similar to H.263. The shape of each
VOP is encoded using context-based arithmetic coding.
FIGURE 1.6
Object-layer-based video coding in MPEG-4.
Comparing MPEG-4 video coding with earlier standards, the major difference lies in the
representation and compression of the shape information. In addition, one activity that distinguishes MPEG-4 from the conventional video coding standards is the synthetic and natural
hybrid coding (SNHC). The target technologies studied by the SNHC subgroup include face
animation, coding and representation of 2D dynamic mesh, wavelet-based static texture coding, view-dependent scalability, and 3D geometry compression. These functionalities used to
be considered only by the computer graphics community. MPEG-4 SNHC successfully brings
these tools into the scope of a video standard, and hence bridges computer graphics and image
processing.
©2001 CRC Press LLC
1.4.4
MPEG-7
MPEG-7 is targeted to produce a standardized description of multimedia material including images, text, graphics, 3D models, audio, speech, analog/digital video, and composition
information. The standardized description will enable fast and efficient search and retrieval
of multimedia content and advance the search mechanism from a text-based approach to a
content-based approach. Currently, feature extraction and the search engine design are considered to be outside of the standard. Nevertheless, when MPEG-7 is finalized and widely
adopted, efficient implementation for feature extraction and search mechanism will be very
important. The applications of MPEG-7 can be categorized into pull and push scenarios. For
the pull scenario, MPEG-7 technologies can be used for information retrieval from a database
or from the Internet. For the push scenario, MPEG-7 can provide the filtering mechanism
applied to multimedia content broadcast from an information provider.
As pointed out earlier in this chapter, instead of trying to extract relevant features, manually
or automatically, from original or compressed video, a better approach for content retrieval
should be to design a new standard in which such features, often referred to as meta-data,
are already available. MPEG-7, an ongoing effort by the Moving Picture Experts Group, is
working exactly toward this goal (i.e., the standardization of meta-data for multimedia content
indexing and retrieval).
MPEG-7 is an activity triggered by the growth of digital audiovisual information. The group
strives to define a “multimedia content description interface” to standardize the description of
various types of multimedia content, including still pictures, graphics, 3D models, audio,
speech, video, and composition information. It may also deal with special cases such as facial
expressions and personal characteristics.
The goal of MPEG-7 is exactly the same as the focus of this chapter (i.e., to enable efficient
search and retrieval of multimedia content). Once finalized, it will transform the text-based
search and retrieval (e.g., keywords), as is done by most of the multimedia databases nowadays,
into a content-based approach (e.g., using color, motion, or shape information). MPEG-7 can
also be thought of as a solution to describing multimedia content. If one looks at PDF (portable
document format) as a standard language to describe text and graphic documents, then MPEG7 will be a standard description for all types of multimedia data, including audio, images, and
video.
Compared with earlier MPEG standards, MPEG-7 possesses some essential differences. For
example, MPEG-1, 2, and 4 all focus on the representation of audiovisual data, but MPEG-7
will focus on representing the meta-data (information about data). MPEG-7, however, may
utilize the results of previous MPEG standards (e.g., the shape information in MPEG-4 or the
motion vector field in MPEG-1 and 2).
Figure 1.7 shows the scope of the MPEG-7 standard. Note that feature extraction is outside
the scope of MPEG-7, as is the search engine. This is owing to one approach constantly
taken by most of the standard activities (i.e., “to standardize the minimum”). Therefore, the
analysis (feature extraction) should not be standardized, so that after MPEG-7 is finalized,
various analysis tools can be further improved over time. This also leaves room for competition among vendors and researchers. This is similar to MPEG-1 not specifying motion
estimation and MPEG-4 not specifying segmentation algorithms. Likewise, the query process
(the search engine) should not be standardized. This allows the design of search engines and
query languages to adapt to different application domains, and also leaves room for further
improvement and competition. Summarizing, MPEG-7 takes the approach of standardizing
only what is necessary so that the description for the same content may adapt to different users
and different application domains.
We now explain a few concepts of MPEG-7. One goal of MPEG-7 is to provide a standardized method of describing features of multimedia data. For images and video, colors or
©2001 CRC Press LLC
FIGURE 1.7
The scope of MPEG-7.
motion are example features that are desirable in many applications. MPEG-7 will define a
certain set of descriptors to describe these features. For example, the color histogram can be a
very suitable descriptor for color characteristics of an image, and motion vectors (commonly
available in compressed video bitstreams) form a useful descriptor for motion characteristics
of a video clip. MPEG-7 also uses the concept of description scheme (DS), which means
a framework that defines the descriptors and their relationships. Hence, the descriptors are
the basis of a description scheme. Description then implies an instantiation of a description
scheme. MPEG-7 not only wants to standardize the description, but it also wants the description to be efficient. Therefore, MPEG-7 also considers compression techniques to turn
descriptions into coded descriptions. Compression reduces the amount of data that need to be
stored or processed. Finally, MPEG-7 will define a description definition language (DDL) that
can be used to define, modify, or combine descriptors and description schemes. Summarizing,
MPEG-7 will standardize a set of descriptors and DSs, a DDL, and methods for coding the
descriptions. Figure 1.8 illustrates the relationship between these concepts in MPEG-7.
FIGURE 1.8
Relationship between elements in MPEG-7.
The process to define MPEG-7 is similar to that of the previous MPEG standards. Since
1996, the group has been working on defining and refining the requirements of MPEG-7 (i.e.,
what MPEG-7 should provide). The MPEG-7 process includes a competitive phase followed
©2001 CRC Press LLC
by a collaborative phase. During the competitive phase, a call for proposals is issued and
participants respond by both submitting written proposals and demonstrating the proposed
techniques. Experts then evaluate the proposals to determine the strength and weakness of
each. During the collaborative phase, MPEG-7 will evolve as a series of experimentation
models (XMs), where each model outperforms the previous one. Eventually, MPEG-7 will
evolve into an international standard. Table 1.5 shows the timetable for MPEG-7 development.
At the time of this writing, the group is going through the definition process of the first XM.
Table 1.5 Timetable of MPEG-7
Call for test material
Call for proposals
Proposals due
First experiment model (XM)
Working draft (WD)
Committee draft (CD)
Final committee draft (FCD)
Draft international standard (DIS)
International standard (IS)
Mar 1998
Oct 1998
Feb 1999
Mar 1999
Dec 1999
Oct 2000
Feb 2001
July 2001
Sep 2001
Once finalized, MPEG-7 will have a large variety of applications, such as digital libraries,
multimedia directory services, broadcast media selection, and multimedia authoring. Here are
some examples. With MPEG-7, the user can draw a few lines on a screen to retrieve a set
of images containing similar graphics. The user can also describe movements and relations
between a number of objects to retrieve a list of video clips containing these objects with
the described temporal and spatial relations. Also, for a given content, the user can describe
actions and then get a list of similar scenarios.
1.5
Standards for Multimedia Communication
In addition to video coding, multimedia communication also involves audio coding, control
and signaling, and the multiplexing of audio, video, data, and control signals. ITU-T specifies a
number of system standards for multimedia communication, as shown in Table 1.6 [8]. Due to
the different characteristics of various network infrastructures, different standards are needed.
Each system standard contains specifications about video coding, audio coding, control and
signaling, and multiplexing.
For multimedia communication over the Internet, the most suitable system standard in
Table 1.6 is H.323. H.323 [9] is designed to specify multimedia communication systems
on networks that do not guarantee QoS, such as ethernet, fast ethernet, FDDI, and token
ring networks. Similar to other system standards, H.323 is an umbrella standard that covers
several other standards. An H.323-compliant multimedia terminal has a structure as shown in
Figure 1.9. For audio coding, it specifies G.711 as the mandatory audio codec, and includes
G.722, G.723.1, G.728, and G.729 as optional choices. For video coding, it specifies H.261
as the mandatory coding algorithm and includes H.263 as an alternative. H.225.0 defines the
multiplexing of audio, video, data, and control signals, synchronization, and the packetization
mechanism. H.245 is used to specify control messages, including call setup and capability
exchange. In addition, T.120 is chosen for data applications. As in Figure 1.9, a receive path
©2001 CRC Press LLC
Table 1.6 ITU-T Multimedia Communication Standards
Network
PSTN
N-ISDN
B-ISDN/ATM
System
Video
Audio
H.324
H.261/263
G.723.1
H.320
H.261
G.7xx
H.321
H.261
G.7xx
H.310 H.261/H.262 G.7xx, MPEG
QoS LAN
H.322
H.261
G.7xx
Non-QoS LAN H.323
H.261
G.7xx
Note:G.7xx represents G.711, G.722, and G.728.
Mux
H.223
H.221
H.221
H.222.0/H.222.1
H.221
H.225.0
Control
H.245
H.242
Q.2931
H.245
H.242
H.245
delay is used to synchronize audio and video (e.g., for lip synchronization) and to control
jitters.
FIGURE 1.9
H.323 terminal equipment.
In addition to terminal definition, H.323 also specifies other components for multimedia
communication over non-QoS networks. These include the gateways and gatekeepers. As
shown in Figure 1.10, the responsibility of a gateway is to provide interoperability between
H.323 terminals and other types of terminals, such as H.320, H.324, H.322, H.321, and H.310.
A gateway provides the translation of call signaling, control messages, and multiplexing mechanisms between the H.323 terminals and other types of terminals. It also needs to support
transcoding when necessary. For example, for the audio codec on an H.324 terminal to interoperate with the audio codec on an H.323 terminal, transcoding between G.723.1 and G.711
is needed. On the other hand, a gatekeeper serves as a network administrator to provide the
address translation service (e.g., translation between telephone numbers and IP addresses)
and to control access to the network by H.323 terminals or gateways. Terminals have to get
permission from the gatekeeper to place or accept a call. The gatekeeper also controls the
bandwidth for each call.
©2001 CRC Press LLC
FIGURE 1.10
Interoperability of H.323.
1.6
Conclusion
In this chapter, we described several emerging video coding and multimedia communication
standards, including H.263, H.26L, MPEG-4, MPEG-7, and H.323. Reviewing the development of video coding, as shown in Figure 1.11, we can see that the progress of video coding
and multimedia standards is tied to the progress in modeling of the information source. The
finer the model, the better we can compress the signals, and with more content accessibility to
FIGURE 1.11
Trend of video coding standards.
the user. At the same time, the price to pay includes higher complexity and less error resilience.
The complexity manifests itself not only in the higher computation power that is required, but
also in higher flexibility. For example, whereas H.261 is a well-defined and self-contained
©2001 CRC Press LLC
compression algorithm, MPEG-4 and MPEG-7 are toolboxes of a large number of different
algorithms.
References
[1] Ahmed, N., Natarajan, T., and Rao, K.R., “Discrete cosine transform,” IEEE Trans. on
Computers, C-23, pp. 90–93, 1974.
[2] Rao, K.R., and Yip, P., Discrete Cosine Transform, Academic Press, New York, 1990.
[3] Netravali, A.N., and Robbins, J.D., “Motion-compensated television coding: Part I,”
Bell Systems Technical Journal, 58(3), pp. 631–670, March 1979.
[4] Netravali, A.N., and Haskell, B.G., Digital Pictures, 2nd ed., Plenum Press, New York,
1995.
[5] ITU-T Recommendation H.263: “Video coding for low bit rate communication,” Version 1, Nov. 1995; Version 2, Jan. 1998.
[6] Chen, T., Swain, C.T., and Haskell, B.G., “Coding of sub-regions for content-based scalable video,” IEEE Trans. on Circuits and Systems for Video Technology, 7(1), pp. 256–
260, February 1997.
[7] Sikora, T., “MPEG digital video coding standards,” IEEE Signal Processing Magazine,
pp. 82–100, Sept. 1997.
[8] Schaphorst, R., Videoconferencing and Videotelephony: Technology and Standards,
Artech House, Boston, 1996.
[9] Thom, G.A., “H.323: The multimedia communications standard for local area networks,” IEEE Communication Magazine (Special Issue on Multimedia Modem), pp. 52–
56, December 1996.
©2001 CRC Press LLC
Chapter 2
An Efficient Algorithm and Architecture for
Real-Time Perspective Image Warping
Yi Kang and Thomas S. Huang
2.1
Introduction
Multimedia applications are among the most important embedded applications. HDTV, 3D
graphics, and video games are a few examples. These applications usually require real-time
processing. The perspective transform used for image warping in MPEG-4 is one of the most
demanding algorithms among real-time multimedia applications. An algorithm is proposed
here for a real-time implementation of MPEG-4 sprite warping; however, it can be useful in
general computer graphics applications as well.
MPEG-4 is a new standard for digital audio–video compression currently being developed by
the ISO (International Standardization Organization) and the IEC (International Electrotechnical Commission). It will attempt to provide greater compression, error robustness, interactiveness, support of hybrid natural and synthetic scenes, and scalability. MPEG-4 will require
more computational power than existing compression standards, and novel architectures will
probably be necessary for high-complexity MPEG-4 systems. Whereas current video compression standards transmit the entire frame in a single bitstream, MPEG-4 will separately
encode a number of irregularly shaped objects in the frame. The objects in the frame can then
be encoded with different spatial or temporal resolutions [1].
By studying the MPEG-4 functions, we find that there are two critical parts for real-time
implementation: one is motion estimation in the encoder and the other is sprite warping in
the decoder. The algorithm for motion estimation in MPEG-4 is similar to those in previous
standards. There has already been plenty of work on algorithms and architectures for real-time
motion estimation. However, there have been few discussions on real-time sprite warping. We
therefore focus on algorithm and architecture development for sprite warping.
Real-time sprite warping involves implementing a perspective transform, a bilinear interpolation, and high-bandwidth memory accesses. It is both computationally expensive and
memory intensive. This poses a serious challenge for designing real-time MPEG-4 architectures. With the goal of real time and cost-effectiveness in mind, we first optimize our algorithm
to reduce the computation burden of the perspective transform by proposing the constant denominator algorithm. This algorithm dramatically reduces divisions and multiplications in
the perspective transform by an order of magnitude. Based on the proposed algorithm, we
designed an architecture which implements the real-time sprite warping. To make our architecture feasible for implementation under current technologies, we address the design of the data
path as well as the memory system according to the real-time requirement of computations and
©2001 CRC Press LLC
memory accesses in the sprite warping. Other related issues for implementation of real-time
sprite warping are also discussed.
2.2
A Fast Algorithm for Perspective Transform
The perspective transform is widely used in image and video processing, but it is computationally expensive. The most expensive part is its huge number of divisions. It is well
known that a division unit has the highest cost and the longest latency among all basic data
path units. The number of divisions in the perspective transform would make its real-time
implementation formidable without any fast algorithm. This motivates us to explore a new
algorithm for real-time perspective transform. The constant denominator method reduces the
number of required division operations to O(N ) while maintaining high accuracy. It also has
fewer multiplications and divisions.
2.2.1
Perspective Transform
Perspective transforms are geometric transformations used to project scenes onto view planes
along lines which converge to a point. The perspective transform which maps two-dimensional
images onto a two-dimensional view plane is defined by
ax + by + c
gx + hy + 1
dx + ey + f
y =
gx + hy + 1
x =
(2.1)
(2.2)
where (x, y) is a coordinate in the reference image, (x , y ) is the corresponding coordinate in
the transformed image, and a, b, c, d, e, f , g, and h are the transform parameters.
The perspective transform has many applications in computer-aided design, scientific visualization, entertainment, advertising, image processing, and video processing [3]. One new
application for the perspective transform is MPEG-4. In MPEG-4 one of the additional functionalities proposed to support is sprite coding [7]. A sprite is a reference image used to
generate different views of an object. The reference image is transmitted once, and future
images are produced by warping the sprite with the perspective transform. Because the transform parameters a, b, c, d, e, f , g, and h are rational numbers, they are not encoded directly.
Instead, the image is encoded using four (x , y ) pairs, since the transform parameters can
be determined from the reference and warped coordinates of four reference points using the
following system of equations:
 
  
x1 y1 1 0 0 0 −x1 x1 −y1 x1
a
x1
 
  
 x2   x2 y2 1 0 0 0 −x2 x2 −y2 x2   b 
 
  
 x   x3 y3 1 0 0 0 −x3 x −y3 x   c 
 3 
3
3 
 
  
 x4   x4 y4 1 0 0 0 −x4 x4 −y4 x4   d 
 
 =
(2.3)
 y   0 0 0 x y 1 −x y −y y   e 
1 1
1 1
1 1 
 1 
 
  
 y   0 0 0 x2 y2 1 −x2 y −y2 y   f 
2
2 
 2 
 
  
 y3   0 0 0 x3 y3 1 −x3 y3 −y3 y3   g 
h
y4
0 0 0 x4 y4 1 −x4 y4 −y4 y4
©2001 CRC Press LLC
High compression is therefore possible using sprite coding, especially for background sprites
and synthetic objects. After the original image is transmitted, the new view on the right can
be described using four points.
The warped image can be transmitted using fewer reference points. If three reference points
are transmitted, the affine transform is used for estimation. The affine transform is equivalent
to the perspective transform, with g and h equal to zero. Only two reference points are required
using an isotropic transformation, where g = h = 0, d = −b, and e = a. If only one reference
point is used, the transformation becomes simple translation, where g = h = 0, a = e = 1,
and b = d = 0. These simpler approximations provide less complexity, but generally provide
a less accurate estimate of the warped image.
To prevent holes or overlap in the warped sprite, backward perspective mapping is used.
Each point (x , y ) in the warped sprite is obtained from point (x, y) in the reference image.
The backward perspective mapping can be obtained from the adjoint and determinant of the
forward transform matrix [10]:
(hf − e)x + (b − hc)y + (ec − bf )
a x + b y + c
=
(eg − dh)x + (ah − bg)y + (db − ae)
g x + h y + i (d − f g)x + (cg − a)y + (af − dc)
d x + e y + f y=
=
(eg − dh)x + (ah − bg)y + (db − ae)
g x + h y + i x=
(2.4)
(2.5)
Though x and y are integers, x and y generally are not. Bilinear interpolation is used to
approximate the pixel value at point (x, y) from the four nearest integer points.
The perspective transform is computationally expensive. Computation of x and y using
equations (2.4) and (2.5) requires one division, eight multiplications, and nine additions per
pixel. The division is especially expensive. Since the transform parameters are not integers,
floating point computations are typically used. For real-time hardware implementations using
high-resolution images, direct computation of the transform is too slow. An approximation
method must be used.
2.2.2
Existing Approximation Methods
The perspective transform can be approximated using polynomials to avoid the expensive
divisions needed to compute the rational functions in equations (2.4) and (2.5). Linear approximation is the simplest and most widely used approximation technique. However, it usually
results in large errors due to the simplicity of the approximation [2, 4]. To achieve greater
accuracy, more complex methods such as quadratic approximation, cubic approximation, biquadratic approximation, and bicubic approximation have been proposed [6, 10]. Additional
methods to reduce aliasing and simplify resampling have also been developed, such as the
two-pass separable algorithm [10].
The Chebyshev approximation is a well-known method in numerical computation that also
has been used to approximate the perspective transform [2]. Its main advantage over other
methods is that its error is evenly distributed [8]. The result thus visually appears closer to the
ideal result. The formula for the Chebyshev approximation is
f (x) ≈
N−1
ck Tk (x) − 0.5c0
(2.6)
N
2 f (xk ) Tj (xk ) ,
N
(2.7)
k=0
where cj ’s are the coefficients computed as
cj =
k=1
©2001 CRC Press LLC
Tj (x) is the j th base function for the approximation, f (x) is the target function to approximate,
and N is the order of the approximation. N = 2 for the quadratic Chebyshev approximation;
N = 3 for the cubic Chebyshev approximation.
Biquadratic and bicubic Chebyshev methods have also been proposed to approximate the
perspective transform [2]. These methods first calculate the Chebyshev control points, then
use transfinite interpolation to approximate the rational functions using polynomials.
All of the above approximation methods require more multiplications and additions than
direct computation of the original rational functions. For complex approximations such as the
Chebyshev methods, the additional multiplications and additions offset the benefit of avoiding
division. Simpler approximations such as linear approximation require fewer additional operations, but often achieve poor quality. These methods also require an initialization procedure
to compute the approximation coefficients on every scan line. This increases the hardware
overhead.
In the following section, a new method to perform the perspective transform is proposed.
This new method does not increase the number of multiplications and additions, has a simple
initialization procedure, and decreases the number of divisions from O(N 2 ) to O(N ).
2.2.3
Constant Denominator Method
Equations (2.4) and (2.5) both contain the same denominator: g x + h y + i . Setting the
denominator equal to a constant value defines a line in the x y plane.
k = g x + h y + i (2.8)
Furthermore, lines defined by different values of k are all parallel and all have slope equal to
−g / h . The constant k for the line with y intercept equal to q can be calculated as
kq = h q + i (2.9)
By calculating the perspective transform along lines of constant denominator, the number of
divisions is reduced from one per pixel to one per constant denominator line.
The constant denominator method begins by calculating (d − f g), (cg − a), (af − dc),
(hf −e), (b−hc), (ec−bf ), (eg −dh), (ah−bg), and (db−ae). These coefficients need only
be calculated once per frame. Next, (eg − dh) and (ah − bg) are used to calculate the slope
m of the constant denominator lines. There are four possible cases: m < −1, −1 ≤ m ≤ 0,
0 < m ≤ 1, and 1 < m. The case determines whether the constant denominator lines are
scanned in the horizontal or vertical direction.
Figure 2.1 illustrates a case where 0 < m < 1. The lines all have slope m = −g / h and
represent constant values of g x + h y + i . The pixels are shaded to indicate which constant
denominator line they approximately fall on. The pixels for the initial line are determined by
starting at the origin and applying Bresenham’s Algorithm. Bresenham’s Algorithm requires
only incremental integer calculations [3]. The result is the table in Figure 2.1, which lists the
corresponding vertical position for every horizontal position on the constant denominator line
that passes through the origin. By storing the table as the difference of subsequent entries, the
number of bits required to store the table is the larger of the width or height of the image.
After the position of the constant denominator line has been determined, the actual warping
is performed. The reciprocal of the denominator is first calculated for the constant denominator
line which crosses the origin:
r=
©2001 CRC Press LLC
1
1
1
= = k0
h ∗0+i
i
(2.10)
FIGURE 2.1
Lines of constant denominator with 0 < slope < 1.
This is the only division required for the first constant denominator line. This reciprocal
is then multiplied by d , e , f , a , b , and c to obtain the coefficients in equations (2.11)
and (2.12).
x = ra x + rb y + rc
y = rd x + re y + rf
(2.11)
(2.12)
The horizontal position x is incremented from 0 to M − 1, where M is the width of the
image. For each value of x , y is obtained from the line table. The current value of the x and
y coordinates, xn and yn , are calculated from the previous values of the x and y coordinates,
xn−1 and yn−1 , using the following equations. If y = 0,
xn = xn−1 + ra yn = yn−1 + rd (2.14)
xn = xn−1 + [ra + rb ]
yn = yn−1 + [rd + re ]
(2.16)
(2.13)
If y = 1,
(2.15)
Only two additions are required to calculate xn and yn for each pixel on the constant denominator line. No multiplications or divisions are required per pixel.
The next constant denominator line is warped by calculating r for point (x , y ) = (0, 1)
using the following equation:
r=
1
1
1
= = k1
h ∗1+i
h + k0
(2.17)
One addition and one division are required to calculate r. The line table is used to trace the
new line, and equations (2.13)–(2.16) are used to warp the pixels on the new line. Every constant
denominator line below the original line is warped, followed by the constant denominator lines
above the original line.
©2001 CRC Press LLC
Because xn and yn are generally not integers, bilinear interpolation is used to calculate the
value of the warped pixel using the four pixels nearest to (xn , yn ) in the original sprite. The
warped pixel P is calculated using the following three equations, as shown in Figure 2.2:
P01 = P0 + (P1 − P0 ) ∗ dx
P23 = P2 + (P3 − P2 ) ∗ dx
P = P01 + (P23 − P01 ) ∗ dy
(2.18)
(2.19)
(2.20)
FIGURE 2.2
Bilinear interpolation.
As shown above, the constant denominator method reduces the number of divisions required
to calculate (x, y) from one per pixel, using equations (2.4) and (2.5) directly, to one per
constant denominator line. For an image M pixels wide and N pixels high, the number of
divisions is reduced from MN using the direct method to, at most, M + N − 1. The number
of multiplications needed to calculate (x, y) is reduced from 8MN to 8(M + N − 1) + 17.
The drastic reduction in divisions and multiplications makes the constant denominator method
suitable for real-time sprite decoding.
In addition, the constant denominator method can be used to calculate the backward affine
transform when only three reference points are transmitted. In this case, r = 1 for every point
in the plane. No divisions and only 14 multiplications per frame are therefore required for the
affine transform.
2.2.4
Simulation Results
To compare the visual quality of the warping approximations, five methods were implemented in C++: direct warping, constant denominator, quadratic, quadratic Chebyshev, and
cubic Chebyshev. The methods were then used to warp the checkerboard image, which is
a standard test image for computer graphics. The checkerboard image is useful because the
perspective transform should preserve straight lines. The parameters are set to a = 1.2, b = 0,
c = −100, d = 0, e = 1.2, f = −20, g = −.0082, and h = 0. The simulation shows that
straight lines in the original image are curved greatly by the quadratic and quadratic Chebyshev
methods. They are curved slightly by the cubic Chebyshev method. The constant denominator
method preserves the straight lines.
To generate test data for a wide range of cases, simulations were conducted varying g and
h over {−.1, −.01, −.001, −.0001, 0, .0001, .001, .01, .1}. Parameters a and e were set to 1,
and the remaining parameters were set to 0. An error image was calculated for each method
using the direct warping image as a reference, and the mean squared error (MSE) was computed
from each error image. The mean, median, and maximum values of mean squared error for
©2001 CRC Press LLC
each method are shown in Table 2.1. A histogram of the MSE for the four methods is shown
in Figure 2.3. The MSE is plotted on a logarithmic scale, and all MSEs less than 1 are plotted
at 1. One third of the simulations for the constant denominator method had MSEs below 1.
The largest error occurred for the case where g = 0.01 and h = −0.1. The other three methods
were significantly less accurate than the constant denominator method.
Error in the constant denominator method occurs because the pixels do not fall exactly on
constant denominator lines. Each pixel can lie a maximum of one-half pixel off the actual
constant denominator line if we treat each pixel as a square. An additional source of error is
from sprite resampling via the bilinear interpolation. Most of the error in Table 2.1 for the
constant denominator method is due to position computation because the direct warped image
with resampling is used as the error reference.
Table 2.1 Checkerboard Mean Squared Error
Table
Method
Mean
Median
Constant denominator
Quadratic
Quadratic Chebyshev
Cubic Chebyshev
73
2,831
2,118
1,822
20
693
457
392
Max
428
15,888
14,313
14,116
FIGURE 2.3
Checkerboard mean squared error histogram.
The constant denominator method was also tested on natural images. Simulation was done
for the a = 1, b = 0, c = 0, d = 0, e = 1, f = 0, g = −0.1, and h = 0.002 case using a
coastguard image. The MSE for the constant denominator method was 0.00043. The error is
so small that it can hardly be picked up by the eyes. Table 2.2 shows a performance comparison
©2001 CRC Press LLC
between the various approximation methods as g and h are varied between −0.1 and 0.1 for
the coastguard image.
Table 2.2 Coastguard Mean Squared Error Table
2.2.5
Method
Mean
Median
Constant denominator
Quadratic
Quadratic Chebyshev
Cubic Chebyshev
73
2,831
2,118
1,822
20
693
457
392
Max
428
15,888
14,313
14,116
Sprite Warping Algorithm
We designed an algorithm to perform sprite warping using the perspective transform as
specified in MPEG-4. The sprite warping algorithm performs the following tasks:
• Step 1: Compute the eight perspective transform parameters a, b, c, d, e, f , g, and h
from the reference coordinates.
• Step 2: Compute the nine backward transform coefficients (d −f g), (cg −a), (af −dc),
(hf − e), (b − hc), (ec − bf ), (eg − dh), (ah − bg), and (db − ae).
• Step 3: Use Bresenham’s Algorithm to calculate the line table for the first constant
denominator line.
• Step 4: Compute the constant r in equation (2.17) using restoring division [5]. Then
compute the coefficients in equations (2.11) and (2.12). This step is performed once per
constant denominator line.
• Step 5: Perform the backward transform for every pixel along the constant denominator
line described above.
• Step 6: Fetch the four neighboring pixels from memory for every warped pixel and
perform bilinear interpolation to obtain the new pixel value.
Step 1 entails solving the system of equations given in equation (2.3). Using LU decomposition, the eight sprite warping parameters can be calculated using 36 divisions, 196 multiplications, and 196 additions. Steps 2 through 5 use the constant denominator method to
perform the perspective transform. The computation of the backward transform coefficients
in step 2 requires 14 multiplications and nine additions. Calculating the line table in step 3
requires three multiplications, one division, and either M or N additions, depending on the
slope of the line. These three steps are performed once per frame. Step 4 requires one division,
eight multiplications, and three additions for every constant denominator line. Step 5 requires
two additions for every pixel. After the warped coordinate has been computed, the bilinear
interpolation in step 6 requires three multiplications and six additions for every pixel.
For gray-scale sprites M pixels wide and N pixels high and with horizontal scanning, the
entire sprite warping process requires at most M + N + 36 divisions, 3MN + 8M + 8N + 205
multiplications, and 8MN + 4M + 3N + 202 additions. Color sprites require additional
operations. For YUV images with 4:2:0 format, sprite warping requires at most a total of
1.5M + 1.5N + 35 divisions, 4.5MN + 12M + 12N + 200 multiplications, and 11.5MN +
6M + 4.5N + 199 additions.
©2001 CRC Press LLC
The computation burden can be reduced by using fixed point instead of floating point operations wherever possible. Steps 1, 2, and 4 are best suited for floating point operations.
However, since steps 1 and 2 are performed once per frame, and step 4 is performed once per
constant denominator line, they consume only a small fraction of the computational power.
Step 3 is also performed once per frame. The additions in step 3 can be performed in fixed
point.
Most of the computations are performed in steps 5 and 6, since these steps are performed
on each pixel. In step 5, a floating point coefficient is multiplied by the integer coordinate x or y . Therefore, instead of using true floating point, the coefficients can be represented in
block floating point format. Fixed point operations can then be used for step 5. After (x, y) is
calculated for each pixel, it is translated to a long fixed point number. Thus, only fixed point
computation is required for the bilinear interpolation in step 6.
By using fixed point operations for steps 5 and 6, the number of floating point multiplications
is reduced to at most 12M + 12N + 196 and the number of floating point additions becomes
4.5M + 4.5N + 199. The number of floating point divisions remains 1.5M + 1.5N + 35.
Almost all of the operations are now fixed point. 4.5MN fixed point multiplications and
11.5MN + 1.5M fixed point additions at most are required for steps 3, 5, and 6. Table 2.3
lists the number of operations required for various full-screen sprites.
Table 2.3 Number of Operations per Second
Required for 30 Frames per Second
2.3
Sprite Size
QCIF
CIF
ITU-R 601
Sprite width
Sprite height
Float. divide
Float. multiply
Float. add
Fixed multiply
Fixed add
176
144
15,000
120,000
49,000
3.4 million
8.8 million
352
288
30,000
240,000
92,000
14 million
35 million
720
576
59,000
470,000
180,000
56 million
140 million
Architecture for Sprite Warping
An MPEG-4 sprite warping architecture is described which uses the constant denominator method. The architecture exploits the spatial locality of pixel accesses and pipelines an
arithmetic logic unit (ALU) with an interpolation unit to perform high-speed sprite warping.
Several other implementation issues (e.g., boundary clipping and error accumulation) are also
discussed.
2.3.1
Implementation Issues
One issue inherent to the perspective transform is aliasing. Subsampling the sprite can
cause aliasing artifacts for perspective scaling. However, sprite warping is intended for video
applications where aliasing is less of a problem due to the motion blur. To address aliasing
in the constant denominator method, techniques such as adaptive supersampling could be
used. Supersampling would be performed when consecutive accesses to the sprite memory
are widely separated.
©2001 CRC Press LLC
Boundary clipping can also be a concern. Sprite warping can attempt to access reference
pixels beyond the boundaries of the reference sprite. If the simple point clipping method is
used, four comparisons per pixel are required. Instead, a hybrid point–line clipping method
can be used with the constant denominator method. For each constant denominator line, the
endpoints are first checked to see if they fall within the boundaries of the reference sprite. If
both endpoints are in the reference sprite, the line is warped. If only one endpoint is outside
the boundary, warping begins with this endpoint using point clipping. Once a point within the
boundary is warped, clipping is turned off, because the remaining points on the line are within
the sprite. If both endpoints lie outside the reference sprite, point clipping is used beginning
with one of the endpoints. Once a point inside the reference sprite is reached, warping switches
to the other endpoint. Point clipping is used until the next point with the sprite is reached,
when point clipping is turned off. Using this method, comparisons are only required when the
reference pixel is out of bounds. Because memory accesses and interpolations are not required
for the out-of-bound pixels, and clipping computations are not required for in-bound pixels,
the clipping procedure does not slow the algorithm.
Error accumulation in the fixed-point, iterative calculation of equations (2.13)–(2.16) must
also be considered. Sufficient precision of the fractional part of xn and yn must be used
to prevent error from accumulating to 1. The number of bits k required for the fractional
part depends on the height N and width M of the warped sprite according to the following
inequality:
k ≥ log2 (MAX[M, N ])
(2.21)
The integral part of xn and yn must contain enough bits to avoid overflow. Because (xn , yn )
is a coordinate in the reference plane, they theoretically have infinite range. Practically, the
number of integral bits j is chosen according to the size of the reference sprite plus additional
bits to prevent overflow. If a is the number of overflow bits and the reference sprite is P × Q
pixels, then
j ≥ log2 (MAX[P , Q]) + a
(2.22)
For example, if the reference and warped sprite are both 720 × 576 pixels and four overflow
bits are used, then a = 4, k = 10, j = 10, and 24 total bits are required for calculating xn and
yn .
2.3.2
Memory Bandwidth Reduction
Memory bandwidth is a concern for high-resolution sprites. Warped pixels are interpolated
from the four nearest pixels in the original sprite. Warping a sprite can therefore require four
reads and one write for every pixel in the sprite. An ITU-R 601 sprite requires 89 MB/s of
memory bandwidth at 30 frames per second.
Figure 2.4 illustrates the memory access pattern for sprite warping using the constant denominator method. It shows lines of slope −g/ h in the original sprite which correspond to
the lines of constant denominator in the warped sprite. While the memory access lines in the
original sprite are parallel to each other, they are not evenly spaced, and memory accesses on
different lines do not have the same spacing. Points in the warped sprite can also map to points
outside the original sprite.
The total memory access time required to warp a sprite can be reduced by either decreasing
the time required for each memory access or decreasing the number of accesses. Unlike scanline algorithms which enjoy the advantage of block memory access in consecutive addresses,
the constant denominator method must contend with diagonal memory access patterns. However, spatial locality inherent in diagonal access can be exploited. Figure 2.4 shows the use
©2001 CRC Press LLC
of spatial locality to reduce the time per access. The original sprite is divided into rectangular
pages, which correspond to pages in the sprite memory. Consecutive accesses on a line will
frequently lie on the same page. Fast page mode can therefore be used to retrieve the data
quickly.
FIGURE 2.4
Example memory access pattern.
A cache can be used to reduce the number of accesses per pixel. Consecutive accesses on a
constant denominator line often reference common pixels in the original sprite. Consecutive
constant denominator lines frequently use many of the same pixels. By retaining pixel values
in a cache, accesses to main memory can be avoided.
Cache effectiveness is dependent on the spacing between memory accesses. In the upper
left area of the example in Figure 2.4, memory access lines are closely spaced. Pixels on
the upper left will be accessed many times, and a cache will save memory accesses. In the
lower right area, however, memory access lines are widely spaced. Pixels are not shared
between consecutive lines, and a cache will not be as effective. However, because the lines are
widely spaced, most of the pixels in the lower right area are not accessed from memory. Many
accesses will instead occur outside the boundaries of the sprite memory and will be resolved
by boundary clipping instead of being retrieved from memory or the cache. The worst-case
memory access situation therefore does not occur for widely spaced lines. The cache should
be designed for line spacings small enough such that most of the reference sprite pixels are
read four times.
A very small cache which holds only four pixels will reduce the number of memory reads
per sprite. By keeping the four pixels used to interpolate the previous point in the cache, the
worst-case number of memory reads per sprite will be reduced from four times the number of
warped pixels to three times the number of warped pixels. The worst case occurs when pixels
on diagonal lines are accessed. If consecutive accesses on the lines are widely spaced, then
the cache will be of no use. However, many pixels on the diagonal lines will not be accessed
and the total number of accesses to sprite memory will be small. This is therefore not the
worst case. Instead, the worst case occurs when consecutive pixels on the diagonal lines are
accessed. One pixel in the cache can be reused; the three remaining pixels must be read in
from memory.
©2001 CRC Press LLC
A larger cache will further reduce the memory bandwidth required. Figure 2.5 illustrates
the use of a cache with a three-line capacity. The cache is three-way set associative to remove
conflict cache misses. For lines with slope greater than 1 or less than −1, as in the figure,
there is one set for every y coordinate in the sprite, and pixels are tagged with the x coordinate.
Shallower lines have one set for every x coordinate and are tagged with the y coordinate.
The three-line cache reduces the worst-case number of reads to one per pixel. For an ITU-R
601 sprite, a three-line cache requires approximately 17 Kbit.
FIGURE 2.5
Cache operation example.
2.3.3
Architecture
The data path for a sprite warping architecture is shown in Figure 2.6. It contains two
processors: an ALU to perform steps 1 through 5 in the sprite warping process and an interpolation unit to perform step 6. Since steps 5 and 6 are the two steps executed per pixel, they
are assigned to different processors.
The ALU performs integer addition and multiplication. It reads reference coordinates from
the coordinate buffer and calculates the perspective transform coefficients, using the small
scratch memory for intermediate storage. The nine backward transform coefficients are then
stored in the floating point coefficient buffer. The ALU uses Bresenham’s Algorithm to compute the incremental line table for the first constant denominator lines. The line table is stored
in the Bresenham shift register, which is simply a line of serially connected, 1-bit flip-flops.
For each line, the ALU computes the six coefficients in equations (2.11) and (2.12). For each
pixel, the coordinates of the corresponding pixel in the original frame are calculated and partitioned into an integer part (xl , yl ) and a fractional part (dx, dy). The integer part is output
to the pixel cache while the fractional part is passed to the interpolation unit.
The pixel cache outputs pixels P0 , P1 , P2 , and P3 . These are the four pixels with coordinates
(xl , yl ), (xl+1 , yl ), (xl , yl+1 ), and (xl+1 , yl+1 ), which are shown in Figure 2.2. If the pixels
©2001 CRC Press LLC
FIGURE 2.6
Sprite warping architecture.
are not in the cache, they are retrieved from memory. The pixels are transmitted serially to the
interpolation unit.
The interpolation unit is based on a design commonly used for half-pixel motion compensation [9]. It is shown in detail in Figure 2.7. The unit reads a new pixel whenever the cache
signals that the value P0123 is ready. It receives dx and dy from the ALU and outputs a bilinearly interpolated pixel after reading every fourth pixel. P0 and P1 are first linearly interpolated
using dx to compute P01 . P2 and P3 are then interpolated using dx to compute P23 . Finally,
the vertical fraction dy is used to linearly interpolate P01 and P23 and obtain the bilinearly
interpolated pixel P . P is then output to the sprite memory.
If the four interpolation pixels are not in the cache, memory access time is critical. Table 2.4
lists the memory requirements for warping sprites with various resolutions. The memory
size listed is for a single sprite buffer. Since the warped sprite and original sprite are stored in
separate areas of sprite memory, two sprite buffers are required. To provide additional memory
bandwidth, the sprite buffers can be stored on separate memory chips. If a single warping unit
is used to warp k sprites, k + 1 sprite buffers are required.
Table 2.4 Memory Requirements for Sprite Warping
Sprite Format
QCIF
CIF
ITU-R 601
Memory size
Pixel reads/frame
Pixel writes/frame
Time/read at 30 fps
297 Kbits
152,000
38,000
219 ns
1,188 Kbits
608,000
152,000
54 ns
4,860 Kbits
2,488,000
622,080
13 ns
Table 2.4 lists the worst-case number of pixel reads and writes required to warp a sprite.
The table also lists the average time per read that must be met if the sprite is to be warped at
30 frames per second. It assumes that the warped sprite and original sprite are contained in
©2001 CRC Press LLC
FIGURE 2.7
Interpolation unit.
separate memories. The times were obtained using
tread =
number of frames
second
number of pixel reads
frame
−1
(2.23)
If the pixels are currently in the cache, they will be transmitted to the interpolation unit
quickly, and the computation time in the interpolation unit becomes critical. A new pixel
cannot be read in until the previous pixel has been linearly interpolated. Assuming the linear
interpolation time, tinterpolate , is less than tpage hit (the time to read from memory on a page hit),
then the average read time is determined by
tread = ctinterpolate + (1 − c)[ptpage hit + (1 − p)tpage miss ]
(2.24)
where c is the cache hit ratio, p is the page hit ratio, and tpage miss is the time to read from
memory on a page miss.
If no cache is used, equation (2.24) reduces to
tread = ptpage hit + (1 − p)tpage miss
(2.25)
Assuming DRAM access times of 20 ns on a page hit and 85 ns on a page miss, the 219-ns
cycle time listed in Table 2.4 for QCIF sprites can be easily obtained without a cache. CIF
sprites can also be warped without a cache, because the 73-ns cycle time can be met for p > .5,
which is a very low page hit ratio. For both sprite sizes, the interpolation unit can be designed
to match the memory access time.
With a four-pixel cache, the average read time equation becomes
tread =
1
3
tinterpolate + [ptpage hit + (1 − p)tpage miss ]
4
4
(2.26)
where c = 41 is the cache hit ratio for the worst case. The four-pixel cache can be used to
warp sprite sizes larger than CIF. It cannot warp ITU-R 601 sprites, because they require the
©2001 CRC Press LLC
short 13-ns cycle from Table 2.4. Instead, the three-line cache is used, where c =
tpage hit = 20 ns and tpage miss = 85 ns, equation (2.24) can then be rewritten as
3
4.
For
3
1
tinterpolate + [ptpage hit + (1 − p)tpage miss ] = (28p − 17.3) nanoseconds (2.27)
4
4
which simplifies to
tread =
tinterpolate < (21.67p − 11) nanoseconds
(2.28)
This equation is satisfied by realistic interpolation times and page hit ratios. For example, an
8.5-ns interpolation time and a 0.9 page hit ratio, or a 6.3-ns linear interpolation time and a
0.8 page hit ratio, can be used for real-time warping of ITU-R 601 sprites with 0.35µ m or
better VLSI technology.
2.4
Conclusion
We have presented a new fast algorithm for computing the perspective transform. The
constant denominator method reduces the number of divisions required from O(N 2 ) to O(N )
and also dramatically reduces multiplications in the computation. The speed of the constant
denominator method does not sacrifice the accuracy of the algorithm. Indeed, it has more than
35 times less error compared with other approximation methods. The algorithm primarily
targets real-time implementation of sprite warping. However, it is generally for speeding
up the perspective transform. Based on this algorithm, an architecture was proposed for
the implementation of sprite warping for MPEG-4. Our architecture is feasible under current
VLSI technology. We also analyzed the real-time requirement of the architecture and addressed
several other implementation issues.
References
[1] CCITT. MPEG-4 video verification model version 11.0. ISO-IEC JTC1/SC29/WG11
MPEG98/N2172, Mar. 1998.
[2] Demirer, M., and Grimsdale, R.L. Approximation techniques for high performance texture mapping. Computer & Graphics 20, 4 (1996).
[3] Hearn, D., and Baker, M.P. Computer Graphics, 2 ed., Prentice-Hall, Englewood Cliffs,
NJ, 1994.
[4] Heckbert, P., and Moreton, H.P. Interpolation for polygon texture mapping and shading.
In State of the Art in Computer Graphics Visualization and Modeling, D.F. Rogers and
R.A. Earnshaw, Eds., Springer-Verlag, Berlin, 1991.
[5] Hennessy, J.L., and Patterson, D.A. Computer Architecture: A Quantitative Approach,
2 ed., Morgan Kaufmann Publishers, 1996.
[6] Kirk, D., and Vorrhies, D. The rendering architecture of the dn10000vs. Computer
Graphics 24 (1990).
©2001 CRC Press LLC
[7] Lee, M.C., Chen, W., Lin, C.B., Gu, C., Markoc, T., Zabinsky, S.I., and Szeliski, R. A
layered video object coding system using sprite and affine motion model. IEEE Transactions on Circuits and Systems for Video Technology 7, 1 (Feb. 1997).
[8] Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. Numerical Recipes in
C, 2 ed., Cambridge University Press, London, 1994.
[9] Sun, M.T. Algorithms and VLSI architectures for motion estimation. In VLSI Implementations for Communications, P. Pirsh, Ed., Elsevier Science Publishers, New York,
1993.
[10] Wolberg, G. Digital Image Warping. IEEE Computer Society Press, 1990.
©2001 CRC Press LLC
Chapter 3
Application-Specific Multimedia Processor
Architecture
Yu Hen Hu and Surin Kittitornkun
3.1
Introduction
Multimedia signal processing concerns the concurrent processing of signals generated from
multiple sources, containing multiple formats and multiple modalities. A key enabling technology for multimedia signal processing is the availability of low-cost, high-performance signal
processing hardware including programmable digital signal processors (PDSPs), applicationspecific integrated circuits (ASICs), reconfigurable processors, and many other variations.
The purposes of this chapter are (1) to survey the micro-architecture of modern multimedia
signal processors, and (2) to investigate the design methodology of dedicated ASIC implementation of multimedia signal processing algorithms.
3.1.1
Requirements of Multimedia Signal Processing (MSP) Hardware
Real-Time Processing
With real-time processing, the results (output) of a signal processing algorithm must be
computed within a fixed, finite duration after the corresponding input signal arrives. In other
words, each computation has a deadline. The real-time requirement is a consequence of the
interactive nature of multimedia applications. The amount of computations per unit time,
also known as the throughput rate, required to achieve real-time processing varies widely
for different types of signals. If the required throughput rate cannot be met by the signal
processing hardware, the quality of service (QoS) will be compromised. Real-time processing
of higher dimensional signals, such as image, video, or 3D visualization, requires an ultra-high
throughput rate.
Concurrent, Multithread Processing
A unique feature of MSP hardware is the need to support concurrent processing of multiple
signal streams. Often more than one type of signal (e.g., video and sound) must be processed concurrently as separate task threads in order to meet deadlines of individual signals.
Synchronization requirements also impose additional constraints.
©2001 CRC Press LLC
Low-Power Processing
Multimedia signal processing devices must support mobile computing to facilitate ominous
accessibility. Low-power processing is the key to wireless mobile computing. Technologies
(TTL vs. CMOS, power supply voltages) are the dominating factor for power consumption.
However, architecture and algorithm also play a significant role in system-wide power consumption reduction.
3.1.2
Strategies: Matching Micro-Architecture and Algorithm
To achieve the performance goal (real-time processing) under the given constraint (low
power consumption), we must seek a close match between the multimedia signal processing
algorithm formulation and the micro-architecture that implements such an algorithm. On the
one hand, micro-architecture must be specialized in order to custom fit to the given algorithm.
On the other hand, alternative algorithm formulations must be explored to exploit its inherent
parallelism so as to take advantage of the power of parallel micro-architecture.
Specialization
Specialized hardware can be customized to execute the algorithm in the most efficient
fashion. It is suitable for low-cost, embedded applications where large-volume manufacturing
reduces the average design cost. Hardware specialization can be accomplished at different
levels of granularity. Special function units such as an array multiplier or multiply-andaccumulator (MAC) have been used in programmable DSPs. Other examples include a bit
reversal unit for fast Fourier transform and so forth.
Another approach of specialization is to use a special type of arithmetic algorithm. For
example, CORDIC arithmetic unit is an efficient alternative when elementary functions such as
trigonometric, exponential, or logarithmic functions are to be implemented. Another example
is the so-called distributed arithmetic, where Boolean logic functions of arithmetic operations
are replaced with table-lookup operations using read-only memory.
At a subsystem level, specialized hardware has also been developed to realize operations that
are awkward to be realized with conventional word-based micro-architecture. For example,
the variable-length entropy-coding unit is often realized as a specialized subsystem.
Specialized hardware consisting of multiple function units to exploit parallelism is also
needed to handle computation-intensive tasks such as motion estimation, discrete cosine transform, and so forth. At the system level, specialized hardware has also been developed to serve
large-volume, low-cost, and embedded consumer applications, such as the MPEG decoder
chip.
Parallelism
Parallelism is the key to achieving a high throughput rate with low power consumption.
To reduce power consumption, power supply voltage must be reduced. Lower power supply
voltage implies lower switching speed. As such, to meet the real-time processing throughput
constraint, more function units must be activated together, taking advantage of the potential
parallelism in the algorithm.
Many MSP algorithms can be formulated as nested iterative loops. For this family of
algorithms, they can be mapped algebraically into regular, locally interconnected pipelined
processing arrays such as the systolic array. Examples include discrete cosine transform, full
search motion estimation, discrete wavelet transform, and discrete Fourier transform.
In addition to the systolic array, parallelism can be exploited in different formats. A vectorbased parallel architecture is capable of performing vector operations efficiently. A specific
©2001 CRC Press LLC
vector-parallel architecture is known as the subword parallelism. It appears as the multimedia
extension (MMX) instructions in general-purpose microprocessors.
Some algorithms do not have a regular structure such as nested iterative loops. However,
since MSP applications often deal with indefinite streams of signals, it is also possible to
develop pipelined special-purpose hardware to exploit the parallelism. Examples include fast
discrete cosine transform (DCT) algorithms.
For programmable DSP processors, instruction-level parallelism (ILP) has dominated modern superscalar microprocessor architecture. A competing ILP approach is known as the very
long instruction word (VLIW) architecture. The main difference between ILP and VLIW
is that ILP architecture relies on a hardware-based instruction issuing unit to exploit the potential parallelism inherent in the instruction stream during the run time, whereas the VLIW
micro-architecture relies heavily on a compiler to exploit ILP during the compile time.
3.2
3.2.1
Systolic Array Structure Micro-Architecture
Systolic Array Design Methodology
Systolic array [1, 2] is an unconventional computer micro-architecture first proposed by
H.T. Kung [3]. It features a regular array of identical, simple processing elements operated
in a pipelined fashion. It can be visualized that data samples and intermediate results are
processed in a systolic array in a manner analogous to how the blood is pumped by the heart —
a phenomenon called systole circulation — which is how this architecture received its name.
A systolic array exhibits characteristics of parallelism (pipelining), regularity, and local
communication. If an algorithm can be described as a nested “do” loop with simple loop body,
specifically known as a regular iterative algorithm, then it can be mapped algebraically onto
a systolic array structure.
A number of multimedia signal processing algorithms can be implemented using systolic
arrays. Examples include two-dimensional DCT (2D DCT), video block motion estimation,
and many others. To illustrate systolic array design methodology, consider the convolution of
a finite length sequence {h(n); 0 ≤ n ≤ M − 1} with an infinite sequence {x(n); n = 0, 1, . . .
yn =
min(n,M−1)
h(k)x(n − k)
n = 0, 1, . . .
(3.1)
k=0
This algorithm is usually implemented with a two-level nested do loop:
Algorithm 1:
For n = 0, 1, 2,...
y(n) = 0
For k = 0 to min(n,M-1),
y(n) = y(n)+h(k)*x(n-k)
end
end
It can be implemented using a systolic array containing M processing elements as depicted
in Figure 3.1. In Figure 3.1, the narrow rectangular box represents delay, and the square
©2001 CRC Press LLC
FIGURE 3.1
Convolution systolic array.
box represents a processing element (PE). Moreover, every PE is identical and performs its
computation in a pipelined fashion. The details of a PE are shown in Figure 3.2. In this
FIGURE 3.2
A processing element of the convolution systolic array.
figure, the circle represents arithmetic operations. The above implementation corresponds to
the following algorithm formulation:
Algorithm 2:
s(n,0) = x(n);
g(n,0) = 0;
g(n,k+1) =g(n,k)+h(k)*s(n,k);
s(n,k+1) = s(n,k);
g(n+1,k+1) = g(n,k+1);
s(n+2,k+1) = s(n,k+1);
y(n) = g(n+M,M) ;
n
n
n
n
n
n
=
=
=
=
=
=
0,
0,
0,
0,
0,
0,
1,
1,
1,
1,
1,
1,
2,...
2,...;
2,...;
2,...;
2,...;
2,...
k
k
k
k
=
=
=
=
0
0
0
0
to
to
to
to
M-1,
M-1,
M-1,
M-1,
In the above formulation, n is the time index and k is the processing element index. It can
be verified manually that such a systolic architecture yields correct convolution results at the
sampling rate of x(n).
Given an algorithm represented as a nested do loop, a systolic array structure can be obtained
by the following three-step procedure:
1. Deduce a localized dependence graph of the computation algorithm. Each node of the
dependence graph represents computation of the innermost loop body of an algorithm
represented in a regular nested loop format. Each arc represents an inter-iteration dependence relation. A more detailed introduction to the dependence graph will be given
later in this chapter.
2. Project each node and each arc of the dependence graph along the direction of a projection vector. The resulting geometry gives the configuration of the systolic array.
©2001 CRC Press LLC
3. Assign each node of the dependence graph to a schedule by projecting them along a
scheduling vector.
To illustrate this idea, let us consider the convolution example above. The dependence
graph of the convolution algorithm is shown in Figure 3.3. In this figure, the input x(n) is
from the bottom. It will propagate its value (unaltered) along the northeast direction. Each of
the coefficients {h(k)} will propagate toward the east. The partial sum of y(n) is computed
at each node and propagated toward the north. If we project this dependence graph along the
[1 0] direction, with a schedule vector [1 1], we obtain the systolic array structure shown on
the right-hand side of the figure. To be more specific, each node at coordinate (n, k) in the
dependence graph is mapped to processing element k in the systolic array. The coefficient
h(k) is stored in each PE. The projection of the dependence vector [1 1] associated with the
propagation of x(n) is mapped to a physical communication link with two delays (labeled by
2D in the right-hand portion of the figure). The dependence vector [0 1] is mapped to the
upward communication link in the systolic array with one delay. Figure 3.1 is identical to the
right side of Figure 3.3 except more details are given.
FIGURE 3.3
Dependence graph of convolution (left) and systolic array projection (right).
The systolic design methodology of mapping a dependence graph into a lower dimensional
systolic array is intimately related to the loop transformation methods developed in parallel
program compilers. A detailed description of loop transform can be found in [4].
3.2.2
Array Structures for Motion Estimation
Block motion estimation in video coding standards such as MPEG-1, 2, and 4, and H.261
and H.263 is perhaps one of the most computation-intensive multimedia operations. Hence it
is also the most implemented algorithm.
We will briefly explain block-based motion estimation using Figure 3.4. A basic assumption
of motion estimation is that there is high temporal correlation between successive frames
in video streams; hence, the content of one frame can be predicted quite well using the
contents of adjacent frames. By exploiting this temporal redundancy, one need not transmit
the predictable portion of the current frame as long as these reference frame(s) have been
successfully transmitted and decoded. Often, it is found that the effectiveness of this scheme
can be greatly enhanced if the basic unit for comparison is reduced from the entire frame to a
much smaller “block.” Often the size of a block is 16 × 16 or 8 × 8 (in the unit of pixels). This
is illustrated on the right-hand side of Figure 3.4. Let us now focus on the “current block” that
has a dotted pattern in the current frame. In the reference frame, we identify a search area that
surrounds a block having the same coordinates as the current block. The hypothesis is that
within this search area, there is an area equal to the size of the current block which best matches
©2001 CRC Press LLC
FIGURE 3.4
Block motion estimation.
(is similar to) the current block. Then, instead of transmitting all the pixels in the current block
of the current frame, all we need is to specify the displacement between the current block
location and the best matched blocking area on the reference frame. Then we cut-and-paste
this area from the reference frame to the locations of the current block on a reconstructed
current frame at the receiving end. Since the reference frame has been transmitted, the current
block at the current frame can be reconstructed this way without transmitting any bit in addition
to the displacement values, provided the match is perfect.
The displacement we specified above is called the motion vector. It inherits this name
from the motion estimation task in computer vision researches. However, there, the motion
estimation is performed on individual pixels, and the objective is to identify object motion
in sequential image frames. Since each pixel within the search area can be the origin of a
matching block, its coordinates become a candidate for a motion vector. If every pixel within
the search area is tested in order to find the best matching block, it is called a full-search blockmatching method. Obviously, a full search block-matching algorithm offers the best match.
But the computation cost is also extremely high. On the other hand, the matching operations
can be written in a regular six-level nested do loop algorithm. Thus, numerous systolic array
or other dedicated array architectures have been proposed. We note that there are also many
fast block-matching algorithms proposed to skip pixels in the search area in order to reduce
computation without significantly compromising matching quality. Unfortunately, most of
these fast search algorithms are too complicated for a systolic array implementation. In this
section, we will survey systolic array structures for the implementation of only the full-search
block-matching motion estimation algorithm. First, we review some notations and formulas
of this algorithm.
FBMA (Full-Search Block-Matching Algorithm)
Assume a current video frame is divided into Nh × Nv blocks in the horizontal and vertical
directions, respectively, with each block containing N × N pixels. The most popular similarity
criterion is the mean absolute difference (MAD), defined as
MAD(m, n) =
N−1 N −1
1 |x(i, j ) − y(i + m, j + n)|
N2
i=0 j =0
©2001 CRC Press LLC
(3.2)
where x(i, j ) and y(i + m, j + n) are the pixels of current frame and previous frame, respectively. The motion vector (MV) corresponding to the minimum MAD within the search area
is given by
MV = arg{min MAD(m, n)}
− p ≤ m, n ≤ p ,
(3.3)
where p is the search range parameter. We focus on the situation where the search area is a
region in the reference frame consisting of (2p + 1)2 pixels.
In the FBMA, MAD distortions between the current block and all (2p + 1)2 candidate
blocks are to be computed. The displacement that yields the minimum MAD among these
(2p + 1)2 positions is chosen as the motion vector corresponding to the present block. For the
entire video frame, this highly regular FBMA can be described as a six-level nested do loop
algorithm, as shown below.
Algorithm 3: Six-level nested do loop of full-search
block-matching motion estimation
Do h=0 to Nh-1
Do v=0 to Nv-1
MV(h,v)=(0,0)
Dmin(h,v)=∞
Do m=-p to p (-1)
Do n=-p to p (-1)
MAD(m,n)=0
Do i=hN to hN+N-1
Do j=vN to vN+N-1
MAD(m,n)= MAD(m,n)+|x(i,j)-y(i+m,j+n)|
End do j
End do i
If Dmin(h,v) > MAD(m,n)
Dmin(h,v)=MAD(m,n)
MV(h,v)=(m,n)
End if
End do n
End do m
End do v
End do h
The frame rate for a particular resolution standard (e.g., MPEG-2, H.261) can be used as a
performance metric. Assuming that time to compute an MV of one block of N × N pixels is
Tblock , then the time to compute the whole video frame is
Tframe = Nh Nv Tblock ,
(3.4)
and the frame rate Fframe is determined by
Fframe =
©2001 CRC Press LLC
1
Tframe
.
(3.5)
Synchronization
Data
Input
Control Unit
Local
Memory
Processing
Array
Result
FIGURE 3.5
MEP block diagram.
Motion Estimation Subsystem Architecture
A generic block diagram of a motion estimation subsystem consists of a processing array,
local (on-chip) memory, and a control unit as shown in Figure 3.5.
The control unit provides the necessary clock timing signals and flags to indicate the beginning and completion in processing the current block. The local memory unit not only acts as
an on-chip cache but also facilitates data reordering. The size of the local memory depends on
the specific systolic mapping performed. Based on the geometry of the processing array (in
conjunction with local memory), existing motion estimation array structures can be roughly
classified into four categories:
• 2D array
• linear array
• tree-type structure (TTS)
• hybrid
We will briefly survey each of these array structures.
2D Array Micro-Architecture
The AB2 architecture [5] shown in Figure 3.6 and its sibling AS2 (not shown) were among
the first motion estimation array structures. Subsequently, AB2 has been modified [6] to scan
the search area data sequentially in raster scan order using shift registers. This reduces the
need for a large number of input–output (I/O) pins. However, the overall processing element
utilization is rather inefficient. An improved AB2-based architecture is presented by [7]. The
movement of search area data is carefully studied so that it can exploit a spiral pattern of data
movement. On average, this processor array is able to compute two MADs in every cycle.
However, it requires a PE that is twice as complicated. This can reduce the computation latency
at the expense of more complicated PE architecture. These earlier array structures are often
derived in an ad hoc manner without employing a formal systolic array mapping strategy.
A modular semisystolic array derived by performing the systolic mapping of a six-level
nested do loop algorithm on an array is presented in [8]. First, we transform the three pairs
of indices (v, h), (m, n), (i, j ) of the six-level nested do loop in Algorithm 3 to a three-level
nested do loop with indices (b, l, k), where b, l, and k represent block, search vector, and
pixel, respectively, of the entire frame. A systolic multiprojection technique [1] is then used
to project the 3D dependence graph (DG) into a linear array. Next, exploiting the fact that the
neighboring search area shares many reference frame pixels, this linear array is further folded
into a spiral 2D array as shown in Figure 3.7. In this configuration, the search area pixel y
©2001 CRC Press LLC
20 10 00
21 11 01
22 12 02
0
0
0
0
AD
00
AD
10
AD
20
AD
01
AD
11
AD
21
AD
02
AD
12
AD
22
A
A
A
M
MV
FIGURE 3.6
AB2 architecture [5]. AD: absolute difference, A: addition, M: memory.
is broadcast to each processing element in the same column, and the current frame pixel x
is propagated along the spiral interconnection links. The constraint of N = 2p is imposed
to achieve a low I/O pin count. A simple PE is composed of only two eight-bit adders and a
comparator, as shown in Figure 3.7.
In [9] the six-level nested do loop is transformed into a two-level nested do loop, which is
then mapped into a linear array and then folded into a 2D spiral array. The resulting design has
better scalability to variable block sizes and search ranges and does not need data broadcasting.
In [10], another 2D array structure is proposed. It uses multiprojection directly to transform
the dependence graph corresponding to the six-level nested do loop into a 2D fully pipelined
systolic array. Two levels of on-chip caches are required to handle the data movements.
Furthermore, it has been shown that the previous motion estimation array architecture [6]
is a special case of this 2D array structure. In the architectures proposed in [11] and [12],
attention is paid to data movement before and after the motion estimation operations. Data
broadcasting is used to yield a semisystolic array [11]. Two sets of shift register arrays are
used to switch back and forth between two consecutive current blocks to ensure 100% PE
utilization (Figure 3.8).
Linear Array Architecture
A linear array configuration uses fewer processing elements but has a lower data throughput
rate. It is suitable for applications with a lower frame rate and lower resolution such as
videoconferencing and/or videophone. The AB1 [5] depicted in Figure 3.9 is an example of
linear array architecture.
The performance of a linear array architecture can be enhanced using data broadcasting to
reduce the pipelining latency in a systolic array where data are propagated only to its nearest
neighboring PE. In [13], it is suggested to broadcast either the current block pixels or the search
area pixels so that PEs that need these data can be computed earlier. Obviously, when the array
size grows, long global interconnection buses will be needed to facilitate data broadcasting.
This may increase the critical path delay and hence slow down the applicable clock frequency.
A hybrid SIMD (single instruction, multiple data) systolic array, consisting of four columns
of 16 PEs, has been proposed by [14]. It is essentially the collection of four independent 16 ×
©2001 CRC Press LLC
D
PE
PE
D
PE
D
PE
MV
D
D
PE
PE
D
PE
D
PE
D
D
PE
PE
D
PE
D
PE
D
D
PE
ctrl 1
x(b,l,k)
PE
MUX
1
0
y(b,l,k)
D
PE
MUX
0
1
D
PE
ctrl 2
y(b,l+N-1,k-(N-1))
FIGURE 3.7
2D array with spiral interconnection (N = 4 and p = 2). PE: processing element, D:
delay, ctrl: control line, MUX: multiplexer.
1 linear arrays; hence, it should be considered as a variant of linear array architecture. More
recently, a linear array structure was reported in [15]. It is based on slicing and tiling of a 4D
DG onto a single 2D plane in order to make the projection easier. Global buses are needed to
broadcast search area data. Additional input buffers are required to reorder the input sequence
into a format suitable for the processing array. On the other hand, modules can be linearly
cascaded for better parallelism or to handle bigger block size as well as a larger search range.
Tree-Type Structure (TTS) Architecture
TTS is suitable for not only FBMA but also irregular block-matching algorithms such as
the three-step hierarchical search. Since each tree level shown in Figure 3.10 can be viewed
as a parallel pipeline stage, the latency is shorter. Nevertheless, the computation time is still
comparable to those of 1D or 2D array architectures. The problem associated with TTS is the
memory bandwidth bottleneck due to the limited number of input pins. This can be alleviated
by a method called 1/M-cut subtree, as proposed in [16], to seek a balance between memory
bandwidth and hardware complexity.
Hybrid Architecture
Several hybrid architectures proposed in the literature are now briefly reviewed.
In [17], two types (type 1 and type 2) of hybrid architectures are proposed. In these architectures, search area data y are injected into a 2D array with tree adders in a meander-like
pattern. The type-1 architecture is similar to the AB2 array [5] shown in Figure 3.6. It imposes
©2001 CRC Press LLC
y
MADa
Min(MADa,MADb)
Com
x
Reg
MADb
x
A
AD
DFF
y
FIGURE 3.8
Diagram of an individual processing element. Reg: register, Com: compare, AD: absolute difference, A: addition, DFF: D flip-flop.
O
x
x
x
AD
AD
AD
y
y
y
A
M
MV
FIGURE 3.9
AB1 architecture [5]. AD: absolute difference, M: memory, -o-: delay.
the constraint that N = 2p + 1. The type-2 architecture is analogous to the AS2 array in [5].
These array architectures have registers on both the top and bottom of the processing array to
support meander-like movement of search area data.
In [17], a hybrid TTS/linear structure has been suggested. This architecture consists of a
parallel tree adder to accumulate all the partial sums calculated by a linear array of PEs. To
achieve the same throughput as a 2D array, clock frequency must be increased n times from the
2D array, where n is the degree of time-sharing. A register ring is added to accumulate SAD
after a tree adder, as reported in [18, 19]. Another hybrid architecture [20] utilizes a linear
array of N 1/2-cut subtrees with systolic accumulation instead of a single 1/32-cut subtree, as
shown in [16].
Performance Comparison
We use the following features to compare different motion estimation array architectures:
• Area and complexity
• Number of I/O ports and memory bandwidth
©2001 CRC Press LLC
MV
M
A
A
A
A
D
x
y
x
D
D
D
y
x
y
x
y
FIGURE 3.10
Tree-type structure [16]. D: absolute difference, A: addition, M: memory.
• Throughput rate of motion vectors
• Scalability to larger block size and search range
• Operating clock frequency
• Dynamic power consumption
• PE utilization
Area and complexity can be represented by the number of PEs, the micro-architecture of
an individual PE, and the number of on-chip memory units such as latches, pipeline registers,
shift registers, etc. Motion vector computation throughput rate can be determined by block
computation time. The memory bandwidth is proportional to the number of I/O ports required
by the processing array. I/O ports include current block, search area data, and motion vector
output ports. A multiple-chip solution provides the ability to support a bigger block size and
search range.
With today’s technology, a single-chip solution or subsystem solution is more practical and
cost-efficient. A few architectures can truly scale well but require a large number of fan-outs
as a result of broadcasting. Block-level PE utilization is taken into consideration rather than
the frame level. Power consumption becomes more and more important to support mobile
communication technology. The block size of N = 16 and search range of p = 8 are used as
common building blocks. In Tables 3.1 and 3.2, the performance parameters are formulated
as functions of N and p.
For simulated or fabricated layouts, important parameters such as maximum operating frequency, die size, transistor count, and power consumption can be used to evaluate the performance of each architecture in Table 3.2. For example, the bigger the die size, the more likely
lower yield becomes, leading to the higher list price. Within a certain amount of broadcasting,
the higher the transistor count, the more power is consumed. Otherwise, power consumed by
the inherent capacitance and inductance of long and wide interconnection may become more
©2001 CRC Press LLC
apparent. This can affect the battery time of a digital video camcorder and/or multimedia
mobile terminal.
3.3
3.3.1
Dedicated Micro-Architecture
Design Methodologies for Dedicated Micro-Architecture
A dedicated micro-architecture is a hardware implementation specifically for a given algorithm. It achieves highest performance through both specialization and parallelism.
Implementation of Nonrecursive Algorithms
Any computing algorithm can be represented by a directed graph where each node represents
a task and each directed arc represents the production and consumption of data. In its most
primitive form, such a graph is called a data flow graph. Let us consider an algorithm with the
following formulation.
Algorithm 4:
tmp0= c4*(-x(3)+x(4));
y(3) = ic6*(x(3) + tmp0);
y(7) = ic2*(-x(3) + tmp0);
It can be translated into a data flow diagram as shown in Figure 3.11. In this algorithm,
three additions and three multiplication operations are performed. There are two input data
samples, x(3) and x(4), and two output data samples, y(3) and y(7). c4, ic2, and
ic6 are precomputed constant coefficients which are stored in memory and will be available
whenever needed. To implement this algorithm, one must have appropriate hardware devices
to perform addition and multiplication operations. Moreover, each device will be assigned to
perform a specific task according to a schedule. The collection of task assignment and schedule
for each of the hardware devices then constitutes an implementation of the algorithm.
FIGURE 3.11
An example of a data flow diagram.
©2001 CRC Press LLC
©2001 CRC Press LLC
Table 3.1 Architecture Comparison for Search Range p = N/2 = 8
Architecture
Search
Range
PE
Computation Time
(cycles)
I/O Ports
(8 bits)
Memory Units
(8 bits)
Komarek and Pirsch [5]
AS1
−p/ + p
2p + 1
N (N + 2p)(2p + 1)
3
10p + 6
AB1
−p/ + p
N
N(N + 2p)(2p + 1)
2N + 1
2N + 1
AS2
−p/ + p
N (2p + 1)
N (N + 2p)
N (N + 2P )
3(N + P )(3N + 2) + 1
AB2
−p/ + p
N2
(N + 2p)(2p + 1)
2N + 1
2N 2 + N + 1
Vos and Stegherr [17]
(2D)
−p/ + p
N2
N2
4
7N 2 + 2Np
2D array (type 1)
Linear array
−p/ + p
N
N(2p + 1)2
4
3N 2 + 2Np
Yang et al. [13]
−p/ + p − 1
N
2p(N 2 + 2p)
4
4N
Hsieh and Lin [6]
−p/ + p
N2
(N + 2p)2 + 5
3
3N 2 + (N − 1)(2p − 1)
Jehng et al. [16]
−p/ + p
N 2 /16
32(2p + 1)2
4
N 2 /16 + 1
Wu and Yeh [14]
−p/ + p
4N
2N (2N + p)
4
N2
2
Nam et al. [18] &
−p/ + p − 1
N
(2p) N + N + log2 N
4
8N + 1
Nam and Lee [19]
−p + 1/ + p
2N
(2p)2 N
6
9N + 4p
Chang et al. [15]
2
Yeo and Hu [8]
−p/ + p − 1
N
N2
4
2N 2
2
2
Pan et al. [7]
−p + 1/ + p − 1
2N
(N + 2p)(p + 3)
N +3
2N + 4N + 1
Chen et al. [20]
−p/ + p
2N 2 + 2 (2p + 2N/M)(2p + 1)
3
2N 2 + 2N + 2
Lee and Lu [11]
−p/ + p − 1
N2
(2p)2
4
5N 2 + 2(N − 1)(N + 2p)
2
You and Lee [12]
−p/ + p − 1
kv
(2pN ) /kv
10
(N + 2p)2
2
2
2
Chen and Kung [10]
−p/ + p
N
N
3
2N + (N + 2p)2
2
2
STi3220 [21]
−p/ + p − 1
N
N + 46
5
2N 2
2
2
Kittitornkun and Hu [9]
−p/ + p
(2p + 1)
N
4
3(2p + 1)2 + N 2
Note: The number of PE corresponds to the number of arithmetic units that perform absolute difference (AD), addition (A), and
comparison (M).
Table 3.2 Parameter Comparison of Fabricated or Simulated Layouts
Architecture
Techno.
(µm)
Max Freq
(MHz)
I/O
Pads
Die size
(mm2 )
Transistor
Count
Power
Consum. (W)
Yang et al. [13]
1.2
25
116
3.15 × 3.13
52,000
Na.
Hsieh and Lin [6]
Wu and Yeh [14]
1.0
0.8
120
23
Na.
65
Na.
5.40 × 4.33
Na.
86,000
Na.
Na.
Chang et al. [15]
0.8
Na.
100
6.44 × 5.26
102,000
Na.
Vos and Schobinger [22]
Nam and Lee [19]
Chen et al. [20]
Lee and Lu [11]
0.6
0.8
0.8
0.8
72
50
30
100
Na.
Na.
97
84
228
Na.
12.0 × 4.3
9.5 × 7.2
1,050,000
Na.
Na.
310,000
Sti3220 [21]
Na.
20
144
Na.
Na.
Na.
Na.
Na.
1.95
@ 50 MHz
2.4
@ 20 MHz
Na.: not available.
Assume that four hardware devices, two adders and two multipliers, are available. The delay
for an addition is one time unit, whereas for a multiplication it is two time units. Furthermore,
assume that after the execution of each task, the result will be stored in a temporary storage
element (e.g., a register) before it is used as the input by a subsequent task. A possible
implementation of Algorithm 4 is illustrated in Table 3.3.
Table 3.3 Implementation # 1 of Algorithm 4
In this table, each column represents one time unit, and each row represents a particular
device. The numerical number in each shaded box corresponds to the particular task in the
data flow graph. Blanked cells indicate that the corresponding device is left idle. Note that
task 2 cannot be commenced before task 1 is completed. This relationship is known as data
dependence. Also note that in time unit 4, tasks 3 and 5 are executed in both adders in parallel.
This is also the case in time units 5 to 6 where tasks 4 and 6 are executed in the two multipliers in
parallel. Thus, with a sufficient number of hardware devices, it is possible to exploit parallelism
to expedite the computation.
Suppose now that only one adder and one multiplier are available; then an implementation
will take longer to execute. An example is given in Table 3.4. Note that the total execution
time is increased from 6 to 8 time units. However, only half the hardware is needed.
Let us consider yet another possible implementation of Algorithm 4 when there is a stream
of data samples to be processed by the hardware.
©2001 CRC Press LLC
Table 3.4 Implementation # 2 of Algorithm 4
Algorithm 5:
for i = 1 to . . .,
tmp0(i)= c4*(x(3,i)+x(4,i));
y(3,i) = ic6*(x(3,i) + tmp0(i));
y(7,i) = ic2*(x(3,i) + tmp0(i));
end
Algorithm 5 contains an infinite loop of the same loop body as Algorithm 4. Since the
output of loop i (tmp0(i), y(3,i), y(7,i)) does not depend on the output of other
iterations, the corresponding DG of Algorithm 5 will contain infinitely many copies of the DG
of a single iteration shown in Figure 3.11. Since the DGs of different iteration index i are
independent, we need to focus on the realization of the DG of a single iteration. Then we may
duplicate the implementation of one iteration to realize other iterations. In particular, if the
input data samples x(3,i) and x(4,i) are sampled sequentially as i increases, multiple
iterations of the this algorithm can be implemented using two adders and three multipliers
(Table 3.5).
Table 3.5 Multi-Iteration Implementation # 1 of Algorithm 5
Note: Cells with the same texture or shade belong to tasks of the same iteration.
In this implementation, each type of box shading corresponds to a particular iteration index i. This implementation differs from the previous two implementations in several ways:
(1) Multiple iterations are realized on the same set of hardware devices. (2) Each adder or
multiplier performs the same task or tasks in every iteration. In other words, each task is
assigned to a hardware device statically, and the schedule is periodic. Also, note that execution of tasks of successive iterations overlap. Thus, we have an overlap schedule. (3) While
each iteration will take seven time units in total to compute, every successive iteration can
be initiated every two time units. Hence, the throughput rate of this implementation is two
©2001 CRC Press LLC
time units per iteration. The average duration between the initiation of successive iterations is
known as the initiation interval.
Comparing these three implementations, clearly there are trade-offs between the amount of
resource utilized (number of hardware devices, for example) and the performance (the total
delay, in this case) achieved. In general, this can be formulated as one of two constrained
optimization problems:
• Resource-constrained synthesis problem — Given the maximum amount of resources,
derive an implementation of an algorithm A such that its performance is maximized.
• Performance-constrained synthesis problem — Given the desired performance objective, derive an implementation of an algorithm A such that the total cost of hardware
resources is minimized.
The resource-constrained synthesis problem has an advantage in that it guarantees a solution as long as the available hardware resource is able to implement every required task in
algorithm A. On the other hand, given the desired performance objective, an implementation
may not exist regardless of how many hardware resources are used. For example, if the performance objective is to compute the output y(3) and y(7) within four time units after input
data x(3) and x(4) are made available, then it is impossible to derive an implementation to
achieve this goal.
Implementation of Recursive Algorithms
Let us consider the following example:
Algorithm 6:
for i = 1 to . . .
y(i) = a*y(i-1) + x(i)
end
This is a recursive algorithm since the execution of the present iteration depends on the output
from the execution of a previous iteration. The data flow graph of this recursive algorithm
is shown in Figure 3.12. The dependence relations are labeled with horizontal arrows. The
FIGURE 3.12
Data flow graph of Algorithm 6.
thick arrows indicate inter-iteration dependence relations. Hence, the execution of the ith
iteration will have to wait for the completion of the (i − 1)th iteration. The data flow graph
can be conveniently expressed as an iterative computation dependence graph (ICDG) that
contains only one iteration, but label the inter-iteration dependence arc with a dependence
distance d, which is a positive integer. This is illustrated in Figure 3.13. We note that for a
nonrecursive algorithm, even if it has an infinite number of iterations (e.g., Algorithm 4), its
complete data flow graph contains separate copies of the DG of each iteration. These DGs
have no inter-iteration dependence arc linking them.
©2001 CRC Press LLC
FIGURE 3.13
ICDG of Algorithm 6.
A challenge in the implementation of a recursive algorithm is that one must consider the interiteration dependence relations. Many design theories have been developed toward this goal [4],
[23]–[25]. The focus of study has been on the feasibility of performance-constrained synthesis.
Given a desired throughput rate (initiation interval), one wants to derive an implementation
that can achieve the desired performance using the minimum number of hardware modules.
Suppose that multiplication takes two clock cycles and addition takes one clock cycle. It is
easy to see that y(i) cannot be computed until three clock cycles after y(i) is computed. In
other words, the minimum initiation interval is (2+1) = 3 clock cycles. In a more complicated
ICDG that contains more than one tightly coupled cycle, the minimum initiation interval can
be found according to the formula
τi (k)
i
Imin = Max
k j (k)
j
where τi (k) is the computation time of the ith node of the kth cycle in the ICDG and j (k) is
the j th inter-iteration dependence distance in the kth cycle. Let us now consider the example
in Figure 3.14. There are two cycles in the ICDG in this figure. The initiation interval can be
FIGURE 3.14
An ICDG containing two cycles.
calculated as follows:
Imin = max{(3 + 1 + 2 + 2)/(1 + 2), (3 + 2 + 2)/2} = max{8/3, 7/2} = 3.5
If the desired initiation interval is larger than the minimum initiation interval, one may consider
any efficient implementation of a single iteration of the ICDG, and then simply duplicate that
implementation to realize computations of different iterations. For example, in the case of
Algorithm 6, one may use a single adder and a multiplier module to implement the algorithm
if, say, the desired throughput rate is one data sample per four clock cycles. The corresponding
implementation is quite straightforward (Table 3.6).
Here we assume that x(i) is available at every fourth clock cycle: 4, 8, 12, . . . . Thus the
addition operation can take place only at these clock cycles. The shaded boxes in the adder
row of Table 3.6 are also labeled with the corresponding y(i) computed at the end of that
©2001 CRC Press LLC
Table 3.6 Implementation of Algorithm 6
Note: Initiation interval = four clock cycles.
clock cycle. The multiplication can then be performed in the immediate next two clock cycles.
However, the addition must wait until x(i) is ready.
Suppose now the desired throughput rate is increased to one sample per two clock cycles,
which is smaller than the minimum initiation interval of three clock cycles. What should
we do? The solution is to use an algorithm transformation technique known as the lookahead transformation. In essence, the look-ahead transformation is to substitute the iteration
expression of one iteration into the next so as to reduce the minimum initiation interval at the
expense of more computations per iteration. For example, Algorithm 6, after applying the
look-ahead transformation once, can be represented as:
Algorithm 7:
for i = 1 to . . .
y(i) = aˆ2*y(i-2) + a*x(i-1) + x(i)
end
The corresponding ICDG is displayed in Figure 3.15. The new minimum initiation interval
FIGURE 3.15
ICDG of Algorithm 7.
now becomes: (2 + 1)/2 = 1.5 < 2 clock cycles, as desired. Next, the question is how
to implement this transformed algorithm with dedicated hardware modules. To address this
question, another algorithm transformation technique called loop unrolling is very useful.
Specifically, we consider splitting the sequence {y(i)} into two subsequences {ye(i)} and
{yo(i)} such that
ye(i) = y(2i) and yo(i) = y(2i + 1).
Then the iterations in Algorithm 7 can be divided into two subiterations with ye(i) and yo(i):
Algorithm 8:
for i = 1 to . . .
©2001 CRC Press LLC
ye(i) = aˆ2*ye(i-1) + a*x(2i-1) + x(2i)
yo(i) = aˆ2*yo(i-1) + a*x(2i) + x(2i+1)
end
To implement Algorithm 8, we denote a new sequence
u(i) = x(i) + a ∗ x(i − 1)
Then one can see that Algorithm 8 corresponds to two independent subloops:
ye(i) = a 2 ∗ ye(i − 1) + u(2i)
yo(i) = a 2 ∗ yo(i − 1) + u(2i + 1)
Each of these subloops will compute at a rate twice as slow as u(i) is computed. Since x(i) is
sampled at a rate of one sample per two clock cycles, ye(i) and yo(i) each will be computed
at a rate of one sample every four clock cycles. Hence, on average, the effective throughput
rate is one sample of y(i) every two clock cycles. A possible implementation is shown in
Table 3.7.
Table 3.7 Implementation of the Loop-Unrolled ICDG of Algorithm 8
In this implementation, u(i) is computed using the adder and multiplier #1. For example,
u(3) is computed after a ∗ x(2) is computed and x(3) is available during the fifth clock cycle.
The two subloops share a common multiplier #2 and the same adder that is used to compute
u(i). Note that a 2 ∗ ye(i) or a 2 ∗ yo(i) is computed right after ye(i) or yo(i) is computed
in the adder. Also note that there are four clock cycles between when ye(1) and ye(2) are
computed. This is also the case between yo(1) and yo(2).
In the rest of this section, we survey a few multimedia algorithms and the corresponding
implementations.
3.3.2
Feed-Forward Direct Synthesis: Fast Discrete Cosine Transform (DCT)
Dedicated Micro-Architecture for 1D Eight-Point DCT
An N-point DCT is defined as:
y(k) = c(k)
N−1
n=0
cos
2π k(2n + 1)
x(n)
4N
(3.6)
√
√
where c(0) = 1/ N and c(k) = (2/N ), 1 ≤ k ≤ N − 1. The inverse DCT can be rewritten
as:
x(n) =
N−1
k=0
©2001 CRC Press LLC
cos
2π k(2n + 1)
c(k)y(k)
4N
(3.7)
For the case of N = 8, the DCT can be written as a matrix vector product [26]
y = C8 x
(3.8)
The 8 × 8 matrix C8 can be factored into the product of three matrices:
C8 = P8 K8 B
(3.9)
where P8 is a permutation matrix, and K8 is a block diagonal matrix


G1

1
G1

K8 = 


G2
2
G4
cos(3π/8) cos(π/8)
with G1 = cos(π/4), G2 =
, and
− cos(π/8) cos(3π/8)


cos(5π/16) cos(9π/16) cos(3π/16) cos(π/16)
 − cos(π/16) cos(5π/16) cos(9π/16) cos(3π/16)

G4 = 
− cos(3π/16) − cos(π/16) cos(5π/16) cos(9π/16)
− cos(9π/16) − cos(3π/16) − cos(π/16) cos(5π/16)
(3.10)
(3.11)
is an anticirculant matrix. Finally, B can be further factored into the product of three matrices
consisting of 0, 1, and −1 as its entries: B = B1 B2 B3 . Based on this factorization, Feig and
Winograd [26] proposed an efficient eight-point DCT algorithm that requires 13 multiplication
operations and 29 additions. An implementation of this algorithm in MatlabTM m-file format
is listed below.
Algorithm 9: Fast DCT Algorithm
function y=fdct(x0);
% implementation of fast DCT algorithm by Feig and Winograd
% IEEE Trans. SP, vol. 40, No. 9, pp. 2174-93, 1992.
% (c) copyright 1998, 1999 by Yu Hen Hu
%
% Note that the array index is changed from 0:7 to 1:8
% These are constants which can be stored as parameters.
C1 = 1/cos(pi/16);
C2=1/cos(pi/8);
C3 =1/cos(3*pi/16);
C4 = cos(pi/4);
C5 = 1/cos(5*pi/16); C6 = 1/cos(3*pi/8); C7 =1/cos(7*pi/16);
% Multiply
A1 = x0(1)
A2 = x0(2)
A3 = x0(3)
A4 = x0(4)
by B3
+ x0(8);
+ x0(7);
+ x0(6);
+ x0(5);
A5
A6
A7
A8
=
=
=
=
x0(1)
x0(2)
x0(3)
x0(4)
% Multiply by B2
A9 = A1 + A4;
A10 = A2 + A3;
A11 = A1 - A4;
A12 = A2 - A3;
©2001 CRC Press LLC
-
x0(8);
x0(7);
x0(6);
x0(5);
% Multiply by B1
A13 = A9 + A10;
A14 = A9 - A10;
% multiply by (1/2) G1
M1 = (1/2)*C4*A13;
% y(1)
M2 = (1/2)*C4*A14;
% y(5)
% multiply by (1/2) G2
A15 = -A12 + A11; M3 = cos(pi/4)*A15;
A20 = A12 + M3;
A21 = -A12 + M3;
M6 = (1/4)*C6*A20;
% y(3)
M7 = (1/4)*C2*A21;
% y(7)
% Now multiply by (1/2)G4
% multiply by H_42
A16 = A8 - A5;
A17 = -A7 + A5;
A18 = A8 + A6;
A19 = -A17 + A18;
% Multiply by 1, G1, G2
M4 = C4*A16;
M5 = C4*A19;
A22 = A17 + M5;
A23 = -A17 + M5;
M8 = (1/2)*C6*A22; M9 = (1/2)*C2*A23;
% Multiply by H_41, then by Dˆ-1, and then 1/2 this is G4
% then multiply by (1/2) to make it (1/2) G4
A24 = - A7 + M4;
A25 = A7 + M4;
A26 = A24 - M8;
A27 = A25 + M9;
A28 = -A24 - M8;
A29 = -A25 + M9;
M10 = -(1/4)*C5*A26; % y(2)
M11 = -(1/4)*C1*A27; % y(4)
M12 = (1/4)*C3*A28;
% y(8)
M13 = -(1/4)*C7*A29; % y(6)
y(1) = M1; y(2) = M10; y(3) = M6; y(4) = M11;
y(5) = M2; y(6) = M13; y(7) = M7; y(8) = M12;
To support high-throughput real-time image and video coding, a DCT algorithm must be
executed at a speed that matches the I/O data rate. For example, in HDTV applications, videos
are processed at a rate of 30 frames per second, with each frame 2048 × 4096 pixels. At a
4:1:1 ratio, there can be as many as
30 × (6/4) × 2048 × 4096 × (2 × 8)/64
= 45 × 211+12+1+3−6 = 94, 371, 840 ≈ 94.5 million 8-point DCT operations
to be performed within one second. Hence, dedicated micro-architecture will be needed in
such an application.
The DG shown in Figure 3.16 completely describes the algorithm and dictates the ordering
of each operation that needs to be performed. In this figure, the inputs x(0) to x(7) are made
available at the left end and the results y(0) to y(7) are computed and made available at the right
end. Each shaded square box represents a multiplication operation, and each shaded circle
represents an addition. The open circles do not correspond to any arithmetic operations, but
are used to depict the routing of data during computation. Since the direction of dependence is
always from left to right, it is omitted in Figure 3.16 in the interests of clarity. From Figure 3.16,
it can be identified that the critical path is from any of the input nodes to M5, and from there
to any of the four output nodes y(1), y(3), y(5), and y(7). The total delay is five additions
and three multiplications.
©2001 CRC Press LLC
FIGURE 3.16
Dependence graph of the fast DCT algorithm.
Once a dependence graph is derived, one may directly map the DG into a dedicated hardware
implementation by (1) designating a hardware module to realize each computation node in the
DG, and (2) interconnecting these hardware modules according to the directed arcs in the DG.
Two types of hardware modules will be used here: an adder module, which takes one clock
cycle to perform an addition, and a multiplier module, which takes two clock cycles to compute
a multiplication. The mapping of the DG into a hardware module is a binding process where
each node of the DG is mapped onto one hardware module which can implement the function
to be performed on that node. A single hardware module may be used to implement one or
more nodes on the DG. As in the previous section, we assume the output of each hardware
module will be held in a register.
a. Performance-Constrained Micro-Architecture Synthesis
Suppose that one may use as many hardware modules as needed. Then, from a theoretical
point of view, one may always derive an implementation to achieve the desired throughput
rate. This is because successive eight-point DCT operations are independent of each other.
For each new arriving eight-point data sample, one can always assign a new set of hardware
modules and initiate the computation immediately. Hence the minimum initiation interval can
be made as small as possible. The only limiting factor would be the speed to redirect data
samples into appropriate hardware modules.
Next, suppose that in addition to the throughput rate, the latency (time between arrival of
data samples and when they are computed) is also bounded. The minimum latency, given
that a sufficient number of hardware modules are available, is equal to the time delay along
the critical path, which includes five addition operations and three multiplication operations.
Thus, the minimum latency is 5 × 1 + 3 × 2 = 11 clock cycles. The maximum latency is
equal to the total computing time, with every operation executed sequentially. Thus, the upper
bound of latency is 29 × 1 + 13 × 2 = 55 clock cycles.
©2001 CRC Press LLC
Table 3.8 shows an implementation that achieves a throughput rate of one 8-point DCT per
clock cycle and a latency of 11 clock cycles. Note that if the clock frequency is greater than
95 MHz, then this implementation can deliver the required throughput rate for HDTV main
profile performance.
The implementation is expressed in a warped format to save space. In this table, each item
Ai(1 ≤ i ≤ 29) or Mj (1 ≤ j ≤ 13) refers to a separate hardware module and should take up
a separate raw in the implementation. In Table 3.8, each entry Ai or Mj gives the schedule of
the particular hardware module corresponding to the same set of eight data samples.
Table 3.8 A Dedicated Implementation of 8-Point DCT
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
A16
A17
A18
A13
A14
A15
M1
M2
M3
M1
M2
M3
M4
M4
A19
M5
A24
A25
M5
A20
A21
A22
A23
M6
M7
M8
M9
M6
M7
M8
M9
A26
A27
A28
A29
M10
M11
M12
M13
M10
M11
M12
M13
Note: Throughput = 1 DCT/clock cycle, latency = 11 clock cycles.
The implementation is shown in a compact format.
In this implementation, 29 full adders and 13 pipelined multipliers are used. By pipelined
multiplier, we require each multiplication to be accomplished in two successive stages, with
each stage taking one clock cycle. A buffer between these two stages will store the intermediate
result. This way, while stage 2 is completing the second half of the multiplication of the present
iteration, stage 1 can start computing the first half of the multiplication of data from the next
iteration. Thus, with two-stage pipelined operation, such a multiplier can achieve a throughput
rate of one multiplication per clock cycle.
On the other hand, if one type of multiplier module which cannot be broken into two pipelined
stages is used, then two multipliers must be used to realize each multiplication operation in
Table 3.6 in an interleaved fashion. This is illustrated in Figure 3.17. The odd number of the
data set will use multiplier #1 while the even number of the data set will use multiplier #2. As
such, on average, two multiplication operations can be performed in two clock cycles. This
translates into an effective throughput rate of one multiplication per clock cycle. However, the
total number of multiplier modules needed will increase to 2 × 13 = 26.
FIGURE 3.17
Illustration of the difference between pipelined and interleaved multiplier implementation.
©2001 CRC Press LLC
Let us consider relaxing the performance constraints by lowering the throughput rate to one
8-point DCT per two clock cycles and allowing longer latency. One possible implementation,
in a compact format, is shown in Table 3.9.
Table 3.9 Eight-Point DCT Implementation
Note: Throughput rate: 1 DCT per 2 clock cycles; latency: 12 clock cycles; 15 adder modules and
13 multipliers are used.
In this implementation, we use only regular multiplier modules. If we use two-stage
pipelined multiplier modules, the number of multipliers can further be reduced to seven. In
order to minimize the number of adder modules, we choose to execute A26 and A27 (as well
as A28 and A29) sequentially. This change accounts for the additional clock cycle of latency.
b. Resource-Constrained Micro-Architecture Synthesis
In a resource-constrained synthesis problem, the number of hardware modules is given. The
objective is to maximize the performance (throughput rate) under this resource constraint. To
illustrate, let us consider the situation where only one adder module and one multiplier module
is available. In Table 3.10, the first row gives the clock-by-clock schedule for the adder module,
Table 3.10 Implementation of 8-Point DCT with 1 Adder and 1 Multiplier
and the second row gives the schedule for the multiplier module. The shaded area (M2, M6,
M7) indicates that those multiplication operations belong to the previous data set. Thus, this
is an overlapped schedule. The initiation interval is 29 clock cycles — the minimum that can
be achieved with only one adder module. The execution of the adder and the multiplier are
completely overlapped. Hence, we can conclude that this is one of the optimal solutions that
maximize the throughput rate (1 DCT in 29 clock cycles), given the resource constraint (one
adder and one multiplier module).
©2001 CRC Press LLC
c. Practical Implementation Considerations
In the above synthesis examples, the complexity of inter-module communication paths
(buses) is not taken into account, nor do we factor in the amount of temporary storage elements
(registers) needed to facilitate such realization.
Furthermore, in practical hardware synthesis, not all modules have the same word length.
Due to the addition and multiplication operations, the dynamic range (number of significant
digits) will increase. The adder at a later stage of computing will need more bits. Therefore,
before commencing a hardware synthesis, it is crucial to study the numerical property of
this fast DCT algorithm and determine its quantization noise level to ensure that it meets the
requirements of the standard.
Generalization to 2D Scaled DCT
In image and video coding standards such as JPEG and MPEG, a 2D DCT is to be performed
on an 8 × 8 image pixel block X:
Y = C8 XCT8
(3.12)
This corresponds to a consecutive matrix–matrix product. An array structure can be developed
to realize this operation using a systolic array. However, it would require many multipliers.
In [26], a different approach is taken. First, we note that the above formulation can be converted
into a matrix–vector product between a 64 × 64 matrix formed by the kroenecker product of
the DCT matrix, C8 ⊗ C8 , and a 64 × 1 vector X formed by concatenating columns of the X
matrix. The result is a 64 × 1 vector Y that gives each column of the Y matrix:
Y = (C8 ⊗ C8 ) X
(3.13)
The C8 matrix can be factorized, in this case, into the product as follows:
C8 = P8 D8 R8,1 M8 R8,2
(3.14)
where P8 is the same permutation matrix as in the 1D eight-point DCT algorithm. D8 is an
8 × 8 diagonal matrix; R8,1 is a matrix containing elements of 0, 1, and −1; and R8,2 is the
product of three matrices, each of which contains 0, 1, and −1 elements only.


1
 1





1




cos(π/8)


M8 = 
(3.15)

1




cos(π/8)



cos(3π/16) cos(π/16) 
− cos(π/16) cos(3π/16)
For the kroenecker product C8 ⊗ C8 , the factorization becomes
C8 ⊗ C8 = P8 D8 R8,1 M8 R8,2 ⊗ P8 D8 R8,1 M8 R8,2
= P8 D8 ⊗ P8 D8 • R8,1 M8 R8,2 ⊗ R8,1 M8 R8,2
(3.16)
= P8 ⊗ P8 • D8 ⊗ D8 • R8,1 ⊗ R8,1 • M8 ⊗ M8 • R8,2 ⊗ R8,2
Hence a fast 2D DCT algorithm can be developed accordingly. The hardware implementation
approach will be similar to that of 1D DCT. However, the complexity will be significantly
greater.
©2001 CRC Press LLC
One advantage of the factorization expression in (3.13) is that a scaled DCT can be performed. Scaled DCT is very useful for JPEG image coding and MPEG intra-frame coding
standards. In these standards, the DCT coefficients will be multiplied element by element
to a quantization matrix to deemphasize visually unimportant frequency components before
applying scalar quantization. Thus, for each block, there will be 64 additional multiplication
operations performed before quantization can be applied. In effect, this quantization matrix
can be formulated as a 64 × 64 diagonal matrix W such that the scaled DCT coefficient vector
( = WY = W • P8 ⊗ P8 • D8 ⊗ D8 • R8,1 ⊗ R8,1 • M8 ⊗ M8 • R8,2 ⊗ R8,2 X
(3.17)
A complicated flow chart of the above algorithm is given in the appendix of [26]. Due to space
limitations, it is not included here. The basic ideas of designing a dedicated micro-architecture
for 2D scaled DCT will be similar to 1D DCT.
3.3.3
Feedback Direct Synthesis: Huffman Coding
In this section, we turn our attention to the dedicated micro-architecture implementation of
a different class of recursive multimedia algorithms, known as the Huffman entropy coding
algorithm.
Huffman coding encodes symbols with variable-length binary streams without a separator
symbol. It is based on the probability of symbol appearances in the vocabulary. Often the
encoding table is designed off line. During encoding, each symbol is presented to the encoder
and a variable-length bitstream is generated accordingly. This is essentially a table-lookup
procedure. The decoding procedure is more complicated: For each bit received, the decoder
must decide whether it is the end of a specific code or it is in the middle of a code. In other
words, the decoder must be realized as a sequential machine. Due to the variable-length
feature, the number of cycles to decode a codeword varies. The throughput in this case is 1 bit
per clock cycle. Let us consider a Huffman decoding example. Assume the coding table is as
in Table 3.11. Then we may derive the Mealy model state diagram, as shown in Figure 3.18.
Table 3.11 A
Huffman Coding Table
Symbol
A
B
C
D
E
F
Codeword
0
10
1100
1101
1110
1111
Usually the total number of states is the total number of symbols minus 1, and the longest
cycle in the state diagram equals the longest codewords. In practical applications, such as in
JPEG or MPEG, there are a large number of symbols and long codewords. For example, in the
JPEG AC Huffman table, there are 162 symbols, and many codewords are as long as 16 bits.
Implementation of Finite State Machine
A general structure of implementing finite state machine is shown in Figure 3.19. The
state variables are implemented with flip-flops. The combinational circuits can be realized
©2001 CRC Press LLC
FIGURE 3.18
State diagram of the Huffman coding algorithm.
FIGURE 3.19
Finite state machine implementation of Huffman decoding algorithm.
with read-only memory (ROM), programmable logic array (PLA), or dedicated logic gates.
The design issues include: (1) how high the clock rate can go, and (2) how complicated the
combinational circuit design will be.
In the above example, there are five states (a, b, c, d, and e), which require at least three
state variables to represent. There are seven output symbols (A, B, C, D, E, F, and ——) to
be encoded in an additional 3 bits. Thus, there are at least six outputs of the combinational
circuit. In other words, the combinational circuit consists of six Boolean functions sharing the
same set of four Boolean variables (3 state variables + 1 bit input). If a ROM is used, it will
have a size of 16 words with each word containing 6 bits. Let us consider yet another example
of the JPEG AC Huffman table. The JPEG AC Huffman code contains 161 symbols and has
a codeword length smaller than or equal to 16 bits. Since the Huffman tree has 161 nodes, it
requires at least eight state variables (28 = 256 > 161). Output symbol encoding will also
require 8 bits. If a ROM is used to realize the combinational circuit, then it will have a size of
29 × (8 + 8) = 512 × 16 = 8K bits.
The above implementation using a finite state machine ensures a constant input rate in that
it consumes 1 bit each clock cycle. The number of symbols produced at the output varies.
However, on average, the number of clock cycles needed to produce a symbol is roughly equal
to the average codeword length Lavg . Asymptotically, Lavg is a good approximation of the
entropy of the underlying symbol probability distribution. If the input throughput rate is to be
increased, we may scan more than 1 bit at each clock cycle provided the input data rate is at
least twice the decoder’s internal clock rate. This will not increase the number of states, but it
will make the state transition more complicated. For example, if each time 2 bits of input data
are scanned, the corresponding state diagram will be as in Figure 3.20.
The size of the state table doubles for each additional bit being scanned in a clock cycle.
If a ROM is used to realize the state table, the number of addresses will double accordingly.
Moreover, since there can be more than one symbol in the output during each clock cycle, the
©2001 CRC Press LLC
FIGURE 3.20
State diagram decoding 2 bits at a time.
word length of the ROM will also be increased accordingly. Hence it is a trade-off between
hardware complexity and throughput rate.
Lei et al. [27] have proposed a constant output rate Huffman decoding method using FSM
realization. This is accomplished by scanning L bits of input at a time, with M being the
maximum codeword length. Each time, exactly one codeword is decoded. The remaining
bits, which are not part of the decoded symbols, then will be realigned and decoded again.
Let us consider the following bitstream 00110010011100100. During decoding, the decoder
scans the first 4 bits (0011) and determines that the first symbol is A(0). Next, it shifts by
1 (since A is encoded by 1 bit) and decodes the second bit as A again. Next, after shifting
another bit, its window contains 1100, which is decoded as C. The next iteration, it will shift
4 bits instead of 1 bit because the entire 1100 is used. Therefore, during each clock cycle, one
symbol is decoded. However, the rate at which the input data stream is consumed depends on
the composition of the given sequence. This process is depicted in Figure 3.21. Each double
arrow line segment indicates the 4 bits being scanned in a clock cycle. 0 : A indicates that the
left-most bit 0 is being decoded to yield the symbol A. Of course, one can be more opportunistic
by allowing more than one symbol to be decoded in each L-bit window and thereby increase
the decoding rate, at the expense of additional hardware complexity.
FIGURE 3.21
Illustration of constant symbol decoding rate Huffman decoder.
Concurrent VLC Decoding [28]
One way to further increase the coding speed is to exploit parallelism by decoding different
segments of a bitstream concurrently. Successive M-bit segments will be overlapped by an
©2001 CRC Press LLC
L-bit window, where M >> L and L is the maximum codeword length. Therefore, there
must be a split of two codewords within this window. In other words, in the successive Mbitstreams, each can have at most L different starting bit positions within that L-bit window. By
comparing the potential starting bit position within this L-bit window of two M-bitstreams,
we can uniquely determine the actual starting point of each stream and therefore decouple
the successive streams to allow concurrent decoding. To illustrate, consider the bitstream in
Figure 3.22 and the partition into M = 10 bitstreams with an L = 4 bits overlapping window:
FIGURE 3.22
Concurrent VLC decoding.
In this figure, the dashed lines within each window indicate the legitimate codeword splitting
positions. The upper dashed lines are identified from the upper stream segments and the lower
dashed lines are from the lower stream segments. If the upper and lower splitting points
overlap, it will be accepted as a likely codeword splitting point. To elaborate, let us consider
window #1, which is the trailing window of the first upper stream segment. We note that if the
splitting point is at the position to the left of the window, then the previous 4 bits (0110) do not
correspond to any 4-bit symbols. They do contain the codeword B (10) as the last 2 bits. But
then the first 2 bits (01) must be part of a 4-bit codeword. In fact, from the Huffman table, they
must be part of the codeword 1101. Unfortunately, the 2 bits to the left of the stream (0110)
are 10 (the first 2 bits from the left). Hence, we conclude that such a split is not valid. In other
words, for each potential split position, we must trace back to the remainder of the bitstream
segment to validate if there is a legitimate split. In practical implementation, for each stream
segment, and each potential codeword splitting position in the leading window, a Huffman
decoding will be performed. If the decoder encounters an illegitimate Huffman code along
the way, the splitting point is deemed infeasible and the next potential splitting point will be
tested. If a splitting point in the leading window is consistent up to a codeword that partially
falls within the trailing window, the corresponding split position at the trailing window will
be recorded together with the splitting point in the leading window of the same segment. The
legitimate splitting points in the same window of the successive stream segments then will
be regarded as true codeword splitting points. After these points are determined, concurrent
decoding of each stream segment will commence.
3.4
Concluding Remarks
In this chapter, we surveyed implementation strategies for application-specific multimedia
signal processors. Using the application of video coding as an example, we illustrated how
each design style is applied to synthesize dedicated realization under different constraints.
Current research efforts have been focused on low-power implementation and reconfigurable
architecture. With these new research efforts, there will be more alternatives for designers to
choose.
©2001 CRC Press LLC
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[14] Wu, C.-M., and D.-K. Yeh, A VLSI motion estimator video image compression. IEEE
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block matching. IEEE Trans. on Circuits Syst. Video Technol., 1995. 5(4): p. 332–343.
[16] Jehng, Y.-S., L.-G. Chen, and T.-D. Chiueh, An efficient and simple VLSI tree architecture for motion estimation algorithms. IEEE Trans. on Signal Processing, 1993. 40(2):
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[17] Vos, L.D., and M. Stegherr, Parametrizable VLSI architecture for the full-search block
matching algorithms. IEEE Trans. on Circuits Syst., 1989. 26(10): p. 1309–1316.
[18] Nam, S.H., J.S. Baek, and M.K. Lee, Flexible VLSI architecture of full search motion
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©2001 CRC Press LLC
[19] Nam, S.H., and M.K. Lee, Flexibility of motion estimator for video image compression.
IEEE Trans. on Circuits Syst., 1996. 43(6): p. 467–470.
[20] Chen, M.-J., L.-G. Chen, K.-N. Cheng, and M.C. Chen, Efficient hybrid tree/linear array
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[22] Vos, L.D., and M. Schobinger, VLSI architecture for a flexible block matching processor.
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[24] Wang, D.J., and Y.H. Hu, Rate optimal scheduling of recursive DSP algorithms by
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[28] Lin, H.D., and D.G. Messerschmitt, Designing high-throughput VLC decoder. Part II —
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©2001 CRC Press LLC
Chapter 4
Superresolution of Images with Learned Multiple
Reconstruction Kernels
Frank M. Candocia and Jose C. Principe
4.1
Introduction
Superresolution is the term given to the signal processing operation that achieves a resolution
higher than the one afforded by the physical sensor. This term is prevalent within the radar
community and involves the ability to distinguish objects separated in space by less than the
resolution afforded by radar. In the domain of optical images the problem is akin to that of
perfect reconstruction[1, 2]. As such, this chapter will address the issue of image magnification
(also referred to as interpolation, zooming, enlargement, etc.) from a finite set of samples.
An example where magnification can aid multimedia applications is video teleconferencing,
where the bit rate constraints limit video throughput. Such restrictions typically result in the
transmission of a highly compressed and size-limited video sequence of low visual quality. In
this context, the task of superresolution thus becomes one of restoring lost information to the
compressed sequence of images so as to result in their magnification as well as providing a
sharper and/or less degraded image. Among the areas in which multimedia can benefit from
superresolution, the focus herein is on the image processing resulting in the superresolution of
still images. The benefits afforded by the proposed architecture will be examined and several
issues related to the methodology will be discussed.
Commonly, image magnification is accomplished through convolution of the image samples
with a single kernel — such as the bilinear, bicubic [3], or cubic B-spline kernels [4] — and
any postprocessing or subsequent image enhancement would typically be performed in an ad
hoc fashion. The mitigation of artifacts, due to either aliasing or other phenomena, by this
type of linear filtering is very limited. More recently, magnification techniques based on image
domain knowledge have been the subject of research. For example, directional methods [5, 6]
examine an image’s local edge content and interpolate in the low-frequency direction (along the
edge) rather than in the high-frequency direction (across the edge). Multiple kernel methods
typically select between a few ad hoc interpolation kernels [7]. Orthogonal transform methods
focus on the use of the discrete cosine transform (DCT) [8, 9] and the wavelet transform [10].
Variational methods formulate the interpolation problem as the constrained minimization of
a functional [11, 12]. An extended literature survey discussing these methods at great length
has been provided by Candocia [1].
The approach presented herein is novel and addresses the ill-posed nature of superresolution
by assuming that similar (correlated) neighborhoods remain similar across scales, and that this
a priori structure can be learned locally from available image samples across scales. Such local
©2001 CRC Press LLC
information extraction has been prominent in image compression schemes for quite some time,
as evidenced by JPEG- [13] and PCA- [14] based approaches, which typically compress the
set of nonoverlapping subblocks of an image. Recent compression approaches also exploit the
interblock correlation between subblocks [15, 16]. The goal is to divide the set of subblocks
into a finite number of disjoint sets that can individually be represented more efficiently than
the original set. Our approach is similar in spirit in that we exploit interblock correlation for
mapping similar overlapping neighborhoods to their high-resolution counterparts. However,
no one before us has proposed using this information to create constraints that can superresolve
images. We further show that a very simple local architecture can learn this structure effectively.
Moreover, our approach is shown to be equivalent to a convolution with a family of kernels
established from available images and “tuned” to their local characteristics, which represents
an extension to conventional sampling theory concepts.
The chapter is divided into sections as follows. Section 4.2 conceptually introduces the superresolution that is discussed herein. Comments and observations are made and the methodology from which the local architecture arises is also described. Section 4.3 presents the
image acquisition model used for synthesizing our low-resolution images. Section 4.4 describes single and multikernel-based approaches to magnification. Section 4.5 details the local
architecture implementing the superresolution methodology. In Section 4.6, several results
are presented illustrating the architecture’s capability. Section 4.7 discusses several issues
regarding the methodology and Section 4.8 provides our conclusions.
4.2
An Approach to Superresolution
The superresolution approach presented here addresses the reconstruction of an image (from
a finite set of samples) beyond the limit imposed by the Shannon sampling theory [17, 18]. For
the sake of simplicity, our development uses one-dimensional (1D) signals, but the extensions
to two dimensions (2D) should be clear.
Let x(t), where −∞ < t < ∞, be a continuous signal with maximum frequency content
c rad/s. Thus, our analysis is based on band-limited signals. We can represent x(t) as a
linear combination of a set of basis functions as
x(t) =
x[n]k(t, n)
(4.1)
n
where the linear weighting is given by the samples in x[n] and k(t, n) represents our set of
basis functions. Here x[n] ≡ x(nTs ), for integers n satisfying −∞ < n < ∞, and sampling
period Ts . The equation describing the perfect reconstruction of a signal in sampling theory is
∞
t
x(t) =
x(nTs ) sin c
−n
(4.2)
Ts
n=−∞
where, by definition, sin c(t) ≡ sin(πt)
π t . We see that our basis functions are given by k(t, n) =
sin c( Tts − n) and the basis functions in this infinite set are orthogonal [19]; that is,
∞
t
t
sin c
− n sin c
− m dt = Ts δ[n − m] .
Ts
Ts
−∞
The perfect reconstruction can be obtained if the sampling period satisfies Ts < T2c where
2π
. For time signals, T2c is the critical sampling rate, also called the Nyquist rate.
Tc = c
©2001 CRC Press LLC
Therefore, every instance of x(t) can be exactly resolved with an infinite set of samples
provided the density of samples is high enough. The sampling period Ts provides the limit to
which our signals can be perfectly reconstructed (resolved) from an orthogonal set of linear
projections and an infinite number of samples. Notice that the sin c bases are the universal set
of linear projections capable of perfectly reconstructing band-limited signals in time (space).
This set of bases is universal in that all appropriately sampled infinite extent band-limited
signals can be reconstructed with them, irrespective of their content.
The case of finite extent data is more realistic, in particular for images. For finite extent
data, equation (4.2) can be expressed as
x̂(t) =
∞
[x(nTs )w[n]] sin c
n=−∞
t
Ts
−n
(4.3)
where w[n] describes samples of our window function
1; 0 < n < N − 1
w[n] =
0; otherwise
and N is the extent of our window. Notice that the hat superscript in x̂ has been used purposely
to denote the approximation afforded by a finite number of data samples. The finite data set
reduces the resolvability of the signal. We can see this by examining equation (4.3) in the
frequency domain. The continuous Fourier transform of equation (4.3) yields
Ts
X̂() =
[X(ω) W (ω)]
s
s
where is the periodic convolution operator, ω = Ts is the angular frequency in radians,
X(ω) is the DTFT of x(nTs ) and is periodic of period 2π in ω and 2π
Ts in (similarly for
W ()), and () = {1 || < 21 ; 0 otherwise}. The effect of the windowing function w[n]
in equation (4.3) is to smear (distort) the true frequency spectrum X(), which results in a
decreased ability to properly resolve the signal x(t). To illustrate this, consider a down-sampled
image of Lena in Figure 4.1a. This image is 128 × 128 samples in size. It is interpolated to
256 × 256 using equation (4.3) and is illustrated in Figure 4.1b. A visual comparison of the
interpolated image with the original 256 × 256 image in Figure 4.1c demonstrates that the
sin c basis set limits the reconstruction performance. This is standard digital signal processing
knowledge: interpolating a digital representation does not improve frequency resolution. In
order to improve the frequency resolution, more samples (either longer windows or a higher
sampling frequency) of the original signal are necessary [18]. In image processing only a
higher sampling frequency will do the job since the window (the scene) is prespecified.
The objective of superresolution is to reconstruct x(t) more faithfully than the resolution
afforded by equation (4.3), that is, the resolution afforded from a finite set of observed data
obtained at a sampling rate Ts .
4.2.1
Comments and Observations
The conditions necessary for perfect signal reconstruction are: (1) there must be no noise
associated with the collected samples (e.g., no quantization error), (2) the sampling rate must be
higher than the Nyquist sampling rate of the signal, and (3) the signal must be of infinite extent.
We can immediately say that in image processing, perfect signal reconstruction is impossible
because an image has finite extent. In the practical sampling of optical images, the issue of
quantization error is usually not critical. The standard use of an 8-bit dynamic range usually
©2001 CRC Press LLC
FIGURE 4.1
Illustrating the resolution limit imposed by interpolating with the sin c bases of the Shannon sampling theory. Artifacts are clearly visible in the sin c interpolated image of (b).
(a) Lena* 128 × 128 image [obtained from (c)]; (b) image interpolated to 256 × 256;
(c) original (desired) Lena 256 × 256 image (*Copyright © 1972 by Playboy magazine).
yields highly acceptable and pleasing images. The issue of sampling frequency is much more
critical. The information of a natural scene has typically very high spatial frequency content.
The sharp contrast we perceive in order to delineate objects (object boundaries) as well as the
textural character of those objects are just two of the attributes inherent to the high-frequency
content of optical images. As such, the sampling frequency used in collecting optical images is
generally not large enough to fully describe a “continuous image” in the sense of the Shannon
theory. An interesting attribute of optical images is their highly structured nature. This structure
appears locally and can be used to characterize objects in these images; that is, portions of
objects can be described as smooth, edgy, etc. Information such as this is not considered in
the sampling theory.
Let us now make a few observations. Equation (4.2) specifies a set of basis functions which
are linearly weighted by the collected samples of the signal x(t). If the samples are collected
at a sampling rate Ts that does not meet the critical sampling rate, the set of sin c bases cannot
be linearly weighted according to equation (4.2) to perfectly reconstruct our data. However,
this does not preclude the existence of other sets of basis functions that could be linearly
combined by samples collected at a rate below the critical sampling rate and still yield the
signal’s perfect reconstruction. In fact, the perfect reconstruction of a signal according to the
Shannon sampling theory only establishes sufficient conditions for the perfect reconstruction
©2001 CRC Press LLC
from samples [17]. If some other knowledge about the signal corresponding to the observed
samples is available, then this can be used to develop bases for superresolving a signal.
The problem is that when these bases are no longer universal, they become signal dependent.
As a simple example, let’s consider the set of piecewise constant time functions where each
constant segment in the signals has duration T seconds (e.g., signals quantized in time). An
illustration of such a function is provided in Figure 4.2a. Note that this signal has infinite
frequency content due to its staircase nature.
T
x(t)
T
t
T
(a)
1
*
(b)
t
T
t
FIGURE 4.2
Simple example illustrating how perfect reconstruction is possible when a priori knowledge of the relation between signal samples and its corresponding continuous function
is known. (a) Piecewise constant continuous function grossly undersampled according
to the Nyquist criterion; (b) recovering the continuous function in (a) from its samples
requires a simple convolution of the samples with a zero-order-hold kernel.
If our observed samples were obtained by sampling this function every T seconds, then
clearly the convolution of a zero-order-hold kernel with the observed samples would be optimal
for recovering the piecewise constant function. That is, it would yield perfect reconstruction
even though the function in question was grossly undersampled according to the Shannon
sampling theory. This convolution is pictured in Figure 4.2b.
The set of piecewise constant signals is not typically encountered in practice, so the basis
set resulting from the zero-order-hold kernel is of limited use. However, this very simple
example illustrates that superresolution could be based on a priori knowledge about the signal
given the observed samples — irrespective of the frequency content of the signal. Therefore,
superresolution is directly associated with methods to acquire extra information about the
signal of interest and derive from it appropriate bases.
Recently, optimal reconstruction of signals sampled below their Nyquist rate was proved
possible by modeling the signal statistics [20]. Ruderman and Bialek derived the optimal filter
(which happens to be linear) for reconstructing a signal x(t), which is assumed band limited,
Gaussian, zero mean, and stationary. Their results also show that the signal statistics play no
role in perfect reconstruction when the Shannon sampling conditions are met.
The great lesson from this work is that a statistical description can be used to superresolve a
signal from a collected set of samples, irrespective of the relation between sampling frequency
and maximum frequency content. However, the analytic result is valid only for stationary
Gaussian signals. In practice, real-world signals are typically nonstationary and have very
complex statistics. The analytical intractability of determining the optimal filters for complicated density functions, as commented by the authors, limits the practical use of this method.
©2001 CRC Press LLC
However, statistical information about signals can also be obtained with adaptive algorithms.
This is the avenue explored in this work.
4.2.2
Finding Bases for Image Representation
Superresolution of images will be possible if the interpolation system uses more efficiently
the information contained in the available image samples. This requires projections onto dataspecific sets of bases instead of the ones established by the sampling theorem. Naturally,
learning or adaptive system theories play a crucial role in this methodology of designing dataspecific projections. The manner in which these models are realized must be consistent with
the information character of images and how this relates to the superresolution problem.
The universality of the perfect reconstruction theories is an amazing result, but the price paid
is a strict limitation on the resulting resolution. The practical problem is to do the best we can
with the available samples in order to superresolve the images. To yield better reconstructed
images, we must find alternative sets of projections from which to reconstruct our images.
It has been shown that perfect reconstruction can be achieved if a priori signal knowledge is
available, but in practice this knowledge is absent. So a central problem is how to capture
statistical knowledge about the domain and effectively use it to design basis functions. In
determining the set of projections to use, we must either make assumptions regarding our data
or learn this a priori knowledge from the available data using nonparametric models. In our
work we are concerned with the latter.
We herein propose a novel technique for image superresolution by working across scales.
From the original image we create a low-resolution version through a down-sampling operation
on the original image. The high-resolution image becomes the desired response to a learning
system that receives the low-resolution image as input. At this point we have two options to
learn a statistical model: either we model the global image statistics or we seek local statistical
models. The highly structured and localized nature of images calls for the development of
local statistical models. In fact, local models arise naturally from the various structures in
images resulting from the objects in the imaged scene. The local models can be practically
implemented by learning the relation between low- and high-resolution versions of an image.
Two particular traits typical to images can be used in this modeling:
• There is much similarity in local structure throughout an image.
• This structure is maintained across scales.
We now discuss the implications of the existence of these traits in images.
Similarity in Local Structure
The first trait can be exemplified by considering an image of a face. Local neighborhoods in
the person’s cheeks and forehead are generally indistinguishable when viewed independently.
We have assumed that the effects of lighting and other “special” attributes (scars, moles,
birthmarks, etc.) are absent in this comparison. An easy method to test this observation is
to locate these similar image portions in an image and randomly swap them to form a new
image. If the new image resembles the original one then our observation is correct. Similarly,
all neighborhoods exhibiting a particular characteristic can be treated in practically the same
manner. These neighborhoods can be considered generated by the same statistical process. It
has been shown that the targets for the neighborhoods can be interpreted as the mean of each
statistical process — one for each model used [1].
Examples of this first trait abound in images. It has recently been exploited to increase
compression gains. The standard schemes for lossy image compression are based around the
©2001 CRC Press LLC
highly redundant information generally present in small image blocks that could be described
in a more efficient and compact manner. In the case of compression via principal component
analysis (PCA), a single representation is established for all of the blocks in the image. The
recent compression approaches exploit the first image trait by grouping the small blocks of
an image into clusters that are most similar (correlated) with one another. In this way, each
cluster can be described with an efficient representation of its own — separate from those
corresponding to other clusters. Overall, this results in a more efficient image representation
than that afforded by the approaches of the standard compression schemes.
Across-Scale Similarity
If a strong similarity exists between homologous and highly correlated regions of low- and
high-resolution images, then it is foreseeable that a simple transformation can associate these
neighborhoods across scales. Experimental analysis has shown that such similar information
exists locally across scales — a similarity we term scale interdependence. In testing for the
existence of such scale interdependence, experiments were performed. The experiments report
on the percentage of homologous neighborhoods that were similarly clustered from the lowand high-resolution counterpart images that were analyzed. If a high percentage of these
neighborhoods is found, then a strong scale interdependence among neighborhoods is said to
exist. A detailed description and analysis of this simple experiment follows.
In brief, the experiment considers a low- and high-resolution version of an image, xl [n1 , n2 ]
and xh [n1 , n2 ], respectively. The homologous structural neighborhoods of xl [n1 , n2 ] and
xh [n1 , n2 ] are then clustered using vector quantization (VQ) [21] to form K disjoint groups. A
confusion matrix is constructed in which the most likely ordering for the K groups is sought.
Finally, a measure of across-scale similarity is obtained from the percentage of neighborhoods
similarly clustered for the most likely ordering obtained. The definition of a homologous
neighborhood will be stated shortly. Also, the definition of a structural neighborhood, as well
as why these neighborhoods were chosen, will be provided in Section 4.2.3. For now, just note
that a structural neighborhood is an affine mapped version of an image neighborhood.
The H1 × H2 neighborhoods in the N1 × N2 image xl [n1 , n2 ] form the set of neighborhoods
X = {xl [m1 : m1 + H1 − 1, m2 : m2 + H2 − 1]} m
1 =0,...,N1 −H1 ,m2 =0,...,N2 −H2
.
The homologous neighborhoods in the high-resolution image are defined as the G1 H1 × G2 H2
neighborhoods in the (M1 = G1 N1 ) × (M2 = G2 N2 ) image xh [n1 , n2 ], which forms the set
D = {xh [G1 m1 : G1 m1 + G1 H1 − 1,
G2 m2 : G2 m2 + G2 H2 − 1]} m
1 =0,...,N1 −H1 ,m2 =0,...,N2 −H2
.
Recall that xl [n1 , n2 ] is simulated from xh [n1 , n2 ] through decimation by a factor of G1 × G2 .
The manner in which we have simulated the across-scale neighborhoods yields regions of support that encompass the same physical region of the scene — with the low-resolution neighborhood having fewer samples to describe this region. The corresponding image acquisition
model is to be presented.
Now the neighborhoods in X are clustered to form K disjoint groups X1 , . . . , XK and the
neighborhoods in D are separately clustered to form K disjoint groups D1 , . . . , DK . If the
homologous neighborhoods in X and D form similar clusters in their respective images, then
the information content of the low- and high-resolution images must be similar in some sense.
To determine how well clustered information from the same image relates across scales, we
form a confusion matrix as shown in Figure 4.3.
©2001 CRC Press LLC
D1 D2
DK
x1
x2
xK
FIGURE 4.3
Confusion matrix for clustered homologous neighborhoods within their low- and highresolution images. The Xj are the disjoint sets of clustered neighborhoods in the lowresolution image and the Dk are the disjoint sets of clustered neighborhoods in the highresolution image, where j, k = 1, . . . , K.
The entry in location (j, k) of the matrix is the number of neighborhoods assigned to cluster
Xj and Dk , j, k = 1, . . . , K. The interdependence across scales is determined as the maximum
number of homologous neighborhoods common to the clusters formed. Since the ordering
of the true clusters or “classes” between Xj and Dk is not known, we can’t just examine
the contents of the confusion matrix’s diagonal. Instead, we must search for the most likely
ordering. This in turn yields a number that reveals a measure of how similar information in the
low- and high-resolution images was clustered. This number is easily found with the following
simple algorithm:
Step 1: Initialize N = 0.
Step 2: Find the largest number L in the confusion matrix and save its row and column
coordinates (r, c).
Step 3: Perform N ← N + L.
Step 4: Remove row r and column c from the confusion matrix to form a new confusion
matrix with one less row and column.
Step 5: If the confusion matrix has no more rows and columns: STOP else Go to step 2.
The variable N represents the number of homologous neighborhoods common to similar
clusters from the low- and high-resolution images. The percentage of such clustered neighN
borhoods is P = (N1 −H1 +1)(N
since there are a total of (N1 − H1 + 1)(N2 − H2 + 1)
2 −H2 +1)
homologous neighborhoods to cluster in each image.
In Figure 4.4 this percentage is plotted as a function of the number of clusters. The highresolution image clustered is 256 × 256 and is pictured in Figure 4.1c (Lena). Two lowresolution counterpart images have been used. One was obtained through (G1 = 2)×(G2 = 2)
decimation of the high-resolution image (Figure 4.1a). The other image uses a DCT com-
©2001 CRC Press LLC
pressed version of the low-resolution image. The plots also report on two different neighborhood sizes tested: H1 × H2 = 3 × 3 and H1 × H2 = 5 × 5.
Scale Interdependencies for the Lena Image
1
0.9
Interdependency (%)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
ROS: 3x3
ROS: 5x5
0.1
0
5
10
20
15
# of Clusters
25
30
FIGURE 4.4
Scale interdependency plot for the Lena image from Figure 4.1. Two low-resolution
images are used in determining the interdependency: a DCT compressed image (lower
two curves) and a noncompressed one (top two curves). The high-resolution image was
256 × 256 and the corresponding low-resolution images were 128 × 128. Two regions
of support (ROSs) are reported for computing the interdependency: 3 × 3 and 5 ×
5. A very high interdependency among homologous neighborhoods is seen between
the noncompressed low-resolution image and its high-resolution counterpart even when
considering K = 30 clusters.
Figure 4.4 illustrates that there is a very strong interdependence of homologous neighborhoods across image scales when the low-resolution (noncompressed) image of Figure 4.1a is
used — even as the number of clusters increases toward K = 30. This is illustrated by the top
two curves in the plot. The interdependency decreases when the compressed low-resolution
image is used. This is shown by the bottom two curves on the plot. Note that the case of K = 1
always yields an interdependency of 1. This is because for K = 1, no clustering is actually
being performed. That is, all neighborhoods are assumed to belong to the same cluster. As
such, the “disjoint” sets (or single set in this case) have all the homologous neighborhoods in
common.
The interdependency generally decreases as the number of clusters increases. This is intuitively expected because an increase in the number of clusters results in the clustering of
information increasingly specific to a particular image and scale. Because the frequency content between the low- and high-resolution counterpart images differs, the greater specialization
of information within an image is expected to result in less interdependency among them.
©2001 CRC Press LLC
4.2.3
Description of the Methodology
As already alluded to, the superresolution problem is one of determining an appropriate
mapping, which is applied to a set of collected samples in order to yield a “better” reconstructed
image. The manner in which this mapping is determined describes the resulting superresolution
process. The methodology presented herein accomplishes superresolution by exploiting the
aforementioned image traits in order to extract the additional information necessary (i.e.,
beyond the collected samples) to obtain a solution to the ill-posed superresolution problem [2].
Rather than assuming smoothness or relying on other typical constraints, we employ the fact
that a given class of images contains similar information locally and that this similarity holds
across scales. So the fundamental problem is to devise a superresolution scheme that will be
able to determine similarity of local information and capture similarities across scales in an
automated fashion.
Such a superresolution approach necessitates establishing
• which neighborhoods of an image are similar in local structure
• how these neighborhoods relate across scale.
To answer the question of which neighborhoods, the image space of local neighborhoods
will be partitioned. As already alluded to, this is accomplished via a VQ algorithm — for
which many are available. To determine how they relate, each Voronoi cell resulting from
the VQ will be linked to a linear associative memory (LAM) trained to find the best mapping
between the low-resolution neighborhoods in that cluster and their homologous high-resolution
neighborhoods, hence capturing the information across scales. In other words, the assumption
we make is that the information embodied in the codebook vectors and LAMs describes the
relation (mapping) between a low-resolution neighborhood and its high-resolution counterpart.
As such, our approach does not require the assumptions typically needed to obtain a reasonable
solution to the ill-posed superresolution problem.
The LAMs can be viewed as reconstruction kernels that relate the image information across
scales. We choose an adaptive scheme to design the kernels because we know how to design
optimal mappers given a representative set of training images. We further expect that, if the
local regions are small enough, the information will generalize across images. When a new
image is presented, the kernel that best reconstructs each local region is selected automatically
and the reconstruction will appear at the output.
One can expect that this methodology will yield better reconstruction than methods based
on the sampling theory. However, unlike the universal character of the sampling theory,
this superresolution method is specific to the character of images. That is, bases obtained
for one class of images may perform poorly when reconstructing another class. Because of
this, establishing the appropriate models with which to compare our data is important to the
successful superresolution of an image.
4.3
Image Acquisition Model
The image acquisition process is modeled in this section. We use this model to synthesize
a low-resolution counterpart to the original image. With this model, the regions of support of
our low- and high-resolution neighborhoods are homologous (i.e., they encompass the same
physical region of the imaged scene). The model herein was used to obtain the 128 × 128 image
of Figure 4.1a, which was sin c interpolated in that figure. In the superresolution architecture,
©2001 CRC Press LLC
the low-resolution synthesis creates an input from which the information across scales can be
modeled (from the pair of images).
Let the function xa (t, t1 , t2 ) represent a continuous, time-varying image impinging on a
sensor plane. The spatial plane is referenced by the t1 , t2 coordinate axes and time is referenced
by the variable t. The imaging sensor plane is assumed to be a grid of N1 × N2 rectangular
sensor elements. These elements serve to sample the spatial plane within the camera’s field
of view. Each of these elements is said to have physical dimensions p1 × p2 . The output of
each element is proportional to the amount of light that impinges on each sensor during a given
time interval. The output of each sensor, given by xl [n1 , n2 ] where n1 = 0, 1, . . . , N1 − 1 and
n2 = 0, 1, . . . , N2 − 1, can then be expressed as
xl [n1 , n2 ] =
1 p1 (n1 +1) p2 (n2 +1)
p1 n1
0
p2 n2
x(t, t1 , t2 ) dt2 dt1 dt
where the integration over time is one time unit in duration. The subscript l is used to denote
a low-resolution image.
To obtain a higher resolution image, a finer sensor grid encompassing the same field of view
used in obtaining xl [n1 , n2 ] would have to be employed. Let the resolution in each spatial
dimension be increased — by a factor of G1 and G2 in their respective spatial dimensions. The
p1
p2
×G
physical size of the sensor elements now becomes G
units of area. The high-resolution
1
2
image is then given by xh [m1 , m2 ], where m1 = 0, 1, . . . , M1 − 1 and m2 = 0, 1, . . . , M2 − 1,
and Mi = Gi Ni (i = 1, 2). The output for each of the M1 × M2 sensor elements for the
high-resolution image can be described by
xh [m1 , m2 ] =
G1 G2
0
p1 (m1 +1)/G1
p1 m1 /G1
p2 (m2 +1)/G2
p2 m2 /G2
x(t, t1 , t2 ) dt2 dt1 dt
Notice that the integration limits over time have been extended from one time unit to G1 G2
time units in order to maintain the average intensity value for each pixel in the image.
The superresolution process is to estimate the high-resolution image xh [m1 , m2 ] from the
low-resolution image xl [n1 , n2 ]. One can notice that the process of acquiring xl [n1 , n2 ] from
xh [m1 , m2 ] is given by
xl [n1 , n2 ] =
1
G1 G2
G1 (n
1 +1)−1 G2 (n
2 +1)−1
m1 =G1 n1
xh [m1 , m2 ]
(4.4)
m2 =G2 n2
The decimation model in the above equation produces a low-resolution image by averaging
the pixels of G1 × G2 nonoverlapping pixel neighborhoods in the high-resolution image.
4.4
Relating Kernel-Based Approaches
This section introduces kernel-based formalisms for the magnification of images. Conventional approaches to magnification utilize a single kernel and interpolate between samples for
increasing the sample density of an image. The superresolution methodology presented herein
is related to the use of a family of kernels. Each kernel is tailored to specific information of
an image across scales.
©2001 CRC Press LLC
4.4.1
Single Kernel
A magnified image can be obtained by expanding the samples of a low-resolution image
xl [n1 , n2 ] and convolving with a sampled interpolation kernel [22]. For an expansion rate of
G1 × G2 , where G1 , G2 are whole numbers greater than 1, the expanded image is given by

 

 xl n1 , n2 n1 = 0, ±G1 , ±2G1 , . . . 
G1 G2 n = 0, ±G , ±2G , . . .
2
2
2
xe [n1 , n2 ] =
(4.5)




0
otherwise
and the corresponding interpolation kernel, obtained by sampling a continuous kernel, is denoted k[n1 , n2 ]. The interpolated image x̂h [n1 , n2 ] that estimates the true image xh [n1 , n2 ]
is
x̂h [n1 , n2 ] = xe [n1 , n2 ] ∗ ∗k[n1 , n2 ]
(4.6)
where ∗∗ denotes 2D convolution. This form of interpolation is a linear filtering that processes
the image similarly throughout (i.e., it uses the same linear combination of image samples in
determining interpolated points — as does the Shannon sampling theory).
4.4.2
Family of Kernels
Reconstruction with a single kernel is a simple operation since the same function is applied
over and over again to every sample. This is not so when we have at our disposal many kernels.
Two fundamental questions must be answered to reconstruct signals with a family of kernels:
how to choose one member of the family and how to design it. We will formalize these issues
next.
The kernel family approach is a scheme in which the kernel used depends on the local
characteristics of the image [23]. This is formulated as
x̂h [n1 , n2 ] = xe [n1 , n2 ] ∗ ∗kc,l [n1 , n2 ]
(4.7)
The subscripts c and l, which are functions of image location, select a kernel based on the
local image characteristics about the point of interest. The family of kernels is given by
{kc,l [n1 , n2 ] : c = 1, . . . , C; l = 1, . . . , L}. C represents the number of established local
image characteristics (features) from which to compare local neighborhood information and L
is the number of kernels created per feature. In summary, equation (4.7) describes a convolution
with a shift-varying kernel. It is a generalization of equation (4.6) and defaults to the standard
convolution of equation (4.6) when C, L = 1.
4.5
Description of the Superresolution Architecture
Figure 4.5 illustrates the proposed architecture for superresolving images using a family of
kernels. As we proceed, the relation between the architecture and equation (4.7) will be elucidated. The purpose of data clustering is to partition the low-resolution image neighborhoods
into a finite number of clusters where the neighborhoods within each cluster are similar in
some sense. Once the clusters are established, a set of kernels can be developed that optimally
transforms each clustered neighborhood into its corresponding high-resolution neighborhood.
The subsections that follow discuss how the kernel family, implemented here as LAMs (see
Figure 4.5), is established and then used for optical image superresolution.
©2001 CRC Press LLC
LAM 1
LAM 2
Low Resolution
Image
Form C Clusters
Extract Neighborhoods
LAM C
Arrange Superresolved
Neighborhoods
into Image
Superresolved
Image
FIGURE 4.5
Superresolution architecture for the kernel family approach. This paradigm performs
the equivalent operation of a convolution with a family of kernels.
4.5.1
The Training Data
Ideally, the low- and high-resolution data sets used to train the LAMs of Figure 4.5 would
each encompass the same scene and have been physically obtained by hardware with different,
but known, resolution settings. Such data collection is not common. Instead, the low-resolution
counterparts of the given images are obtained via decimation using the image acquisition model
discussed earlier. Once established, the training of the superresolution architecture proceeds
as described in Figure 4.6. Note that the decimation is represented by the ↓ G1 × G2 block
in the figure. The data preprocessing and high-resolution construction sections of this figure
will now be explained.
High Resolution
Neighborhoods
LAM 1
Extract Neighborhoods
High Resolution
Image
LAM 2
G 1 x G2
Extract Neighborhoods
Self Organize the
Neighborhoods to
Form C Clusters
LAM C
Low Resolution
Image
Low Resolution
Neighborhoods
C Neighborhood
Clusters
Association of
Corresponding
Subblocks
FIGURE 4.6
Training architecture for the superresolution of images via the kernel family approach.
4.5.2
Clustering of Data
The neighborhoods considered consist of all the overlapping H1 × H2 neighborhoods of
the low-resolution image xl [n1 , n2 ]. The set of these N = (N1 − H1 + 1)(N2 − H2 + 1)
©2001 CRC Press LLC
neighborhoods in the low-resolution image is given by
X = xl [m1 : m1 + H1 − 1, m2 : m2 + H2 − 1] (4.8)
m1 =0,...,N1 −H1 ,m2 =0,...,N2 −H2
and can be represented by the matrix X ∈ H1 H2 ×N whose columns are the set of vectors
{xr }N
r=1 where xr is a “vectorized” 2D neighborhood. Each low-resolution neighborhood is
paired with its (2G1 −1)×(2G2 −1) homologous high-resolution neighborhood. Specifically,
these high-resolution neighborhoods are described by
xh [G1 m1 + φ1 + 1 : G1 (m1 + 2) + φ1 − 1, (4.9)
S=
G2 m2 + φ2 + 1 : G2 (m2 + 2) + φ2 − 1] m =0,...,N −H ,m =0,...,N −H
1
1
1
2
2
2
Gi (Hi −3)
,
2
where φi =
and i = 1, 2.
Notice that the set of neighborhoods to be clustered here is different from the set used for
arriving at the across-scale similarity measure. The previous set of neighborhoods resulted
from the nonoverlapping neighborhoods in the low- and high-resolution counterpart images.
The set now consists of overlapping neighborhoods. The reason for the overlap is to obtain
multiple estimates of a high-resolution sample. In this way, the final high-resolution sample
can be estimated more reliably.
The neighborhoods in S can be represented by a matrix S ∈ (2G1 −1)(2G2 −1)×N similar to
the representation used in X. These low- and high-resolution neighborhoods are depicted in
Figure 4.7, where the shaded circles represent a low-resolution neighborhood. For the case
of G1 = G2 = 2 in Figure 4.7a, the shaded circles are used to construct the crossed circles
about the center of the low-resolution neighborhood. Note that if we elect not to construct the
center pixel, we will be interpolating locally about the observed image samples. If we elect
to construct the center pixel (along with the other crossed circles), we are allowing for the
ability to change a “noisy” observed sample. Figure 4.7b similarly illustrates this for the case
of G1 = G2 = 3.
In establishing our family of kernels, we have chosen to associate the structure between
the neighborhoods in X and S, not the observed samples themselves. The structure of a
neighborhood is defined as the neighborhood with its mean subtracted out; each neighborhood
thus becomes a vector whose component mean is zero. This kind of preprocessing allows us to
categorize neighborhoods sharing a particular characteristic (i.e., they could be smooth, edgy at
a particular orientation, etc.) as belonging to the same class regardless of the average intensity
of the neighborhood. The structure pr of neighborhood xr is obtained through multiplication
with the square matrix Z ∈ H1 H2 ×H1 H2 (i.e., pr = Zxr for a single neighborhood or P = ZX
for all the input neighborhoods), where


−1
...
−1
H1 H2 − 1


..

1 
H 1 H2 − 1
.
 −1
 .
(4.10)
Z=


.
..
H1 H2 
..
.
−1 
−1
...
−1 H1 H2 − 1
The desired exemplars associated with P are contained in matrix D. Each column in D is
obtained by subtracting the mean of xr from its corresponding neighborhood sr in S. This
is done to compensate for the low-resolution neighborhood mean, which has been subtracted
from xr and must be added back after the high-resolution neighborhood structure is created.
Specifically, D = S−AX, where A ∈ (2G1 −1)(2G2 −1)×H1 H2 is a constant matrix with elements
1
H1 H2 .
The clusters are formed by performing a VQ on the space of structural neighborhoods in P.
This clustering is based on the interblock correlation among the neighborhoods in P [1]. The
©2001 CRC Press LLC
1
2
3
4
5
6
7
8
9
(a)
(b)
FIGURE 4.7
Local image neighborhoods and the pixels they superresolve. Each circle represents a
2D high-resolution image pixel. The shaded circles are the low-resolution image pixels
obtained via decimation of the high-resolution image. The gray pixel is the center of the
low-resolution neighborhood. Each H1 × H2 low-resolution neighborhood constructs a
(2G1 − 1) × (2G2 − 1) high-resolution neighborhood about the low-resolution neighborhood’s center — these are depicted by the crossed circles. The numbers are a convention
used to distinguish between constructed pixels in this neighborhood. (a) Decimation
factor G1 = G2 = 2; (b) decimation factor G1 = G2 = 3.
VQ is accomplished using Kohonen’s self-organizing map [24] for reasons discussed later.
The VQ operation results in a set of C feature vectors {fc }C
c=1 , where usually C << N . The C
clusters Kc , c = 1, 2, . . . , C, formed by our neighborhood, and feature vectors are given by
(4.11)
Kc = pr : pr − fc 2 < pr − fb 2 ; b = 1, 2, . . . , C; b = c; r = 1, 2, . . . , N
4.5.3
Neighborhood Association
The superresolution methodology herein is piecewise local in nature — inherent to the fact
that we consider neighborhoods. A mapping is required to produce high-resolution image
samples from the low-resolution ones that are available. This mapping could be linear (or
affine) or nonlinear. A description of how these mappings have been implemented within the
methodology now follows.
The input–output relationship of a LAM [21] is an affine transformation described by
yr = Wpr + b
(4.12)
where W is a weight matrix that specifies the network connectivity of the LAM, b is a bias
vector, and pr is the input vector (neighborhood structure). Note that yr contains a vector
representation of a superresolved 2D neighborhood structure. The neighborhoods in P and D
are associated in the least square sense to determine the values of the W and b parameters.
These parameters can be obtained recursively via the least mean squares (LMS) algorithm
update equation [21]
W(n + 1) = W(n) + µ(D − Y)PT
(4.13)
where T denotes matrix transposition and µ is the learning rate. They can equivalently be
©2001 CRC Press LLC
obtained in closed form via the pseudo-inverse [21]
W = DPT (PPT )−1 .
(4.14)
We have assumed in equations (4.13) and (4.14) that W is actually the augmented matrix [W|b]
and P is the augmented matrix [PT |v]T , where v is a column vector of ones of appropriate
dimensions.
Nonlinear associative memories (NLAMs) can be used as a substitute for the LAMs of
Figures 4.5 and 4.6. The parameterized nonlinear relation between the input and output can
be achieved using a multilayer perceptron (MLP) [21] and is given by
yr = α ({Wk } , {bk } , pr )
(4.15)
where, in general, α( · ) is a nonlinear function of a set of weight matrices, bias vectors, and
the neighborhood structure, and k describes a layer in the MLP feed-forward configuration.
The NLAM parameters are readily obtained with back-propagation learning [21]. It is well
established as a supervised training method for neural networks. This method generalizes the
LMS training algorithm for linear networks to MLPs. The on-line weight update is similar to
that of the LMS algorithm. At each time step the weight matrix Wk , for layer k of the MLP,
is updated as follows
T
Wk (n + 1) = Wk (n) + µgk (n)yr,k−1
(4.16)
where gk (n) is the local gradient and yr,k−1 is the postneural activity of the previous layer
(hence the k − 1) due to input vector xr . Please note here that the subscript k describes the
weight layer of a feed-forward NLAM with several layers. We could describe the kth layer of
the cth NLAM by Wc,k . The postneural activity of an NLAM at a given layer is recursively
defined as yr,k = Wk (n)yr,k−1 . This is because the output of one layer serves as the input to
the next layer in the feed-forward configuration. Note that the postneural activity for layer 0
at time step n is defined as just the input vector at that time step (i.e., yr,0 ≡ xr ).
If layer k is the output layer, then
gk (n) = dr − yr,k • ϕ yr,k
(4.17)
where • represents the element-by-element multiplication of two matrices (or vectors) and
ϕ ( · ) is the first derivative of ϕ( · ), a differentiable squashing function. If layer k is other than
the output layer, then
!
"
T
(4.18)
(n)gk+1 (n) • ϕ yr,k .
gk (n) = Wk+1
There are C NLAMs to be trained. Each corresponds to a particular cluster of the input data.
NLAM c associates the neighborhoods xr ∈ Kc with its corresponding samples in dr .
4.5.4
Superresolving Images
The construction of a high-resolution image, as depicted in Figure 4.5, results from transforming the neighborhood structure of the low-resolution input image with the parameters
obtained in the training phase. The mean of the neighborhood is subsequently added back to
the transformation. When LAMs are used, the superresolution of a low-resolution neighborhood xr can be expressed as
ŝr = Wc Zxr + bc + Axr
©2001 CRC Press LLC
for
(pr = Zxr ) ∈ Kc
(4.19)
where Wc and bc are the weight matrix and bias vector, respectively, associated with the cth
LAM. As discussed before, there is a direct relation between equation (4.19) and equation (4.7).
Equation (4.19) constructs the high-resolution neighborhoods’ structure ŝr . The subscript r
refers to the neighborhood being constructed. The constructed pixels that overlap are averaged
and the high-resolution image is thus constructed. Averaging several high-resolution samples
improves the reliability of the final high-resolution sample. Equation (4.19) can be equivalently
expressed as
x̂h [n1 , n2 ] =
L
#
$
xe [n1 , n2 ] ∗ ∗ kc,l [n1 , n2 ] + a[n1 , n2 ] · b[n1 , n2 ]
(4.20)
l=1
where L = (2G1 − 1)(2G2 − 1); xe is the expanded low-resolution image; the kernel was
created with the values Wc Z(l, :) and bc (l) (i.e., row l of Wc Z and bc ); a is a constant kernel
with the same extent as kc,l , that averages a low-resolution neighborhood (its impulse response
samples equal H11H2 ); and b[n1 , n2 ] = b1 [n1 ]b2 [n2 ] is responsible for averaging multiple
estimates of superresolved samples. Specifically,
1 ni mod Gi = 0
bi [ni ] =
(4.21)
1
2 otherwise
for i = 1, 2. Notice that the index l refers to a specific convolution pass that is constructing the
corresponding enumerated crossed circle associated with each low-resolution neighborhood
in that pass. Please refer to Figure 4.7a for the case of G1 = G2 = 2.
The NLAM case differs only by the presence of the nested nonlinearities. The construction,
for an M layer MLP topology, is expressed as
ŝr = ϕ Wc,M ϕ . . . ϕ Wc,l Zxr + bc,l + bc,M + Axr for (pr = Zxr ) ∈ Kc (4.22)
where Wc,k and bc,k are the weight matrix and bias vector, respectively, at layer k of the cth
NLAM, and ϕ denotes the squashing function at each layer of the feed-forward structure.
4.6
Results
The results illustrated in this section make use of the Peppers image for training and the Lena
image for testing. The Lena image has already been illustrated; the Peppers image can be found
in several references (e.g., [1, 23]). The LAM-based results were compared against several
kernel-based interpolation results including the subpixel edge localization and interpolation
(SEL) technique [6], which fits an ideal step edge through those image regions where an edge
is deemed to exist and otherwise uses a bilinear interpolation. The parameters for the SEL
technique were the same as those reported in [6].
Table 4.1 reports on the peak signal-to-noise ratio (PSNR) resulting from kernel-based
interpolation of the Lena and Peppers 128 × 128 images by a factor of 2 in each dimension.
2 ) where
The PSNR is defined as PSNR ≡ −10 log10 (erms
2
erms
=
M
1 −1 M
2 −1
2
1
xh [m1 , m2 ] − x̂h [m1 , m2 ]
M1 M2
m1 =0 m2 =0
and xh and x̂h take values in [0, 1].
©2001 CRC Press LLC
(4.23)
Table 4.1 PSNR for Magnified Images
Zero Bilinear Bicubic
Cubic
SEL Traina Testa Trainb
Testb
Order
B-Spline
Lena
27.00
27.26
27.45
27.43
27.48 32.63 31.78 32.23
31.71
Peppers 27.48
27.51
27.74
27.74
27.79 34.58 32.90 34.03
32.74
a Used 30 LAMs.
b Used 30 NLAMs.
Note: The interpolation factor was 2 in each image axis from the listed 128 × 128 images. The
training and test cases of the kernel family approach utilized 30 features and a 3 × 3 region of
support (ROS). In the test cases, the parameters obtained in training to reconstruct Lena were
used for the Peppers and vice versa.
The plot in Figure 4.8 illustrates the PSNR when superresolving the Lena 128 × 128 image
with varying numbers of LAMs by a factor of 2 in each dimension. The system parameters
(feature vectors, weights, and biases) were trained using the Peppers 256 × 256 image (i.e.,
a different image). The solid and dashed lines in Figure 4.8 denote training and test set
reconstruction performance, respectively, using regions of support (ROSs) 3 × 3 and 5 × 5.
In general, the PSNR of the training set increased as the number of LAMs increased. This is
intuitively expected because an increase in the number of LAMs yields a greater specialization
to particular image features, hence a more accurate image reconstruction. The feature set
extracted using a 5 × 5 ROS yields more macroscopic image characteristics than does a 3 ×
3 ROS. This results in greater specialization of the characteristics particular to the image of
interest and generally to a more faithful image reconstruction on the training set.
In the test set, however, the larger ROS tended to show a drop in PSNR performance as the
degree of specialization to image features increased. This general trend was encountered in all
the tests we have run. It suggests that the similarity between features, as the system specializes
more (uses more LAMs), tends to occur at a more microscopic level. It can also be observed
that the kernel family approach yielded higher PSNR than those methods listed in Table 4.1.
A visual comparison of the results, utilizing the common approaches and the kernel family
approach for the Lena image, can be observed in Figure 4.9. The training and testing images
shown in each of these figures were created using 30 LAMs and an ROS of 3 × 3. They
correspond to those points in Figure 4.8 marked by a circle. In Figure 4.10 we see the 30 features
extracted from the Peppers 128 × 128 image that were used in reconstructing the test image of
Figure 4.9. Notice how “regular” and edgy these features are. The features extracted are image
dependent, and we would expect a different set to result from texture images, for example. The
number below each feature signifies the maximum gray-level difference between the largest
and smallest value present in each feature. Therefore, the feature with the “1” below it can be
considered a constant feature, that is, one that contains practically no structure. This feature
corresponds to those image portions that are very smooth. The features have not been scaled
here; instead, the constant feature (which is the zero vector) is represented by gray (128 in an
8-bit scale). Positive feature values become lighter and negative feature values are represented
by a proportionately darker shade of gray.
The superresolved training and testing images of Figure 4.9 were of similar quality. The kernel family superresolved images appear crisper than those obtained with the other approaches
presented here. In Figure 4.11 we have shown the magnitude spectra of the reconstructed images corresponding to Figure 4.9. The spectra here are for the full reconstructed image, not just
the portion shown in the figure. It is evident from viewing these spectra that the LAM-based
approach is recovering information above half the sampling frequency and reproducing better
the high-frequency information characteristic of the original images. The SEL approach is
©2001 CRC Press LLC
Lena Superresolution Results
33.5
ROS:5x5
33
PSNR (dB)
ROS:3x3
32.5
32
ROS:3x3
ROS:5x5
31.5
31
0
5
10
15
# of LAMs
20
25
30
FIGURE 4.8
Training and testing superresolution results for the Lena image considering two different
regions of support (ROSs). The solid lines correspond to training set results and the
dashed lines are test set results. The curves related to the training data result from
superresolving the Lena 128 × 128 image with the systems (features and LAMs) trained
to reconstruct the Lena 256 × 256 image from the Lena 128 × 128 image. The curves
related to testing result from superresolving the Lena 128 × 128 image with the systems
trained to reconstruct the Peppers 256 × 256 image from the Peppers 128 × 128 image.
Superresolved images corresponding to the two circled points are shown in Figure 4.9.
also able to reproduce high-frequency information. This is because, as mentioned earlier, the
SEL approach fits an ideal step edge through those image regions it deems are edges. The low
PSNR of the SEL approach can be attributed to its performance in smoothly varying image
regions. This is because, in these regions, the SEL approach uses the bilinear kernel for its
interpolation. In summary, the superresolved images of this work generally appear more crisp
than those obtained with the other approaches presented here. Edges seemed to be preserved
well with our approach, and the higher PSNR obtained with our methodology is evidence of the
accuracy of reconstruction in smoothly varying image regions relative to the other approaches
reported herein.
To test our hypothesis that the system captures well redundancy across scales, we illustrate in
Figure 4.12 the superresolution of the Lena 128 × 128 image using two successive G1 = G2 =
2 reconstruction stages with the same codebook and LAMs. In other words, the resulting “test”
image of Figure 4.9 is fed through the system of Figure 4.5 twice with the same parameters
used in the first superresolution stage for a total superresolution factor of 4 in each dimension.
Notice that the LAM reconstructed image is crisper than the other expanded images. This
also supports our claim regarding the similarity of image neighborhoods across scales —
which we exploit for superresolution. Figure 4.13 illustrates a case of what can happen when
inappropriate sets of bases are used for the image reconstruction. In this figure, the Lena 128
©2001 CRC Press LLC
©2001 CRC Press LLC
FIGURE 4.9
Visual comparison of the reconstruction results for the Lena* image. The 128 × 128 image was reconstructed to 256 × 256. A zoomed section (using
nearest neighbor replication) of the reconstructed results is displayed. The “training” reconstruction utilized the 30 features and corresponding
LAMs obtained in training to reconstruct the Lena 256 × 256 image from the Lena 128 × 128 image with an ROS of 3 × 3. The “testing”
reconstruction utilized the 30 features and corresponding LAMs obtained in training to reconstruct the Peppers 256 × 256 image from the
Peppers 128 × 128 image with an ROS of 3 × 3. (* Copyright © 1972 by Playboy magazine. Reproduced by special permission . )
FIGURE 4.10
Features extracted from the Peppers 128 × 128 image. They were used in superresolving
the Lena 128 × 128 image to a size of 256 × 256; these results are given in Figure 4.9.
Notice the largely edgy nature of these features. The features have not been scaled
here; instead, the constant feature is represented by gray (128 in an 8-bit scale). The
number below each feature represents the maximum 8-bit gray-level difference between
the largest and smallest value of that feature.
× 128 image is being reconstructed to 256 × 256. The desired image is given in Figure 4.13a.
The image reconstructed from the 30 features and LAMs obtained in training to reconstruct
the Peppers 256 × 256 image from the Peppers 128 × 128 image is given in Figure 4.13b, and
the image reconstructed from the 30 features and LAMs obtained in training to reconstruct the
Pentagon 256 × 256 image from the Pentagon 128 × 128 image is given in Figure 4.13c. The
Pentagon images have not been pictured here. The systems of the Peppers are appropriate for
the reconstruction of Lena. However, the systems of the Pentagon are not as appropriate. This
is seen particularly by the reconstruction performance about the right portion of the forehead
and hat in Figure 4.13c. Incorporating the correct a priori information into the reconstruction
process can be beneficial to superresolution, but introducing the wrong information can have
the opposite effect. In Figure 4.13d we compensate for the lack of proper bases by incorporating
the appropriate bases obtained from the Peppers image used in reconstructing Figure 4.13b.
The appropriate bases were simply “appended” to the inappropriate set from the Pentagon that
was used in this example. In this manner, we did not have to retrain a system from scratch in
order to produce an appropriately reconstructed image. Available bases for reconstruction can
simply be incorporated into an existing system to produce adequately reconstructed images.
This is possible because of the hard partitioning scheme our procedure is implementing.
Figure 4.14 illustrates results when NLAMs are used in place of the LAMs. Again, we
superresolve the Lena 128 × 128 image using 30 LAMs and an ROS of 3 × 3. The NLAMs
used a single hidden layer and had approximately the same number of free parameters as did
the LAMs. The LAM and NLAM results are very similar (both visually and in PSNR). In
this and several other tests we have run with superresolution factors of 2 and 3, the added
complexity and flexibility afforded by the NLAMs seems unwarranted. This makes sense
since many nonlinear mappings are reasonably well approximated locally by linear models.
Figure 4.15 illustrates the superresolution of a DCT-compressed Lena image from 256 × 256
to 512 × 512. The parameters used in our local architecture were those trained to superresolve
the compressed Peppers 256 × 256 image to the original peppers 512 × 512 image. The
system used 15 LAMs and an ROS of 5 × 5. The compressed images were obtained by inverse
©2001 CRC Press LLC
©2001 CRC Press LLC
FIGURE 4.11
Magnitude spectra of the reconstructed Lena images in Figure 4.9. The spectra here are for the corresponding full reconstructed images, not just the
zoomed sections pictured in Figure 4.9. The spectra F [n1 , n2 ] of each image have been enhanced via the log scaling log (|F2 [0,0]|) log10 (|F [n1 , n2 ]|).
10
The LAM and SEL approaches are better able to produce higher frequency information relative to the other methods compared.
©2001 CRC Press LLC
FIGURE 4.12
Example of reconstruction of Lena* 128 × 128 image by a factor of 16 (a factor of 4 along each image axis). The reconstruction was accomplished
using two successive stages of reconstruction, each by a factor of 4. The same features and LAMs were used in each stage. The training image
was reconstructed using the features and LAMs trained to reconstruct the Lena 256 × 256 image from the Lena 128 × 128 image with an ROS
of 3 × 3. The test image was reconstructed using the features and LAMs trained to reconstruct the Peppers 256 × 256 image from the Peppers
128 × 128 image with an ROS of 3 × 3. (* Copyright 1972 by Playboy magazine. Reproduced by special permission.)
FIGURE 4.13
Compensating for the effects of reconstruction with “inappropriate” bases. The results
displayed show a portion of the Lena* 256 × 256 image reconstructed from the Lena 128
× 128 image. (a) Original. (b) Reconstructed with the 30 features and LAMs used in
reconstructing the Peppers 256 × 256 image from the Peppers 128 × 128 image; this
yielded a good reconstruction. (c) Reconstructed with the 30 features and LAMs used
in reconstructing the Pentagon 256 × 256 image from the Pentagon 128 × 128 image.
The inappropriate reconstruction is most noticeable in the right portion of the forehead
and on portions of the hat. (d) Reconstructed with 60 features and LAMs: 30 from the
Pentagon image used in (c) and 30 from the Peppers image used in (b). We did not have
to retrain our system in establishing an appropriate set of bases. We simply “append”
the appropriate features and LAMs of the Peppers image to the existing set from the
Pentagon image to reconstruct an adequate image(*Copyright 1972 by Playboy magazine).
transforming each nonoverlapping 8 × 8 subblock of the original 256 × 256 images with only
the 3 × 3 low-frequency DCT coefficients — the others were set to zero. The compression
results in the loss of information within the borders of each subblock and the introduction
of edge artifacts along the borders of the compressed subblocks. Our system was able to
substantially suppress the blocking artifacts in the superresolved image with no explicit prior
©2001 CRC Press LLC
FIGURE 4.14
Comparing the reconstruction of Lena* using LAMs and NLAMs. The overall PSNR
performance of the LAM- and NLAM-based results were very similar. Here the Lena
128 × 128 image is reconstructed by a factor of two in each image axis. The 30 features
and LAMs (NLAMs, respectively) used in the reconstruction were those obtained from
training to reconstruct the Peppers 256 × 256 image from the Peppers 128 × 128 image
with an ROS of 3 × 3. (*Copyright 1972 by Playboy magazine. With permission.)
knowledge of the existence or location of artifacts. This suppression of artifacts is obviously
not possible with any of the kernel-based interpolation approaches.
Recall that the scale interdependence between the compressed image and its uncompressed
counterpart is significantly reduced relative to using a noncompressed low-resolution image.
This reduction in scale interdependence reduces the reliability of the multiple estimates obtained for a high-resolution sample. This limits the extent to which our superresolution can
produce a sharp image. However, the averaging of multiple estimates by considering overlapping neighborhoods in the superresolution architecture is responsible for filtering out the
effects of blockiness. We can notice from Figures 4.9 and 4.12 that if strong scale interdependencies exist, then our multiple estimates of an image sample are relatively reliable and
their averaging does not result in discernible low-pass filtering. This results in crisper images
compared to the kernel-based techniques.
4.7
Issues and Notes
Although the preliminary results are very promising, there are many issues requiring further
analysis. Noteworthy issues pertaining to the superresolution process herein are:
• The feature vectors and LAMs are established in a manner that is not driven directly by
the error rate of superresolution. This is potentially suboptimal. However, because the
function defining our input space partition (the clustering stage) is not differentiable,
this issue is not easily addressed. We have tested our approach using the hierarchical
mixture of experts [25], which trains to minimize the error rate [affine experts (LAMs)
©2001 CRC Press LLC
FIGURE 4.15
Results of superresolution on a compressed image with visible blocking artifacts. (a) 128
× 128 portion of compressed image to magnify. It is shown here as a 256 × 256 image
by using zero-order hold interpolation. (b) Cubic B-spline interpolated result. (c) Superresolution using 15 LAMs and an ROS of 5 × 5. The architecture was trained to
reconstruct the original Peppers 512 × 512 image with its 256 × 256 down-sampled and
DCT-compressed image. (Photo: Copyright 1972 by Playboy magazine. )
and affine transformations for the gating structure were used], and our method trained
faster and consistently produced higher PSNRs in the reconstructed images [1].
• The topological mapping property of Kohonen’s self-organizing map (SOM) was not
used for the results presented here. We used the SOM because of its efficient training approach and its tendency for full codebook utilization. We have performed the
clustering with the Neural Gas algorithm [26] and have not noticed performance differences [1]. The incorporation of the topological information of the SOM to improve the
superresolution is the subject of future research.
• Nonlinear associative memories showed no improvement with respect to LAM performance for the parameters utilized in these experiments. Since the neighborhood sizes
and the superresolution factors were small, a linear mapper seems to capture well the
©2001 CRC Press LLC
redundancy across scales. However, for larger superresolution factors the mapping will
tend to be more and more nonlinear, so NLAMs may yield a performance advantage.
• The superresolution approach herein also allows for noninteger (rational) magnification
factors. The size of the images associated (as well as what local samples are to be
constructed) determines this factor for the feature vectors and LAMs established. Thus
different feature and LAM sets must be established for different magnification factors.
• The low-resolution neighborhood size used is a trade-off between the amount of local
support considered and how much information is to be constructed. As a rule of thumb,
we suggest setting Hi ≥ 2Gi − 1 but keeping Hi (i = 1, 2) reasonably small. The
number of free parameters is determined by the low-resolution support specified by Hi .
Note that as Gi increases, there is more missing information to construct; hence, more
low-resolution sample support is needed.
• The results presented here use a single image for training — the Peppers. However, multiple images can easily be (and have been [23]) used for training the system parameters
in Figure 4.5. Our experiments have revealed that there is much similar local structure
among images, which might not be apparent when images are casually viewed.
• The number of input vectors should be much larger than the dimensionality of the input
space for proper LAM training. This results in the solution of an overdetermined problem
rather than an underdetermined one.
• The system in Figure 4.5 lends itself to the incorporation of new or additional features
(and LAMs) and does not require retraining of the existing parameters. This allows for
quick amending of the bases used for reconstruction.
• The methodology presented for superresolution is general and has been used in the
superresolution of synthetic aperture radar (SAR) imagery [1, 27]. Due to the nature
of these signals, the processing accounts for local information in the frequency domain
— which necessarily implies the learning of nonlocal basis functions upon which our
collected samples are projected. This is in contrast to the local processing performed in
the spatial domain of the optical images in this chapter.
4.8
Conclusions
A local architecture has been presented for the superresolution of optical images. The procedure was shown to be equivalent to convolution of the image with a family of kernels developed
from a training image. The ill-posed superresolution problem was addressed by determining
locally the optimal least-squares projections across scales for image neighborhoods of similar
character. The similarity between neighborhoods was characterized by their interblock correlation. The key assumption of our approach was that this similarity of neighborhoods in the
low-resolution image also held across scales — an assumption that we’ve noticed experimentally to be very reasonable. The use of LAMs for the local transformation is interesting in
that the relation between correlated neighborhoods’ structure across scales seems reasonably
modeled by an affine mapping. This simplifies the training and eases the need for establishing
more complicated nonlinear transformations.
Several interesting traits were demonstrated which favor the use of this architecture. These
include: the real-time implementation of the architecture due to its highly parallel nature, the
©2001 CRC Press LLC
incorporation of new bases into the reconstruction without having to retrain the system, and
the inherent ability to regulate errors made in the reconstruction through smoothing. This last
trait results from considering overlapping blocks from which multiple sample estimates can be
averaged if they are not reliable — this reduces the possibility of introducing artifacts into the
image. This bodes well when superresolving images exhibit “blockiness” due to compression.
Finally, the need for an analysis that mathematically supports the assumptions we’ve observed
to be reasonable is warranted and has been left for future research.
References
[1] F.M. Candocia, “A Unified Superresolution Approach for Optical and Synthetic Aperture
Radar Images,” Ph.D. dissertation, University of Florida, Gainesville, 1998.
[2] A.M. Tekalp, Digital Video Processing, Ch. 17, Upper Saddle River, NJ: Prentice-Hall,
1995.
[3] A.N. Netravali and B.G. Haskell, Digital Pictures: Representation, Compression and
Standards, 2nd ed., New York: Plenum Press, 1995.
[4] M. Unser, A. Aldroubi, and M. Eden, “Fast B-Spline Transforms for Continuous Image
Representation and Interpolation,” IEEE Trans. Pattern Anal. Mach. Int., vol. 13, no. 3,
pp. 277–285, 1991.
[5] S.D. Bayrakeri and R.M. Mersereau, “A New Method for Directional Image Interpolation,” Proc. Int. Conf. Acoustics, Speech, Sig. Process., vol. 4, pp. 2383–2386, 1995.
[6] K. Jensen and D. Anastassiou, “Subpixel Edge Localization and the Interpolation of Still
Images,” IEEE Trans. Image Process., vol. 4, no. 3, pp. 285–295, 1995.
[7] A.M. Darwish and M.S. Bedair, “An Adaptive Resampling Algorithm for Image Zooming,” Proc. SPIE, vol. 2666, pp. 131–144, 1996.
[8] S.A. Martucci, “Image Resizing in the Discrete Cosine Transform Domain,” Proc. Int.
Conf. Image Process., vol. 2, pp. 244–247, 1995.
[9] E. Shinbori and M. Takagi, “High Quality Image Magnification Applying the GerchbergPapoulis Iterative Algorithm with DCT,” Systems and Computers in Japan, vol. 25, no. 6,
pp. 80–90, 1994.
[10] S.G. Chang, Z. Cvetkovic, and M. Vetterli, “Resolution Enhancement of Images Using
Wavelet Transform Extrema Extrapolation,” Proc. Int. Conf. Acoustics, Speech, Sig.
Process., vol. 4, pp. 2379–2382, 1995.
[11] N.B. Karayiannis and A.N. Venetsanopoulos, “Image Interpolation Based on Variational
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[12] R.R. Schultz and R.L. Stevenson, “A Bayesian Approach to Image Expansion for Improved Definition,” IEEE Trans. Image Process., vol. 3, no. 3, pp. 233–242, 1994.
[13] G.K. Wallace, “The JPEG Still Image Compression Standard,” Commun. ACM, vol. 34,
no. 4, pp. 30–44, 1991.
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[14] R.D. Dony and S. Haykin, “Neural Network Approaches to Image Compression,” Proc.
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Proc., vol. 4, no. 10, pp. 1358–1370, 1995.
[16] N. Kambhatla and T. Leen, “Dimension Reduction by Local Principal Component Analysis,” Neural Computation, vol. 9, pp. 1493–1516, 1997.
[17] R.J. Marks, Introduction to Shannon Sampling and Interpolation Theory, New York:
Springer-Verlag, 1991.
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[19] P.M. Woodward, Probability and Information Theory, with Applications to Radar, 2nd
ed., Oxford, NY: Pergamon Press, 1964.
[20] D.L. Ruderman and W. Bialek, “Seeing Beyond the Nyquist Limit,” Neural Computation,
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1994.
[22] R.W. Schafer and L.R. Rabiner, “A Digital Signal Processing Approach to Signal Interpolation,” Proc. IEEE, vol. 61, no. 6, pp. 692–702, 1973.
[23] F.M. Candocia and J.C. Principe, “A Neural Implementation of Interpolation with a
Family of Kernels,” Proc. Int. Conf. Neural Networks, vol. 3, pp. 1506–1510, 1997.
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©2001 CRC Press LLC
Chapter 5
Image Processing Techniques for Multimedia
Processing
N. Herodotou, K.N. Plataniotis, and A.N. Venetsanopoulos
5.1
Introduction
Multimedia data processing refers to a combined processing of multiple data streams of
various types. Recent advances in hardware, software, and digital signal processing allow for
the integration of different data streams which may include voice, digital video, graphics, and
text within a single platform. A simple example may be the simultaneous use of audio, video,
and closed-caption data for content-based searching and browsing of multimedia databases or
the merging of vector graphics, text, and digital video. This rapid development is the driving
force behind the convergence of the computing, telecommunications, broadcast, and entertainment technologies. The field is developing rapidly and emerging multimedia applications,
such as intelligent visual search engines, multimedia databases, Internet/mobile audiovisual
communication, and desktop video conferencing will all have a profound impact on modern
professional life, health care, education, and entertainment.
The full development and consumer acceptance of multimedia will create a host of new
products and services including new business opportunities for innovative companies. However, in order for these possibilities to be realized, a number of technological problems must
be considered. Some of these include, but are not limited to, the following:
1. Novel methods to process multimedia signals in order to meet quality of service requirements must be developed. In the majority of multimedia applications, the devices
used to capture and display information vary considerably. Data acquired by optical,
electro-optical, or electronic means are likely to be degraded by the sensing environment. For example, a typical photograph may have excessive film grain noise, suffer
from various types of blurring (motion or focus blur), or have unnatural shifts in hue,
saturation, or brightness. Noise introduced by the recording media degrades the quality
of the resulting images. It is anticipated that the use of digital processing techniques,
such as filtering and signal enhancement, will improve the performance of the system.
2. Efficient compression and coding of multimedia signals, in particular, visual signals
with an emphasis on negotiable quality of service contracts, must be considered. Rich
©2001 CRC Press LLC
data types such as digital images and video signals have enormous storage and bandwidth requirements. Techniques that allow images to be stored and transmitted in more
compact formats are of great importance. Multimedia applications are putting higher
demands on both the achieved image quality and compression ratios.
Quality is the primary consideration in applications such as DVD drives, interactive
HDTV, and digital libraries. Existing techniques achieve compression ratios from 10:1
to 15:1, while maintaining reasonable image quality. However, higher compression
ratios can reduce the high cost of storage and transmission and also lead to the advent
of new applications (i.e., future display terminals with photo-quality resolution, or the
simultaneous broadcast of a larger number of visual programs).
3. Innovative techniques for indexing and searching multimedia data must be developed.
Multimedia information is difficult to handle in terms of both its size and the scarcity
of tools available for navigation and retrieval. A key problem is the effective representation of this data in an environment in which users from different backgrounds can
retrieve and handle information without specialized training. Unlike alphanumeric data,
multimedia information does not have any semantic structure. Thus, conventional information management systems cannot be directly used to manage multimedia data.
Content-based approaches seem to be a natural choice where audio information along
with visual indices of color, shape, and motion are more appropriate descriptions. A
set of effective quality measures are also necessary in order to measure the success of
different techniques and algorithms.
In each of these areas, a great deal of progress has been made in the past few years, driven
in part by the availability of increased computing power and the introduction of new standards
for multimedia services. For example, the emergence of the MPEG-7 multimedia standard
demands an increased level of intelligence that will allow the efficient processing of raw
information; recognition of dominant features; extraction of objects of interest; and the interpretation and interaction of multimedia data. Thus, effective multimedia signal processing
techniques can offer promising solutions in all of the aforementioned areas.
This chapter focuses on the intelligent processing of visual information within the research
domain of multimedia signal processing using color image processing techniques in conjunction with fuzzy concepts. More specifically, the framework presented includes filtering,
segmentation, and meta-data concepts using adaptive techniques for a number of application
areas. The organization of the chapter is as follows. Section 5.2 reviews some of the key issues
of color imaging with an emphasis on the models needed to support the efficient representation
of color information among various devices in a multimedia system. Section 5.3 focuses on
the problem of color image filtering for the improvement and enhancement of image quality.
New filtering schemes are introduced to meet the challenging high quality of standards necessary in the multimedia era. Color image processing applications demand digital filters that are
suitable for complex nonlinear problems, have a reduced complexity, are numerically robust,
and are computationally attractive. In Section 5.4, the problem of color image segmentation
is addressed for the purposes of audiovisual coding in object-based compression schemes.
The segmented regions can be used to form a nonuniform mesh structure which allows for a
more accurate motion estimation and compensation in contrast to the conventional block-based
methods. Section 5.5 explores the application of color segmentation and fuzzy analysis for
the automatic localization of the facial region in an image or video sequence. The extraction
process can be utilized for a more efficient coding or for indexing and retrieval in multimedia
databases. Lastly, some open technical issues and promising application trends are suggested
in the concluding section.
©2001 CRC Press LLC
5.2
Color in Multimedia Processing
Color is a key feature used to understand and recollect the contents within a scene. It
is found to be a highly reliable attribute for image retrieval because it is generally invariant
to translation, rotation, and scale changes [1]. Several color coordinate systems have come
into existence for establishing a numerical description of color. The representation of color
is based on the classical three-color theory whereby any color can be reproduced by mixing
an appropriate set of three primary colors [2]. In this way, the numerical representation of a
particular color can be specified by its three component vectors within the 3D color coordinate
system. The set of all colors form a vector space called the color space or color model.
Color information is commonly represented in the widely used RGB (red, green, blue)
Cartesian coordinate system. This basis is hardware oriented and is suitable for acquisition or
display devices but not particularly applicable in describing the perception of colors. In this
coordinate space, the RGB primaries are additive in that the individual contributions of each
primary are added to form the overall result. The YIQ (Y is the luminance and I and Q are the
chrominance components) and CMYK color models are also hardware-based systems and are
utilized for different application purposes. The former is used in color television broadcasting
and is a recoding of the RGB components for transmission efficiency and downward compatibility with the earlier monochrome TV standards. The CMYK color space, on the other hand,
is important in dealing with printing devices where subtractive primaries are relevant. Colors
are specified in this latter model by what is removed or subtracted from white light, rather than
by what is added to black.
The need to formulate a simple yet accurate perceptual color distance prompted the development of a perceptually uniform color space [3]. The Commission Internationale de l’Eclairage
(CIE) standardized the perceptually uniform L∗ u∗ v∗ and L∗ a∗ b∗ coordinate systems, which are
derived by a nonlinear transformation of the RGB values. These color models define a uniform
metric space representation of color so that a perceptual color difference is represented by the
Euclidean distance. The L∗ a∗ b∗ cube-root color coordinate system was essentially developed
to provide a quantitative expression for the Munsell system of color classification [4]. The
following transformation equations can be used to convert a set of RGB vector values to the
L∗ a∗ b∗ space
  
 
X
0.490 0.310 0.200
R
 Y  =  0.177 0.813 0.011   G 
(5.1)
Z
0.000 0.010 0.990
B
1
100Y 3
∗
− 16
(5.2)
L = 25
Y0
1
1
X 3
Y 3
∗
(5.3)
−
a = 500
X0
Y0
1
1 Y 3
Z 3
∗
(5.4)
−
b = 200
Y0
Z0
where the constraint 1 ≤ 100Y ≤ 100 must be satisfied, which is indeed the case for most
practical purposes [5]. The intermediate values [XY Z]T are the CIE XYZ tristimulus values,
and the [X0 Y0 Z0 ]T triplet is the reference white. In equations (5.2)–(5.4), L∗ is correlated
with brightness, a ∗ with the red-green content, and b∗ with the yellow-blue content within
the image. A similar set of nonlinear expressions can be found for the L∗ u∗ v∗ coordinate
©2001 CRC Press LLC
system. The computational complexity of the cube-root expressions above, however, may
render the perceptually uniform spaces unsuitable for real-time applications. Comprehensive
descriptions of the numerous color coordinate systems can be found in [5, 6, 7] along with
their appropriate transformation equations.
The HSV (hue, saturation, value) and the TekHVC (hue, value, chroma) color models belong
to a group of hue-oriented color coordinate systems that correspond more closely to the human
perception of color. These user-oriented color spaces are based on the intuitive appeal of the
artist’s tint, shade, and tone. The proprietary TekHVC model was developed by Tektronix
as a modification of the CIE L∗ u∗ v∗ perceptually uniform color space described earlier. The
HSV coordinate system, originally proposed by Smith [8], is cylindrical and is conveniently
represented by the hexcone model shown in Figure 5.1.
FIGURE 5.1
HSV hexcone color model.
The hue (H) is measured by the angle around the vertical axis and has a range of values
between 0 and 360◦ beginning with red at 0◦ . It gives us a measure of the spectral composition
of a color. The saturation (S) is a ratio that ranges from 0 (i.e., on the V axis), extending radially
outward to a maximum value of 1 on the triangular sides of the hexcone. This component refers
to the proportion of pure light of the dominant wavelength and indicates how far a color is from
a gray of equal brightness. The value (V) also ranges between 0 and 1 and is a measure of the
relative brightness. At the origin, V=0 and this point corresponds to black. At this particular
value, both H and S are undefined and meaningless. As we traverse upward along the V axis we
perceive different shades of gray until the endpoint is reached (where V=1 and S=0), which
is considered to be white. At any point along the V axis the saturation component is zero and
the hue is undefined. This singularity occurs whenever R=G=B. The set of equations below
can be used to transform a point in the RGB coordinate system to the appropriate value in the
©2001 CRC Press LLC
HSV space:
H1 = cos
−1
H = H1 ,
1
[(R − G) + (R − B)]
2
(R − G)2 + (R − B)(G − B)
if B ≤ G
◦
H = 360 − H1 , if B > G
Max (R,G,B) − Min (R,G,B)
S=
Max (R,G,B)
Max (R,G,B)
V=
255
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
In the expressions above, the Max and Min operators select the maximum and minimum values
of the operand, respectively, and R, G, and B range between 0 and 255. A fast algorithm used
here to convert the set of RGB values to the HSV color space is provided in [6].
5.3
Color Image Filtering
Filtering of multichannel images has received increased attention due to its importance in
processing color images. Numerous filtering techniques have been proposed to date for multichannel image processing. Nonlinear filters applied to images are required to preserve edges
and details and remove impulsive and Gaussian noise. On the other hand, vector processing
of multichannel images constitutes one of the most effective methods for filtering and edge
detection [9, 10]. Nonlinear filters based on order statistics (OS) have been extensively used in
the past to smooth and restore images corrupted by noise. Recently, a number of multichannel
filters which utilize correlation among multivariate vectors using distance measures have been
proposed for image filtering. Among them are the vector median filter (VMF) [11], the vector
directional filter (VDF) [12], the fuzzy vector filter (FVF) [13, 14, 15], and different versions
of the weighted mean filter [16, 17].
Apart from nonlinear multichannel filters based on order statistics, a number of fuzzy operators have been developed lately for image processing [18, 19]. Local correlation in the
data is utilized by applying the fuzzy rules directly on the pixels that lie within the operational
window. The output of the fuzzy processing depends on the fuzzy rule and the defuzzification
process, which combines the effects of the different rules into an output value. However, there
is no optimal way to determine the number and type of fuzzy rules required for the fuzzy image
operation. Usually, a large number of rules are necessary and the designer has to compromise
between quality and number of rules, because for even a moderate processing window a large
number of rules are required [20].
The large number of filters available poses some difficulties to the practitioner, since most of
them are designed to perform well in a specific application and their performance deteriorates
rapidly under different operation scenarios. Thus, a nonlinear adaptive filter that performs
equally well in a wide variety of applications is of great importance. Our goal is to devise
a simple, computationally efficient and reliable filter structure, which will deliver acceptable
results without making any assumption about signal or noise characteristics. Fuzzy operators
are utilized to assist us in this task. Consequently, a second objective is to examine aggregation
operators, analyze their properties, and justify their applicability to the design of multichannel
filters.
©2001 CRC Press LLC
5.3.1
Fuzzy Multichannel Filters
The Filtering Structure
Let y(x) : Z l → Z m represent a multichannel image and let W ⊂Z l be a window of
finite size n (filter length). The noisy image pixels inside the window W are denoted as
xj , j = 1, 2, . . . , n. The general form of the filter class is given as a fuzzy weighted average
of the input vectors inside the window W . The uncorrupted multichannel signal is estimated
by determining the center of gravity of the cluster of vectors inside the processing window.
Therefore, the filter’s output at the window center is:
ŷ =
ŷ =
where ξ =
w
n j
j =1 wj
n
ξj x j ,
j =1
n
j =1 wj xj
n
j =1 wj
(5.10)
.
(5.11)
.
The weights of the filter are determined adaptively using transformations of a distance
criterion at each image position. These weighting coefficients are transformations of the sum
of distances between the center of the window (pixel under consideration) and all samples
inside the filter window. The transformation has the meaning of membership function with
respect to the specific window component. Thus, the fuzzy weights provide the degree to
which an input vector contributes to the output, making the filter structure data dependent.
From such a viewpoint, a fuzzy clustering approach is introduced to determine the cluster
center considering the ambiguity of the multichannel signal. The filter structure proposed
here combines distance concepts with data-dependent filters and fuzzy membership functions.
Through the normalization procedure, two constraints necessary to ensure that the output is
an unbiased estimator are satisfied, namely:
• Each weight is a positive number, ξj ≥ 0.
• The summation of all the weights is equal to one,
n
j =1 ξj
= 1.
In multichannel filtering it is desirable to perform smoothing on all vectors that are from
the same region as the vector at the window center. At edges and lines the filter must only
smooth pixels at the same side of the edge as the vector at the window center. The proposed
algorithm assigns to a given point inside the window some membership function defined on
the set of vectors and then uses these membership values to calculate the final output. The
fuzzy weights represent the confidence that the vectors under consideration come from the
same region. It is therefore reasonable to make the weights proportional to the difference, in
terms of a distance measure, between a given vector and its neighbors inside the operational
window. In this way, whenever the current pixel is close to an area with high detail, the vectors
with the relatively large distance values will be assigned smaller weights and will contribute
less to the final filter estimate. Thus, edge or line detection operations prior to filtering can be
avoided, with considerable savings in terms of computational effort.
The filtering structure presented can be considered as an R-ordering-based multichannel filter
because distances inside the operational window are used. However, unlike any R-orderingbased filter, the distances are not used to rank the vectors. Rather, they are used to weight the
vectors such that negligible weights are assigned to outliers. The structure in (5.10) has the
familiar form of an adaptive filter, where the value of the noisy vector at the window center is
replaced by a weighted average value of all the points inside the operational window. It can also
be viewed as a generalization of existing linear or nonlinear averaging filters. Specifically, if the
©2001 CRC Press LLC
weighting coefficients are fixed, a linear shift invariant finite impulse response filter is devised.
Such a filter smoothes the signal but at the same time blurs signal boundaries (e.g., image edges).
In order to alleviate the problem, adaptive methodologies have been introduced, namely filter
structures with adaptively determined coefficients [21, 22]. However, a priori knowledge about
the signal and the desired response is required. Then the coefficients of the adaptive filter can
be optimized for a specific noise distribution with respect to a specific error criterion. However,
such information is not available in realistic signal processing applications. Learning schemes
based on training signals are iterative processes with heavy computational requirements. Their
real-time implementation is usually not feasible. Other adaptive filters are based on different
forms of the Wiener filter with variable coefficients. These filters, however, are based on
the assumption that the input signal and the available desired response are stationary ergodic
processes. This is not true for many practical applications. Other approaches use local statistics
on part of the signal to adaptively calculate the weights [9, 23, 24]. In these designs noise
statistics are often assumed ergodic in order to justify the use of the sample mean and sample
noise covariance in the calculations, although it is known that assumption does not always
hold. In summary, these filters are more perplexing than useful for engineers faced with real
image processing problems. On the contrary, the nonlinear scheme proposed here is simple.
It is adaptive but its coefficients are not calculated using complex iterative procedures.
In the last 5 years weighted mean filters with adaptively determined coefficients have been
proposed for robust multichannel estimation. In [13], a filter structure which uses a sigmoidal
fuzzy transformation to adaptively calculate data-dependent weights was proposed. The measure suggested to calculate distances among the vectors under consideration was the angle
between the vectors. In [14] ordered weights based on the same distance criterion as above
were used to generate the final filter output. Similarly, a multichannel filter that uses the inverse of the Euclidean distance to weight the vectors in the final output was proposed in [16].
This filter extends to multichannel signals the methodology introduced in [17] for univariate
input signals. However, weights based on multichannel distance measures can be constructed
in more than one way because there is no unique way to define the distance between two
multichannel signals. Depending on the distance criteria used and the transformations applied
to them, a number of different adaptive filters can be devised. Although it is not clear how
to select the appropriate distance-based weight, it is known from experimental results that its
form is of paramount importance for the performance of the filter. This work addresses the
problem of the selection of the appropriate weight form. Fuzzy connectives are utilized to
provide weight transformations that can be considered as a generalization of the transforms
already in use.
Before we introduce our methodology to construct a generalized weight function, we will
discuss common distance measures and their corresponding fuzzy transformations.
Distances and Fuzzy Weights
The most crucial step in the filter’s design is the development of the membership functions.
Despite past efforts, a unified form of fuzzy membership functions has not yet been derived [25].
In most cases, it is assumed that somehow they are available. Here, the weights ξj in (5.10) are
determined using fuzzy membership functions based on selected distance criteria. The fuzzy
transformation is not unique. The different fuzzy functions must meet a number of desirable
characteristics but mainly are required to have a smooth finite output over the entire input
range. Several candidate functions can meet the above specification. According to [25], the
most commonly used shapes for membership functions are triangular, trapezoidal, piecewise
linear, and Gaussian-like functions. These functions are chosen by the designer arbitrarily,
based on experience, problem specifications, and computational constraints imposed by the
design. Because the choice of the membership function form is very much problem dependent,
©2001 CRC Press LLC
the only applicable a priori rule is that designers must confine themselves to those functions
that are continuous and monotonic [26].
We devote our attention to fuzzy transformations that are suitable for two important distance
measures extensively used for nonlinear filter design.
The objective in the design is to select an appropriate fuzzy transformation, so that the pixel
with the minimum distance will be assigned the maximum weight.
The first criterion used to judge similarity (distance) between two vectors is the so-called
vector angle criterion. This criterion considers the angle between two vectors as their distance.
The distance associated with the noisy vector xi inside the processing window of length n can
be defined as:
ai =
n
A xi , xj
(5.12)
j =1
with
A xi , xj = cos
−1
xiT xj
|xi ||xj |
(5.13)
This similarity measure was introduced to measure distances between color vectors [12].
Because in the RGB color space, color is defined as relative values in the trichromatic channel
and not as a triplet of absolute intensity values, it was argued in [12] that the distance measure
must respond to relative intensity differences (chromaticity) and not absolute intensity differences (luminance). Thus, the orientation difference between two color vectors was selected as
their distance measure, because it correlates well with their spectral ratio difference.
A number of different shapes can be used to generate a membership function based on the
vector angle criterion. However, in the neural network and fuzzy systems literature [25], a
sigmoidal transformation is usually associated with inner product type distances. Therefore,
if the sum of angles is selected as the similarity measure, a sigmoidal membership function
should be utilized.
The fuzzy weight wi has the following form:
w1i =
β
(1 + exp(ai ))r
(5.14)
where β and r are parameters to be determined. The value of r is used to adjust the weighting
effect of the membership function, and β is a weight scale threshold. Since, by definition,
the vector angle distance criterion delivers a positive number in the interval [0, nπ ] [12],
the output of the fuzzy transformation introduced above produces a membership value in the
β
β
interval [ (1+exp((nπ))
r , 2r ]. However, even for a moderate size window, such as a 3×3 or
5×5 window, the lower limit of the above interval should safely be considered zero. As an
example, for a modest 3×3 window and with r = 1 and β = 2, the corresponding interval
is [1.4 × 10−12 , 1] and for a 5×5 window the interval becomes [1.5 × 10−35 , 1]. Therefore,
we can consider the above membership function as having values in the interval [0, 1]. It can
easily be seen through simple calculations that the above transformation satisfies the design
objectives.
The generalized Minkowski norm (Lp metric) can also be used to measure the distances
between two multichannel vectors [14]. The Lp is defined as:
1
m
p p
dp (i, j ) =
xik − xjk k=1
©2001 CRC Press LLC
(5.15)
where m is the dimension of the vector xi . Using this norm the scalar distance measure
dp (i) =
n
dp (i, j )
(5.16)
j =1
is associated with the noisy vector xi inside a filter window of length n. For such a distance
an appropriate membership function is the exponential (Gaussian-like) form:
dp (i)r
,
(5.17)
w2i = exp −
β
where r is a positive constant and β is a distance threshold. The actual values of the parameters
vary with the application. The above parameters correspond to the denominational and exponential fuzzy generators controlling the amount of fuzziness in the fuzzy weight. It is obvious
that since the distance measure is always a positive number, the output of this fuzzy membership function lies in the interval [0, 1]. The fuzzy transformation is such that the higher the
distance value, the lower the fuzzy weight becomes. It can easily be seen that the membership
function is one (maximum value) when the distance value is zero and becomes zero (minimum
value) when the distance value is infinite.
5.3.2
The Membership Functions
Both membership functions can be used to derive the fuzzy weights introduced in the filter structure of (5.10). However, the shape and the parameters of the functions were chosen
intuitively based on our experience and the distance criterion selected. More recently, membership functions have been designed using optimization procedures [25]. The general idea is
to tune the shape and the parameters of the membership function using a training signal. The
form of the fuzzy membership function is usually fixed ahead of time. Then a set of available
training pairs (input, membership values) is used to tune the parameters of the assumed membership function. The most commonly used procedure exploits the mean squared error (MSE)
criterion. In addition, since most of the used shapes are nonlinear, iterative schemes (e.g.,
back-propagation) are used in the calculations [26]. However, in an application such as image
processing, in order for the membership function to be tuned adaptively, the original image or
an image with properties similar to those of the original must be available. Unfortunately, this
is seldom the case in real-time image processing applications, where the uncorrupted original
image or knowledge about the noise characteristics is not available. Therefore, alternative
ways to obtain the “best” fuzzy transformation must be explored.
To this end, an approach is introduced here in which instead of “training” one membership
function, a bank of candidate membership functions are determined in parallel using different
distance measures. Then, a generalized nonlinear operator is used to determine the final
optimized membership function, which is employed to calculate the fuzzy weights. This
method of generating the overall function is closely related to the essence of computations
with fuzzy logic. By choosing the appropriate operator, the generalized membership function
can meet any specific objective requested by the design. As an example, if a minimum operator
is selected, the designer pays more attention to the objectives that are satisfied poorly by the
elemental functions and selects the overall value based on the worst of the properties. On the
contrary, when using a maximum operator the positive properties of the alternative membership
functions are emphasized. Finally, a mean-like operator provides a trade-off among different,
possibly incompatible, objectives.
Using the previous setting, the problem of determining the overall function is transformed
into a decision-making problem where the designer has to choose among a set of alternatives
©2001 CRC Press LLC
after considering several criteria. We discuss here only discrete solution spaces since distinct
membership function alternatives are available. As in any decision problem, where satisfaction
of an objective is required, two steps can be defined, namely, (1) the determination of the
efficient solution, and (2) the determination of an optimal compromise solution.
The compromise solution can be defined as the one preferred by the designer to all other
solutions, taking into consideration the objective and all the constraints imposed by the design.
The designer can specify the nonlinear operator used to combine elemental functions in advance
and use this operator to single out the final value from the set of available different solutions.
This is the approach followed here. An aggregator (fuzzy connective), whose shape is defined
a priori, will be used to combine the different elemental functions in order to produce the final
weights at each position.
In fuzzy decision making, connectives or aggregators are defined as mappings from
[0, 1]φ → [0, 1] and are often requested to be monotonic with respect to each argument.
The subclass of aggregation operators which are continuous, neutral, and monotonic is called
the class of CNM operators [27]. An averaging operator is a member of the class of compensative CNM operators but different from min or max operators. Averaging operators M can
be characterized under several natural properties, such as monotonicity and neutrality [28]. It
is widely accepted that an averaging operator verifies the following properties:
M : [0, 1]φ → [0, 1]
(i) Idempotency: ∀α, M(α, α, . . . , α) = α
(ii) Neutrality: the order of arguments is unimportant
(iii) M is nondecreasing in each place
The above implies that the averaging operator lies between min and max. However, aggregation operators are in general nonassociative or decomposable since associativity may
conflict with idempotence [29]. An example of averaging operators is the arithmetic mean,
the geometric mean, the harmonic mean, or the root-power mean. The problem of choosing
operators for logical combination of criteria is a difficult one. Experiments in decision making
indicate that aggregation among criteria is neither a conjunctive or disjunctive type of operation. Thus, compensatory connectives which mix both conjunctive and disjunctive behavior
were introduced in [30].
In this work a compensative operator, first introduced in [31], is utilized to generate the final
membership function. Following the results in [31], the operator is defined as the weighted
mean of a (logical AND) and a (logical OR) operator:
1−γ γ
A
B= A B
· A B
(5.18)
γ
where A, B are sets defined on the same space and represented by their membership functions.
Different t-norms and t-conorms can be used to express a conjunctive or a disjunctive attitude.
If the product of membership functions is utilized to determine intersection (logical AND) and
the possibilistic sum for union (logical OR), the form of the operator for several sets is as
follows [30]:
γ
(1−γ ) 
φ
φ
1 −
1 − wj i 
wj i
(5.19)
wci =
j =1
j =1
where wci is the overall membership function for the sample at pixel i, wj i is the j th elemental
membership value, and γ ∈ [0, 1]. The weighting parameter γ is interpreted as the grade of
©2001 CRC Press LLC
compensation, taking values in the range of [0, 1] [31]. In this work a constant value of 0.5 is
used for γ .
The product and the possibilistic sum are not the only operators that can be used in (5.18). A
simple and useful t-norm function is the min operator. In this chapter, we also use this t-norm
to represent intersection. Subsequently, the max operator is the corresponding t-conorm [25].
In such a case, the compensative operator of (5.18) has the following form:
(1−γ ) γ
φ
φ
wci = min wj i
max wj i
j =1
j =1
(5.20)
The form of the compensative operator is not unique. A number of other mathematical
models can be used to represent the AND aggregation. An alternative operator, which combines
the averaging properties of the arithmetic mean (member of the averaging operator class) with
a logical AND operator (conjunctive operator) was proposed also in [30].


m
φ
wci = γ min wj i + (1 − γ ) φ −1
(5.21)
wj i 
j =1
j =1
where wci is the overall membership function for the sample at pixel i and the parameter
γ ∈ [0, 1] is interpreted as the grade of compensation. In this equation the min t-norm stands
for the logical AND. Alternatively, the product of membership functions can be used instead
of the min operator in the above equation. The arithmetic mean is used to prevent higher
elemental weights with extreme values from dominating the final outcome. The operator is
computationally simple and possesses a number of desirable characteristics.
Compensatory operators are intuitively appealing but are based on ad hoc definitions and
properties, such as monotonicity, neutrality, or idempotency, that cannot always be verified.
However, despite these drawbacks, these methods are still appealing in that they can express
compensatory effects or interactions between design objectives. For this reason, we utilize
them in the next subsection to construct the overall fuzzy weights in our adaptive filter designs.
5.3.3
A Combined Fuzzy Directional and Fuzzy Median Filter
In our adaptive filter, we intend to assign higher weights to those samples that are more
centrally located (inside the filter window). However, as we have seen in Section 5.3.2 for
multichannel data, the concept of vector ordering has more than one interpretation and the
vector median inside the processing window can be defined in more than one way. Therefore,
the determination of the most centrally positioned vector heavily depends on the distance
measure used. Each distance measure described in Section 5.3.2 selects a different most
centrally located vector. Since multichannel ordering has no natural basis, it is anticipated
that we should expect better filtering results combining ranking criteria which utilize different
distances.
Let us assume that the adaptive multichannel filter of (5.10) must be used and the weights
wi ∀i inside the operational window must be assigned. Consider the design objective: The xi
is centrally located as measured with the angle criterion and xi is centrally located using the
Minkowski distance. We intend to establish a fuzzy membership function for this statement.
The first step is to realize that this statement is a composition between two design objectives,
which can be realized using elemental membership functions, such as the ones discussed in
the previous section. Then, utilizing the compensative operator, the overall function can be
obtained. At this point, we must clarify the effect of the compensatory operator in our filter. In
the above design objective, the same degree of attractiveness can be reached by having a less
centrally located vector according to the Euclidean distance, but more central using the angle
©2001 CRC Press LLC
criterion and vice versa. That is, the higher value of “with the angle criterion” compensates
for the lower value of membership in “using the Minkowski distance.”
For the specific case of two elemental membership functions and equal exponents, the
compensative operator defined in (5.18) has the form of a weighted membership product.
Thus, depending on the t-norm or t-conorm used, the overall fuzzy function can be defined as:
a
wci
= (w1i w2i )0.5
(5.22)
a is the overall membership function for the sample at pixel i, or
where wci
0.5
a
wci
= (w1i w2i )0.5 1 − 1 − w1j 1 − w2j
(5.23)
It can easily be seen from (5.21) that using the min and max operators and for equal powers
the operator in (5.18) actually has the form of the geometric mean, a member of the averaging
operators family.
The alternative operator introduced in (5.21) has, for this specific case, the following form:
b
wci
2
= 0.5min wj i + 0.25
j =1
2
wj i
(5.24)
j =1
or
b
wci
= 0.5(w1i ∗ w2i ) + 0.25
2
wj i
(5.25)
j =1
In general, additional weighting factors which will absorb possible scale differences in
the definition of the elemental membership functions must be used. However, since the two
elemental functions used here take values in the interval [0, 1], no such weighting factor is
required.
The averaging operator defined in (5.22), and the two compensative operators defined
in (5.23) and (5.24), can be used to define the fuzzy weights in (5.10) provided that the
elemental fuzzy transforms of (5.14) and (5.17) have been used to construct the elemental
weights. However, in order for our results to be meaningful, the nonlinear operator applied
must satisfy some properties that will guarantee that its application will not alter in any manner
the elemental decisions about the weights. In the literature, there are a number of properties
that all the aggregation or compensative operators must satisfy. In this subsection we will
examine whether the operators we intend to use to calculate the adaptive weights satisfy these
properties [28].
The requisite properties are listed below:
1. Convexity:
• The mean operator in (5.22) is convex.
Proof:
a
wci =
min wki ≤
k
©2001 CRC Press LLC
0.5
min wki max wki
k=1,2
wci a ≤
k=1,2
max wki
k
(5.26)
(5.27)
• The operator introduced in (5.24) is convex.
Proof:
b
= 0.75min wki + 0.25max wki
wcj
k
k
(5.28)
Then, we can conclude that:
b
min wki ≤ wci
≤ max wki
k
k
(5.29)
2. Monotonicity: The property of monotonicity guarantees that the stronger piece of evidence (larger elemental membership value) generates a stronger support in the final
membership function.
• The operator introduced in (5.22) is monotonous.
Proof:
a∗
a
≥ wci
wci
(5.30)
a∗ = (w w )0.5 , w a = (w w )0.5 , and ∀w ≥ w .
where wci
1i ki
1i j i
ki
ji
ci
• The operator introduced in (5.24) is monotonous.
Proof:
For w1i and ∀wki ≥ wj i , min (w1i , wki ) ≥ min (w1i , wj i ), so using (5.24),
b∗
b
≥ wci
wci
(5.31)
3. Idempotence: This property guarantees that the outcome of the overall function generates
the same value with each elemental value if all functions report the same result.
• The operator introduced in (5.22) is idempotent.
Proof:
a
wci
= (ww)0.5 = w
(5.32)
• The operator introduced in (5.24) is idempotent.
Proof:
b
wci
= 0.5w + 0.25(w + w) = w
(5.33)
It can easily be seen from (5.23) that this operator is not idempotent. However, the operator
is symmetric and satisfies the monotonicity requirement, namely,
a∗
a
wci
≥ wci
(5.34)
a∗
= (w1i wki )0.5 (1 − ((1 − w1i ) (1 − wki )))0.5
wci
(5.35)
0.5 0.5
a
1 − (1 − w1i ) 1 − wj i
= w1i wj i
wci
(5.36)
(1 − wki ) ≤ 1 − wj i
(1 − ((1 − w1i ) (1 − wki ))) ≥ 1 − (1 − w1i ) 1 − wj i
(5.37)
where
and
If ∀wki ≥ wj i , then
©2001 CRC Press LLC
(5.38)
Combining (5.34)–(5.38), we can conclude that the operator defined in (5.23) satisfies the
monotonicity requirement.
In addition, it is not hard to see that the operators introduced here are symmetric (neutral).
This property guarantees that the order of presentation for the elemental functions does not
affect the overall membership value.
In summary, we have proven that the compensatory operators we intend to use for the fuzzy
weights calculations in (5.10) correspond to an aggregation class which satisfies a number of
natural properties, such as neutrality and monotonicity.
The decision to utilize a fuzzy aggregator to construct the overall weight is not arbitrary. On
the contrary, it is anticipated that the operator will help us to accomplish the design objective.
The introduction of a combination of different distances in the weight determination procedure
is expected to enhance the filter performance. Each one of the above defined operators can
generate a final membership function, which is sensitive to relative changes in the elemental
membership values and helps us to accomplish our objective. A fuzzy filter, which utilizes
this form of membership function for its fuzzy weights, constitutes a fuzzy generalization of
a combined VMF and VDF.
It must be emphasized that through this design the problem of determining the appropriate
membership function is transformed into the problem of combining a collection of possible
functions. This constitutes a problem of considerably reduced complexity, since admissible
membership functions may be known from physical considerations or design specifications.
The proposed adaptive design is a scalable one. The designer controls the complexity of the
final membership function by determining the number and form of the individual membership
functions. Depending on the problem specification and the computational constraints, the
designer can select the appropriate number of elemental functions to be used in the final
weighting function. The shape of the membership function (e.g., sigmoidal or exponential) is
not the only parameter that differentiates between possible elemental fuzzy transformations.
The designer may decide to use the same form for the elemental functions and assign different
parameter values to them (e.g., different r or β). Then, an overall membership function can
be devised using an appropriate combination of the individual functions. The computational
efficiency of the proposed filter depends not only on the form of the membership function
selected or the operator used for aggregation, but on both of them.
This parallel, adaptive on-line determination of the membership function allows for a fast
design without time-consuming iterative processes. The filter’s output is calculated in one pass
without any recursion. Thus, our filter does not depend on a “good” initial estimate. On the
contrary, it is well known that iterative learning filters starting from certain initial value are
likely to be trapped in local optima with profound consequences to the filter’s performance.
Furthermore, in our design there is no requirement for the training signal needed to assist
learning in iterative adaptive designs. The final fuzzy membership function is determined
without any suboptimal local noise or signal statistic evaluation since such approaches usually
lead to biased solutions. Thus, our adaptive multichannel filters can be used in real-time image
applications, in contrast to other “trainable” multichannel filters, which are based on unrealistic
assumptions about the availability of training sequences.
5.3.4
Application to Color Images
The performance of the new filters introduced here is evaluated below (see also Table 5.1).
The evaluation is carried out using a color test image and their performance is measured against
popular vector processing filters, such as the VMF, the basic vector directional filter (BVDF),
the generalized vector directional filter (GVDF), the arithmetic mean filter (AMF), and the
hybrid filters of [33]. Since our objective is not to develop all the different adaptive filters
©2001 CRC Press LLC
based on fuzzy transformations of the distance but to demonstrate the improvement introduced
in terms of performance using a fuzzy aggregator or compensator, we construct five different
filters based on the distance criteria and elemental transforms described previously. The
following notation is used for convenience.
Table 5.1 Filters Compared
Notation
BVDF
GVDF
AMF
VMF
HF
AHF
FVF1
FVF2
FVF3
FVF4
FVF5
Filter
Basic vector directional filter
Generalized vector directional filter
Arithmetic mean filter
Vector median filter
Hybrid directional filter
Adaptive hybrid directional filter
Fuzzy vector directional filter
2
with weights determined through w1j = 1+exp(a
j)
Fuzzy vector directional filter
with weights determined through w2j = exp[−dp (j )0.5 ]
Fuzzy vector directional filter
with weights determined through w1j , w2j , (5.13)
Fuzzy vector directional filter
with weights determined through w1j , w2j , (5.13)
Fuzzy vector directional filter
with weights determined through w1j , w2j , (5.15)
Ref.
[12, 32]
[12, 32]
[32]
[11]
[32, 33]
[32, 33]
[15, 32]
[15]
[15]
[15]
[15]
The filters are applied to the widely used 512 × 480 RGB color image Lena. The test image
has been contaminated using various noise source models in order to assess the performance of
the filters under different scenarios. The test image is contaminated with correlated Gaussian
noise, and a percentage of the image samples are replaced by outliers, which have very high
or low signal values with equal probability (see Table 5.2).
Table 5.2 Noise Distributions
Number
1
2
3
4
Noise Model
Gaussian (σ = 30)
Impulsive (4%)
Gaussian (σ = 15), impulsive (2%)
Gaussian (σ = 30), impulsive (4%)
The normalized mean squared error (NMSE) has been used as a quantitative measure for
evaluation purposes. It is computed as:
N1 N2
NMSE =
j =0 (y(i, j ) − ŷ(i, j )
N1 N2
2
i=0
j =0 (y(i, j )
i=0
2
,
(5.39)
where N1 and N 2 are the image dimensions, and y(i, j ) and ŷ(i, j ) denote the original image
vector and the estimation at pixel (i, j ), respectively. Table 5.3 summarizes the results obtained
for the Lena test image for a 3 × 3 processing window. The results obtained using a 5 × 5
filter window are given in Table 5.4. The GVDF uses the appropriate gray-scale operator at
the magnitude processing module to obtain the best possible result. It must be emphasized
that these modules are noise dependent. The designer must know a priori the actual noise
©2001 CRC Press LLC
characteristics. This is hardly the case in a real-time image processing situation. In contrast,
the FVF family does not require any information about noise characteristics. However, despite
the fact that the GVDF utilizes more information, we select the best filter from the GVDF
family for the comparisons below [12].
Table 5.3 NMSE (×10−2 ) for the Lena
Image (3 × 3 Window)
Filter
Noise Model
1
2
3
4
None 4.2083
5.1694
3.6600
9.0724
BVDF 2.8962
0.3448
0.4630
1.1354
GVDF 1.4600
0.3000
0.6334
1.9820
AMF 0.6963
0.8186
0.6160
1.298
HF
1.3192
0.2182
0.5158
1.6912
AHF 1.0585
0.2017
0.4636
1.4355
FVF1 0.735
0.2481
0.401
1.039
FVF2 0.9812
0.1663∗ 0.3826
1.1744
FVF3 0.6940∗ 0.2161
0.3310
0.9130∗
∗
FVF4 0.7335
0.1908
0.3234
0.9445
FVF5 0.7201
0.244
0.3511
0.9903
∗ Best filter performance in the corresponding row.
Table 5.4 NMSE (×10−2 ) for the RGB Lena
Image (5×5 Window)
Filter
Noise Model
1
2
3
4
None 4.2083
5.1694
3.6600
9.0724
BVDF 2.800
0.7318
0.6850
1.3557
GVDF 1.0800
0.5400
0.4590
1.1044
AMF 0.5977∗ 0.6656
0.572
0.8896
HF
0.7700
0.3841
0.4890
1.1417
AHF 0.6762
0.3772
0.4367
0.7528
FVF1 0.7549
0.3087
0.4076
0.9550
FVF2 0.6718
0.3040
0.4031
0.7491
FVF3 0.6178
0.3042
0.3813∗ 0.7224
FVF4 0.6584
0.2984∗ 0.3817
0.7444
FVF5 0.6239
0.3069
0.387
0.7074∗
∗ Best filter performance in the corresponding row.
From the results listed in the tables, it can be easily seen that our adaptive design with the
generalized membership function provides consistently good results in every type of noise
situation. The different fuzzy filters attenuate both impulsive and correlated Gaussian noise
with or without outliers present in the test image. It must be noted that if no assumption about
the noise characteristics is made, the fuzzy filter with the generalized membership weights
provides results better than the results obtained by any other filter under consideration. Results
also indicate that our fuzzy techniques are less sensitive to the window length, compared to
the GVDF or the VMF. As an example, it can be seen that our adaptive fuzzy filters do not
suffer from VMF’s inefficiency in a nonimpulsive noise scenario and small filtering window.
©2001 CRC Press LLC
Finally, considering the number of computations, the computationally intensive part of the
fuzzy algorithm is the distance calculation part. However, this step is common in all multichannel algorithms considered here. More than that, the different elemental membership functions
can be calculated in parallel, thus reducing the execution time and making our filters suitable
for real-time implementation with digital signal processors. The adaptation procedure used to
evaluate the generalized membership function does not introduce any additional computational
cost. To the best of our knowledge, the adaptation mechanism introduced in this work is the
only one capable of providing this form of parallel processing capability.
In conclusion, our adaptive design is simple, scalable, does not increase the numerical
complexity of the fuzzy algorithm, and delivers excellent results for complicated multichannel
signals, such as real color images. Moreover, as can easily be seen from the attached images,
the new filters preserve the chromaticity component, which is very important in the visual
perception of color images.
5.4
Color Image Segmentation
Image compression is essential in numerous multimedia applications due to the enormous
bandwidth and storage requirements, as mentioned previously. Conventional coding standards
such as H.261 and MPEG-1 and -2 fail to adequately model object motion within the scene
and also suffer from the familiar blocking artifacts. Furthermore, these schemes deal with
video exclusively at the frame level, thereby preventing the manipulation of individual objects
within the bitstream. Recently, however, greater attention has been paid to a newer generation
of coding schemes that are object based [34, 35]. These methods rely on the techniques of
image analysis and computer graphics to represent the image signals using their structural
features such as contours and regions. In this latter approach, the input video sequence must
first be segmented into an appropriate set of arbitrarily shaped regions [36]. Thus, the success
of any object-based method depends largely on this segmentation process. This not only
improves the coding efficiency, but it can also support various content-based functionalities.
In this section, we focus our attention on the color segmentation problem. A fast color
segmentation algorithm is presented that employs the perceptual HSV color space model to
partition an image into arbitrarily shaped regions. This is carried out by employing a recursive
1D histogram thresholding procedure. The proposed technique is robust, suitable for real-time
implementation (i.e., due to the 1D histogram approach), and very intuitive in describing the
color/intensity content of a region.
The hue component of the HSV color model can be effectively employed to segment the
color content within a scene. However, the hue attribute is ineffective and unreliable when the
saturation or value components are low. Therefore, we partition the image into the following
three primary regions so that an appropriate segmentation scheme can be applied within each
region: (1) an achromatic, (2) a chromatic, and (3) a transitional area. The achromatic regions
are characterized by low values of saturation and value and consist of the black, white, and
gray areas within the scene. Threshold values of S ≤ 10% and V ≤ 20% were used to define
the achromatic sector of the HSV space. A similar saturation threshold was selected in [37]
to partition the achromatic sector of the HVC space without enforcing an intensity restriction.
The intensity information, however, is important [38], and erroneous results may be obtained
if this latter restriction is not imposed [39]. The value component (i.e., the brightness) is
used to segment the achromatic regions of the image. The chromatic region (region 2), on
the other hand, is described by high values of saturation and value where the hue has great
©2001 CRC Press LLC
discriminating power and can be effectively used to segment the chromatic parts of the image.
Threshold values of S ≥ 20% and V > 20% were selected in defining this second region.
Finally, the third region separates the chromatic and achromatic areas and is referred to as the
transition region. Thresholds of 10% < S < 20% and V > 20% were chosen for this latter
region. Slices of this solid correspond to annular rings in the HSV model. The hue component
in this transition region is once again unreliable. Pixel values within this region have very
little chroma and, thus, are better characterized by the value component. This partitioning
of the HSV hexcone model into the three primary regions is summarized in Table 5.5. A
simple two-region model has also been proposed for segmentation purposes in the similar HSI
space [40]. In this scheme, the original image is split into only two regions (chromatic and
achromatic) by using the average value of the peaks found in the saturation histogram as a
threshold value. There are two problems associated with this approach: (1) threshold values
may be over- or underestimated due to the averaging process, which may result in an incorrect
partition of the chromatic and achromatic regions, and (2) no intensity information is taken
into account, which may lead to erroneous results due to the low intensity value pixels.
Table 5.5 Partitioning of the HSV Hexcone Model
Region
Achromatic
Transitional
Chromatic
Bounding Thresholds
S ≤ 10%
V ≤ 20%
10% < S < 20% V > 20%
S ≥ 20%
V > 20%
Segmentation Cue
Value
Value
Hue
Once the image has been partitioned into the three primary regions above, then a histogram
thresholding procedure is carried out within each region using the appropriate cue.
5.4.1
Histogram Thresholding
Segmentation within the achromatic region is performed by using the histogram of the
value component. The value histogram is first formed and smoothened by the scale–space
filtering approach [41]. The largest peak is then selected and the valleys are subsequently
found on either side of this peak. Pixel values within the two valley points are classified as a
uniform area. A set of binary operations which include median filtering and region removal
are used to remove isolated pixels and small regions (i.e., less than a predefined threshold),
respectively. This process is repeated recursively until all the pixels within the achromatic
region are segmented into significant areas of uniformity (i.e., no more regions can be further
extracted from the histogram after the small region removal step).
The procedure just described is also carried out using the value histogram of the pixels within
the transitional region. Areas within this region appear to have some chroma component and,
therefore, are kept disjoint from the achromatic region.
Finally, the chromatic region is segmented by using the hue histogram of the chroma pixels,
as defined in Table 5.5. However, we have found that subdividing the chromatic area further
into subregions yields an improvement in the segmentation results. This division is carried out
at the valleys (i.e., between peaks) of the smoothened saturation histogram of the chromatic
region. In effect, this partitions the chromatic areas into varying levels of saturation for
improved results (i.e., two areas with the same hue but different saturation values are not
grouped together). Segmentation is performed within each of these chromatic subregions by
using the histogram of the hue component (i.e., as done with the value component above).
©2001 CRC Press LLC
5.4.2
Postprocessing and Region Merging
The recursive histogram procedure described in the previous subsection is applied to each
of the three primary regions, until no areas of uniformity can be further extracted. However,
a number of pixels will still remain unclassified as a result of this process (i.e., due to small
region removal, median filtering, etc.). These pixels are subsequently combined into the best
matching region (within a spatially local window) from the set of regions obtained in the initial
histogram extraction process as follows. The image is progressively scanned (in a raster scan
fashion) and a 3 × 3 window is formed for each unclassified pixel that borders at least one pixel
(in the 8th connected nearest neighbor sense) from an initially segmented area. The L2 norm
is computed for each pixel in the window, with respect to the central pixel (i.e., the unclassified
pixel). The smallest value is taken and compared to a predefined threshold. If it is less than the
threshold value, then the central pixel is incorporated into the area where the corresponding
pixel (i.e., the one with the smallest L2 norm) belongs. If it exceeds the threshold value, then
the central pixel is left unclassified. Pixel sites are revisited through a number of iterations
until all the unclassified pixels are grouped to an appropriate region. When no groupings are
made within a particular iteration, then the threshold values are increased so that the process
converges. The selection of the initial threshold value is quite small and is gradually relaxed
(i.e., increased) until all pixels are classified. This process is very fast because there are usually
a small number of pixels (typically at the borders of regions) requiring few iterations.
Once all of the pixels have been classified, a series of binary morphological operations are
used to refine the extracted regions [42]. A binary morphological opening operation is first
used to remove small spurs and thin channels, followed by a binary morphological closing
operation to fill in small holes and gaps.
At this stage, the segmentation of the image into a set of refined, uniform regions is complete.
However, an oversegmented region may result if the threshold for small region removal is set
too low. Region merging is used to overcome this situation by joining bordering regions with
a similar average hue value. Adjacent regions are merged if the Euclidean distance of the
average RGB values of two regions is less than a set threshold. Region merging is performed
in the RGB space due to the lack of an appropriate distance metric in the HSV color space.
Regions can be merged so that the smallest region is of some minimum size, or a specific
number of regions is obtained. Here, we select a fixed threshold based on experimental values
to reduce the computational complexity. Setting an appropriate threshold can also reduce the
regions so that they coincide with semantically meaningful objects.
5.4.3
Experimental Results
The performance of the proposed segmentation scheme was tested with a number of different
video sequences, and the results of the Carphone and Claire sequences are displayed below. In
Figure 5.2a and b, the results of the Carphone QCIF (176 × 144) sequence are shown. Part a
illustrates frame 80 of the original image, whereas part b shows the segmentation results after
region merging in which adjacent areas are joined and the number of regions is reduced. Small
regions were removed if their perimeter was less than 30 pixels. Setting a smaller threshold
generates many additional smaller regions, which may result in an oversegmented image. At
the same time, however, this may capture some additional detail present within the scene. The
postprocessing operations included a 5 × 5 binary median filter and a circular morphological
structuring element. In Figure 5.2c and d, the results of the Claire CIF (360 ×288) sequence are
displayed. Once again, an arbitrary set of regions is effectively extracted in this less detailed
scene.
The effectiveness of this segmentation scheme, and its potential for a more suitable contentbased representation, is encouraging for future object-based video coding environments. This
©2001 CRC Press LLC
FIGURE 5.2
(a) Original frame 80 of the Carphone sequence. (b) Final segmentation of (a) after
region merging. (c) Original frame 100 of the Claire sequence. (d) Final segmentation of
(c) after region merging.
approach is being further enhanced to incorporate motion information so that regions can be
associated with semantically meaningful objects.
5.5
Facial Image Segmentation
The recognition of human faces is currently an active area of research in computer vision [43]–[46]. The task of recognizing human faces is essentially a two-step process: (1) the
detection and automatic location of the human face, and (2) the automatic identification of the
face based on the extracted features. Most of the research to date has been directed toward the
identification phase, with less emphasis being placed on the initial localization stage. However,
the first step is critical to the success of the second and the overall recognition system. Thus,
the importance of obtaining an accurate localization of the face is clear and vital in numerous
multimedia applications including human recognition for security purposes, human–computer
interfaces, and more recently, for video coding, multimedia databases, and video on demand.
Nevertheless, determining the location of a face of unknown size in a scene with a complex or
moving background still remains a difficult problem that is relatively unexplored.
©2001 CRC Press LLC
Several techniques based on shape and motion information have been proposed recently
for the automatic location of the facial region [47]–[49]. The former two are related to video
coding applications, whereas the latter is part of a facial recognition system. The shape-based
approach in [47] models the contours of the face as an ellipse. The location of the facial
region is determined by performing an ellipse fitting task to a thresholded binary edge image.
In [48], a generic 3D face model is adapted to the extracted facial outline from a videophonetype scene for the case where only one person is talking against a stationary background.
In this application, a hierarchical localization scheme is utilized to isolate the facial area.
The technique is based on the shape of the extracted head-and-shoulders silhouette, which is
obtained using the thresholded frame differences. Finally, in [49], a motion detection algorithm
is used to segment the facial area from a complex background. The proposed method locates
the facial region by assuming that the object having the greatest motion in the video sequence
is the face to be detected. This assumption, however, may limit the success of the approach in
applications with nonstationary backgrounds (e.g., mobile videophones) and/or other moving
objects in the scene. The authors also acknowledge potential problems caused by noise or
other objects moving in the background and also suggest a modification in their technique to
better handle the case of tilted or turned faces.
5.5.1
Extraction of Skin-Tone Regions
The identification and tracking of the facial region is determined by utilizing a priori knowledge of the skin tone distributions in the HSV color space outlined earlier. It has been found
that skin-colored clusters form within a rather well-defined region in chromaticity space [50],
and also within the HSV hexcone model [51], for a variety of different skin types. In the
HSV space in particular, the skin distribution was found to lie predominantly within the limited hue range between 0◦ and 50◦ (red–yellow), and in certain cases between 340◦ and 360◦
(magenta–red) for darker skin types [39]. The saturation component suggests that skin colors
are somewhat saturated, but not deeply saturated, with varying levels of intensity.
The hue component is the most significant feature in defining the characteristics of the skin
clusters. However, as mentioned earlier, the hue can be unreliable when: (1) the level of
brightness (i.e., value) in the scene is low, or (2) the regions under consideration have low
saturation values [39]. The first condition can occur in areas of the image where there are
shadows, or generally under low lighting levels. In the second case, low values of saturation
are found in the achromatic regions of a scene. Thus, we must define appropriate thresholds
for the value and saturation components where the hue attribute is reliable. We have defined
the following polyhedron with appropriate threshold values that correspond to the skin-colored
clusters with well-defined saturation and value components, based on a large sample set [39]:
Thue1 = 340◦ ≤ H ≤ Thue2 = 360◦
Thue3 = 0◦ ≤ H ≤ Thue4 = 50◦
S ≥ Tsat1 = 20%
V ≥ Tval = 35%
(5.40)
(5.41)
(5.42)
(5.43)
The extent of the above hue range is purposely designed to be quite wide so that a variety of
different skin types can be modeled. As a result of this, however, other objects in the scene
with skin-like colors may also be extracted. Nevertheless, these objects can be separated by
analyzing the hue histogram of the extracted pixels. The valleys between the peaks are used to
identify the various objects that possess different hue ranges (i.e., facial region and different
colored objects). Scale–space filtering [52] is used to smoothen the histogram and obtain
the meaningful peaks and valleys. This process is carried out by convolving the original hue
histogram, fh (x), with a Gaussian function g(x, τ ) of zero mean and standard deviation τ as
©2001 CRC Press LLC
follows:
−(x − u)2
exp
du
fh (u) √
Fh (x, τ ) = fh (x) ∗ g(x, τ ) =
2τ 2
2π τ
−∞
∞
1
(5.44)
where Fh (x, τ ) represents the smooth histogram. The peaks and valleys are determined by
examining the first and second derivatives of Fh above. In the remote case that another object
matches the skin color of the facial area (i.e., separation is not possible by the scale–space filter),
the shape analysis module that follows provides the necessary discriminatory functionality.
A series of postprocessing operations which include median filtering and region
filling/removal is subsequently used to refine the regions obtained from the initial extraction
stage.
5.5.2
Postprocessing
Median filtering is the first of two postprocessing operations that are performed after the
initial color extraction stage. The median operation is introduced in order to smoothen the
segmented object silhouettes and also eliminate any isolated misclassified pixels that may
appear as impulsive-type noise. Square filter windows of size 5×5 and 7×7 provide a good
balance between adequate noise suppression and sufficient detail preservation. This operation
is computationally inexpensive because it is carried out on the bilevel images (i.e., object
silhouettes).
The result of the median operation is successful in removing any misclassified noise-like
pixels; however, small isolated regions and small holes within object areas may remain after
this step. Thus, we follow the application of median filtering by region filling and removal.
This second postprocessing operation fills in small holes within objects which may occur due
to color differences (e.g., eyes and mouth of the facial skin region), extreme shadows, or any
unusual lighting effects (specular reflection). At the same time, any erroneous small regions
are also eliminated as candidate object areas.
We have found that the hue attribute is reliable when the saturation component is greater
than 20% and meaningless when it is less than 10% [39]. Similar results have also been
confirmed in the HVC color model [37]. Saturation values between 0 and 10% correspond
to the achromatic areas within a scene, whereas those greater than 20% correspond to the
chromatic ones. The range between 10 and 20% represents a sort of transition region from
the achromatic to the chromatic areas. We have observed that, in certain cases, the addition
of a select number of pixels within this 10 to 20% range can improve the results of the initial
extraction process. In particular, the initial segmentation may not capture smaller areas of
the face when the saturation component is decreased due to the lighting conditions. Thus,
pixels within this transition region are selected accordingly [39] and merged with the initially
extracted objects. A pixel within the transitional region is added to a particular object if its
distance is within a threshold of the closest object. A reasonable selection can be made if
the threshold is set to a factor between 1.0 and 1.5 of the distance from the centroid of the
object to its most distant point. The results from this step are once again refined by the two
postprocessing operations described earlier.
At this point, one or more of the extracted objects corresponds to the facial regions. In certain
video sequences, however, we have found gaps or holes around the eyes of the segmented facial
area. This occurs in sequences where the forehead is covered by hair and, as a result, the eyes
fail to be included in the segmentation. We utilize two morphological operators to overcome
this problem and at the same time smoothen the facial contours. A morphological closing
operation is first used to fill in small holes and gaps, followed by a morphological opening
operation to remove small spurs and thin channels [42]. Both of these operations maintain
©2001 CRC Press LLC
the original shapes and sizes of the objects. A compact structuring element such as a circle or
square without holes can be used to implement these operations and also help to smoothen the
object contours. Furthermore, these binary morphological operations can be implemented by
low-complexity hit-or-miss transformations [42].
The morphological stage is the final step prior to analysis of the extracted objects. The
results at this point contain one or more objects that correspond to the facial areas within the
scene. The block diagram in Figure 5.3 summarizes the proposed face localization procedure.
The shape and color analysis unit, described next, provides the mechanism to correctly identify
the facial regions.
FIGURE 5.3
Overall scheme to extract the facial regions within a scene.
5.5.3
Shape and Color Analysis
The input to the shape and color analysis module may contain objects other than the facial
areas. Thus, the function of this module is to identify the actual facial regions from the set of
candidate objects. To achieve this, a number of expected facial characteristics such as shape,
color, symmetry, and location are used in the selection process. Fuzzy membership functions
are constructed in order to quantify the expected values of each characteristic. Thus, the value
of a particular membership function gives us an indication of the goodness of fit of the object
under consideration with the corresponding feature. An overall goodness of fit value can finally
be derived for each object by combining the measures obtained from the individual primitives.
In our segmentation and localization scheme we utilize a set of features that are suitable for
our application purposes. In facial image databases (employees, models, etc.) or videophonetype sequences (video archives of newscasts, interviews, etc.), the scene consists of predominantly upright faces that are contained within the image (i.e., not typically at the edges of
the image). Thus, we utilize features such as the location of the face, its orientation from the
vertical axis, and its aspect ratio to assist with the recognition task. These features can be
determined in a simple and fast manner, as opposed to measurements based on facial features
such as the eyes, nose, and mouth, which may be difficult to compute (i.e., in certain images
the features may be small or occluded). More specifically, we consider the following four
primitives in our face localization system:
1. Deviation from the average hue value of the different skin-type categories. The average
hue value for different skin types varies among humans and depends on the race, gender,
and age of the person. However, the average hue of different skin types falls within a
more restricted range than the wider one defined by equations (5.40) and (5.41) [39].
The deviation of an object’s expected hue value from this restricted range gives us an
indication of its similarity to skin tone colors.
2. Face aspect ratio. Given the geometry and shape of the human face, it is reasonable to
expect that the ratio of height to width falls within a specific range. If the dimensions
©2001 CRC Press LLC
of a segmented object fit the commonly accepted dimensions of the human face then it
can be classified as a facial area.
3. Vertical orientation. The location of an object in a scene depends largely on the viewing
angle of the camera and the acquisition devices. For the intended applications it is
assumed that only reasonable rotations of the head are allowed in the image plane. This
corresponds to a small deviation of the facial symmetry axis from the vertical direction.
4. Relative position of the facial region in the image plane. By similar reasoning to
(3) above, it is probable that the face will not be located right at the edges of the image
but, rather, within a central window of the image.
5.5.4
Fuzzy Membership Functions
A number of membership function models can be constructed and empirically evaluated. A
trapezoidal function model is utilized here for each primitive in order to keep the complexity
of the overall scheme to a minimum. This type of membership function attains the maximum
value only over a limited range of input values. Symmetric or asymmetrical trapezoidal shapes
can be obtained depending on the selected parameter values. The membership function can
assume any value in the interval [0, 1], including both of the extreme values. A value of 0
in the function above indicates that the event is impossible. On the contrary, the maximum
membership value of 1 represents total certainty. The intermediate values are used to quantify
variable degrees of uncertainty. The estimates for the four membership functions are obtained
by a collection of physical measurements of each primitive from a database of facial images
and sequences [39].
The hue characteristics of the facial region (for different skin-type categories) were used
to form the first membership function. This function is built using the discrete universe of
discourse [−20◦ , 50◦ ] (i.e., −20◦ = 340◦ ). The lower bound of the average hue observed in
the image database is approximately 8◦ (African-American distribution), whereas the upper
bound average value is around 30◦ (Asian distribution) [39]. A range is formed using these
values, where an object is accepted as a skin tone color with probability 1 if its average hue value
falls within these bounds. Thus, the membership function associated with the first primitive is
defined as follows:

(x + 20)


, if − 20◦ ≤x≤8◦


28

µ(x) = 1
, if 8◦ ≤x≤30◦



(50
−
x)


, if 30◦ ≤x≤50◦
20
(5.45)
Experimentation with a wide variety of facial images has led us to the conclusion that the
aspect ratio (height/width) of the human face has a nominal value of approximately 1.5. This
finding confirms previous results reported in the open literature [49]. However, in certain
images we must also compensate for the inclusion of the neck area, which has similar skin
tone characteristics to the facial region. This has the effect of slightly increasing the aspect
ratio. Using this information along with the observed aspect ratios from our database, we can
tune the parameters of the trapezoidal function for this second primitive. The final form of the
©2001 CRC Press LLC
function is given by

(x − 0.75)


, if 0.75≤x≤1.25


0.5



1
, if 1.25≤x≤1.75
µ(x) = (2.25 − x)


, if 1.75≤x≤2.25



0.5



0
, otherwise
(5.46)
The vertical orientation of the face in the image is the third primitive used in our shape
recognition system. As mentioned previously, the orientation of the facial area (i.e., deviation
of the facial symmetry axis from the vertical axis) is more likely to be aligned toward the
vertical due to the type of applications considered. A reasonable threshold selection of 30◦
can be made for valid head rotations also observed within our database. Thus, a membership
value of 1 is returned if the orientation angle is less than this threshold. The membership
function for this primitive is defined as follows:

1
, if 0◦ ≤x≤30◦
(5.47)
µ(x) = (90 − x)

, if 30◦ ≤x≤90◦
60
The last primitive used in our knowledge-based system refers to the relative position of the
face in the image. Due to the nature of the applications considered, we would like to assign
a smaller weighting to objects that appear closer to the edges and corners of the images. For
this purpose, we construct two membership functions. The first one returns a confidence value
for the location of the segmented object with respect to the x axis. Similarly, the second
one quantifies our knowledge about the location of the object with respect to the y axis. The
following membership function has been defined for the position of a candidate object with
respect to either the x or y axis:

3d
(x − (d))


, if d≤x≤

d

2




2



3d
5d

1
, if
≤x≤
2
2
µ(x) =
(5.48)

((3d)
−
x)
5d


, if
≤x≤3d


d

2




2



0
, otherwise
The membership function for the x axis is determined by letting d = D4x , where Dx represents
the horizontal dimensions of the image (i.e., in the x direction). In a similar way, the y axis
D
membership function is found by letting d = 4y , where Dy represents the vertical dimensions
of the image (i.e., in the y direction).
The individual membership functions expressed above must be appropriately combined to
form an overall decision. To this end, we utilize the fuzzy aggregators used in Section 5.3.2
to form the overall function used in the filter. In particular, the compensative operator (i.e.,
overall fuzzy membership function), which assumes the form of a weighted product as follows
0.5
m
m
µc =
min µj
max µj
j =1
©2001 CRC Press LLC
j =1
(5.49)
was selected because it provides a good compromise of conjunctive and disjunctive behavior.
The aggregation operator defined in (5.49) is used to form the final decision based on the
designed primitives.
5.5.5
Meta-Data Features
Multimedia databases are composed of a number of different media types, such as images and
video that are binary by nature, and hence are unstructured. An appropriate set of interpretations
must be derived for these media objects in order to allow for content-based functionalities which
include storage and retrieval. These interpretations, or meta-data, are generated by applying
a set of feature-extracting functions on the contained media objects [53]. These functions
are media dependent (i.e., audio, video, images) and are unique even within each media type
(i.e., satellite images, facial images). The following four steps are necessary in extracting the
features from image object types: (1) object locator design, (2) feature selection, (3) classifier
design, and (4) classifier training. The function of the object locator is to isolate the individual
objects of interest within the image through a suitable segmentation algorithm. In the second
step, specific features are selected to identify the different types of objects that might occur
within the images of interest. The classifier design stage is then used to establish a mathematical
basis for distinguishing the different objects based on the designed features. Finally, the last
step is used to train and update the classifier module by adjusting various parameters. In the
previous sections we have designed the object locator to automatically isolate and track the
facial area within a facial image database or a videophone-type sequence. Now, we propose
the use of a set of features that may be used in constructing a meta-data feature vector for the
classifier design and training stages.
Having determined the facial regions within the image, we can construct an n-dimensional
feature vector, f = (f1 , f2 , . . . , fn ), that may be used for content-based storage and retrieval
purposes. We present several features that may be incorporated within a more detailed metadata feature vector. More specifically, we propose the use of hair and skin color and face
location and size as a preliminary set.
Hair color is a significant human characteristic that can be effectively employed in user
queries to retrieve particular facial images. We have determined a scheme to categorize black,
gray/white, brown, and blonde hair colors within the HSV space. First, the H, S, and V
component histograms of the hair regions are formed and smoothened using the scale–space
filter defined earlier. The peak values from each histogram are subsequently determined and
used to form the appropriate classification. The following regions were suitably found from
our large sample set for the various categories of hair color:
(1) Black
Vp < 15%
(2) Gray
Sp < 20% ∩ Vp > 50%
(3) Brown
Sp ≥ 20% ∩ 15 ≤ Vp < 40%
(4) Blonde
20◦ < Hp < 50◦ ∩ Sp ≥ 20% ∩ Vp ≥ 40%
where Hp , Sp , and Vp denote the peaks of the corresponding histograms. Thus, dark or black
hair is characterized by low-intensity values and gray or white hair by low saturation and highintensity values. On the other hand, brown or blonde hair colors are typically well saturated
but differ in their intensity values. The expected value component of dark brown hair lies at
approximately Vp ≈ 20%, lighter brown at around Vp ≈ 35%, and blonde hair at higher
values, Vp ≥ 40%. Therefore, we can use this information to appropriately categorize the
facial regions extracted earlier. We use a suitably sized template above each facial area for the
©2001 CRC Press LLC
classification process as shown in Figure 5.4. The template consists of regions R1 + R2 + R3 .
This provides a fast yet good approximation to the overall description.
FIGURE 5.4
Template for hair color classification = R1 + R2 + R3 .
The next feature we propose to use is the average hue value of the facial area. We have
found that darker skin types tend to shift toward 0◦ (i.e., average hue = 8◦ for the darker skintype sample set), whereas lighter colored skin types move toward 30◦ [39]. In certain cases,
however, lighter skin types with a reddish appearance may also have a slightly reduced average
hue value (i.e., 15◦ ). Nevertheless, the hue sector can be partitioned to discriminate between
lighter and darker skin types as follows: (1) darker colored skin, H < 15◦ , and (2) lighter, H
≥ 15◦ . This can give us a reasonable approximation; however, we believe that the saturation
and value components can improve upon these results.
Finally, the location and size of each facial area (i.e., centroid location and size relative to
the image) can provide very useful information in a retrieval system. These combined features
can give us an indication of whether the face is a portrait shot or if perhaps the body is included.
In addition to this, it can also provide information about the spatial relationships of a particular
facial region with other objects or faces within the scene. Further work is being done in this
latter area.
5.5.6
Experimental Results
The scheme outlined in Figure 5.3 was used to locate and track the facial region in a number
of still images and video sequences. The results from three videophone-type sequences (i.e.,
newscast or interview-type sequences) are presented below: (1) Carphone, (2) Miss America,
and (3) Akiyo.
The segmentation results in Figure 5.5 illustrate the robustness of the technique to the various
cases of object/background motion, lighting, and scale variations. A parameter selection
of τ = 2 was made in the Gaussian function of equation (5.44) in order to smoothen the
histograms. This provided adequate smoothing and was found to be appropriate for the skin
tone distribution models [39]. A similar value [37] has also been suggested in the HVC space.
The shape and color analysis module was used to identify the facial regions from the set of
©2001 CRC Press LLC
FIGURE 5.5
Location and tracking of the facial region for the following video sequences: (a) Carphone, (b) Carphone frames 20–85, (c) Miss America, (d) Miss America frames 20–120,
(e) Akiyo, and (f) Akiyo frames 20–110.
candidate objects. An object was classified as a facial region if its overall membership function,
µc , exceeded a predefined threshold of 0.75. In the QCIF Carphone sequence of Figure 5.5a,
only one candidate region was extracted by the localization procedure in Figure 5.3, which
indeed corresponded to the facial area. In Figure 5.5c, a similar procedure was followed with
the CIF Miss America sequence. In this case, three objects of significant size were extracted,
and the results of these are summarized in Table 5.6.
Only the first object was selected, based on the aggregation of the membership function
values. The objects O2 and O3 were rejected because they scored poorly in their mean hue
value and location and had reduced membership values in the orientation primitive. Finally, in
©2001 CRC Press LLC
Table 5.6 Miss America (Width ×
Height =360×288): Shape and Color
Analysis
Attributes
O1
Objects
O2
O3
Centroid location
x
µ1
y
µ2
177
1
188
1
245
0
120
1
244
0
269
0.02
4.92
1
47.74
0.7
44
0.77
1.61
1
1.16
0.82
1.32
1
µ5
20
1
-6
0.5
-5
0.54
Aggregation
1.0
0.0
0.0
Orientation
θ◦
µ3
Object ratio
r
µ4
Mean hue
Hm ( ◦ )
Figure 5.5e, the facial region was successfully identified and tracked for the Akiyo sequence.
Two candidate objects were extracted in this case and, once again, the face was correctly
selected based on the aggregation values.
Once the facial region is identified, the proposed meta-data features can be computed according to the methodology provided in the previous section. The feature values for each of
the image sequences are summarized in Table 5.7. The average hue value of the facial area
(i.e., skin) is in all three cases greater than 20◦ , which puts them in the lighter skin category,
as expected. Next, we observe the Sp and Vp values of the hair region obtained from our
constructed template. According to our classification scheme, the tabulated values indicate
that the facial image in the Carphone sequence has brown hair, whereas the other two have
black hair. These fuzzy descriptions are appropriate representations of the images shown in
Figure 5.5. Finally, the last two features give us an indication of the location and size of the
face within the scene. In all cases, the facial region is relatively close to the center of the image
(location is with respect to the top left corner) and is of significant size (i.e., a closeup).
5.6
Conclusions
The tremendous advances in both software and hardware have brought about the integration
of multiple media types within a unified framework. This has allowed the merging of video,
audio, text, and graphics with enormous possibilities for new applications. This integration
is at the forefront in the convergence of the computer, telecommunications, and broadcast
industries. The realization of these new technologies and applications, however, demands a
new way of processing audiovisual information. We have shifted from pixel-based models
(pulse code modulation) to statistically dependent pixel models (transform coding) to the
©2001 CRC Press LLC
Table 5.7 Proposed Meta-Data Feature Values
Meta-Data
Skin color
H (◦ )
Carphone
Miss America
Akiyo
24
20
23
Hair color
S (%)
V (%)
Description
29
30
Brown
38
15
Black
16
12
Black
Centroid location
Horizontal (%)
Vertical (%)
40
42
53
60
51
47
Image height/face height
1.7
1.5
1.8
current audiovisual object-based approaches (MPEG-7).
In this chapter we have focused on several aspects of the intelligent processing of visual
information using color imaging techniques and fuzzy concepts. We have applied this methodology to three problem areas, namely: (1) color image filtering, (2) color segmentation, and
(3) automatic face localization and meta-data generation.
Digital images and video signals suffer from several degradations and artifacts. These may
include sensor noise and lens aberrations from commercial camcorders or artifacts from the
digitization process from analog sources. In some cases this may be acceptable, but for highresolution multimedia applications they become objectionable. In Section 5.3 we presented a
new class of filters based on a fuzzy multichannel filtering structure. Our new adaptive design
was computationally efficient, scalable, nonrecursive, and did not require a training signal.
The application of our fuzzy filters to a number of noise-contaminated color images indicated
a performance that improved upon the conventional vector processing filters.
One of the challenges in the representation of visual information is to decompose a video
sequence into its elementary parts. Temporal segmentation refers to finding shot boundaries,
whereas spatial segmentation corresponds to the extraction of visual objects in each frame. In
Section 5.4 we have addressed the spatial segmentation problem using the visual cue of color.
The proposed color segmentation scheme utilized the perceptual HSV color model to effectively partition a sequence into a set of arbitrarily shaped regions. The method was found to be
robust and of relatively low computational complexity due to the 1D histogram procedure and
the binary nature of the postprocessing operations involved. Thresholds for merging regions
and removing small areas were used to avoid the problems associated with oversegmented
results. The effectiveness of this technique and its potential for a suitable content-based representation is encouraging for future object-based video coding environments. The incorporation
of motion information into the segmentation problem holds more promising results. Automatic
segmentation methods are challenged in their quest to extract semantically meaningful objects
as opposed to simple regions. The combination of spatial and temporal segmentation with
object tracking algorithms remains an active area of research. These solutions will enable true
object-based compression schemes and effective indexing and retrieval of the content.
The automatic extraction of facial images in digital pictures is vital in numerous multimedia
applications, including multimedia databases, video on demand, human–computer interfaces,
and video coding. In Section 5.5, a novel technique was introduced to locate and track the
facial area in videophone-type sequences. The proposed method essentially consisted of two
components: (1) a color processing unit and (2) a knowledge-based shape and color analysis
module. The color processing component utilized the HSV color space, while the shape module
©2001 CRC Press LLC
employed a number of fuzzy membership functions to correctly identify the facial region. The
suggested approach was robust with regard to different skin types and various types of object
or background motion within the scene. Having determined the facial regions within an image,
we then constructed a meta-data feature vector that could be used for content-based storage and
retrieval purposes. Meta-data features such as hair and skin color and face location and size
were utilized as a preliminary set. The results of our findings were encouraging in extracting
vital information from facial images. Efforts for content-based video description are an active
research topic. It is highly desirable to index multimedia data using visual features such as
color, texture, and shape; sound features such as audio and speech; and textual features such
as script and closed captioning. It is also of great interest to have the capabilities to browse
and search for this content using compressed data since most video data will likely be stored in
compressed formats. Another area of interest is in temporal segmentation, where it is important
to extract shots, scenes, or objects. Furthermore, higher level descriptions for the direction and
magnitude of dominant object motion and the entry and exit instances of objects of interest
are highly desirable. These are all future research areas to be investigated and fueled with the
upcoming MPEG-7 standard.
In this chapter we have examined the concepts of adaptive fuzzy systems and color processing
for several multimedia applications. More specifically, the algorithms and architectures were
examined; however, further analysis is warranted to address issues of real-time architectures
and realizations, modularity, software portability, and system robustness.
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©2001 CRC Press LLC
Chapter 6
Intelligent Multimedia Processing
Ling Guan, Sun-Yuan Kung, and Jenq-Neng Hwang
6.1
Introduction
The way we access information, conduct business, communicate, educate, learn, and entertain will be profoundly changed by the rapid development of multimedia technologies [17,
79, 124]. Multimedia technologies also represent a new opportunity for research interactions
among a variety of media such as speech, audio, image, video, text, and graphics. As digitization and encoding of images and video have become more affordable, computer and Web
database systems are starting to store voluminous image and video data. Consequently, massive amounts of visual information online have become closer to a reality. This promises a
quantum elevation of the level of tomorrow’s world in entertainment and business. However,
as the data acquisition technology advances rapidly, we have now substantially fallen behind
in terms of technologies for indexing and retrieval of visual information in large archives.
For example, it would be desirable to have a tool that efficiently searches the Web for a
desired picture (or video clip) and/or audio clip by using as a query a shot of multimedia
information [119, 135]. Nowadays, some popular queries might look like: “Find frames
with 30% blue on top and 70% green in bottom” or “Find the images or clips similar to this
drawing.” In contrast to the above similarity-based queries, it has been argued that a so-called
“subject-based” query [135] might be more likely to be used — for example, “Find Reagan
speaking to the Congress.” The subject-based query offers a more user-friendly interface, but
it also introduces a greater technical challenge, which calls for advances in two distinctive
research frontiers [17]:
• Computer networking technology. Novel communication and networking technologies are critical for multimedia database systems to support interactive dynamic interfaces. A truly integrated media system must connect with individual users and contentaddressable multimedia databases. This will involve both logical connection to support
information sharing and physical connection via computer networks and data transfer.
• Information processing technology. To advance the technologies of indexing and retrieval of visual information in large archives, multimedia content-based indexing would
complement well the text-based search. Online and real-time visual information retrieving and display systems would provide popular services to professionals such as business
traders, researchers and librarians as well as general users such as students and housewives. Such systems must successfully combine digital video and audio, text animation,
graphics, and knowledge about such information units and their interrelationships in real
time.
©2001 CRC Press LLC
This chapter addresses mainly emerging issues closely related to the research frontier on
information processing technology.
Because speech, image, and video are playing increasingly dominant roles in multimedia
information processing, content-based retrieval has a broad spectrum of applications. Hence,
quick and easy access of large speech, image, and video databases must be incorporated as an
integral part of many near-future multimedia applications. Future multimedia technologies will
need to handle information with an increasing level of intelligence (i.e., automatic extraction,
recognition, interpretation, and interactions of multimodal signals). This will lead to what can
be called intelligent multimedia processing (IMP) technology.
Indeed, the technology frontier of information processing is shifting from coding (MPEG1 [83], MPEG-2 [84], and MPEG-4 [85]) to automatic recognition — a trend precipitated by a
new member of the MPEG family, MPEG-7 [86, 87], which focuses on the “multimedia content
description interface.” Its research domain will cover techniques for object-based tracking and
segmentation, pattern detection and recognition, content-based indexing and retrieval, and
fusion of multimodal signals. For these, neural networks (NNs), sometimes in combination
with two other branches of computational intelligence (CI), fuzzy system (FS) and evolutionary
computation (EC), can offer a very promising horizon.
6.1.1
Neural Networks and Multimedia Processing
The main reason CI is perceived as a critical core technology for IMP hinges on its learning,
adaptation, reasoning, and evolution capability [4, 57], which enables machines to be taught
to interpret possible variations of the same object or pattern (e.g., scale, orientation, and
perspective).
More specifically, to build an IMP system, the emerging synthesis of various techniques is
required. Each technique plays a specific role in IMP systems. The main characteristics of NNs
are to recognize patterns and to classify input, and to adapt themselves to dynamic environments
by learning; but the mapping structure of an NN is a black box. The resulting NN behavior
is difficult to understand. An FS, on the other hand, can cope easily with human knowledge
and can perform inference, but it does not fundamentally incorporate the learning mechanism.
Neuro-fuzzy computing has developed for overcoming their respective disadvantages [46, 137].
In general, the neural network part is used for learning, whereas the fuzzy logic part is used
for representing knowledge. The learning is fundamentally performed as a necessary change
such as incremental learning, back-propagation, and unsupervised learning schemes. EC can
also tune NNs and FSs. Furthermore, EC has been used for the structure optimization of NNs
and FSs [46, 137]. However, evolution can be defined as a resultant change, not a necessary
change, because EC cannot predict and estimate the effect of the change. To summarize, an
IMP system can quickly adapt to a dynamically changing environment by NNs and FSs, and
the structure of the system can globally evolve by ECs. The capability concerning adaptation
and evolution can construct more advanced IMP systems.
Among the three branches of CI, NNs have been the most popular tool for IMP because
• Neural networks offer unsupervised clustering and/or supervised learning mechanisms
for recognition of objects which are deformed or have incomplete information. Therefore, NNs can be “trained” to see or hear, to recognize objects or speech, or to perceive
human gestures.
• Neural networks are powerful pattern classifiers that appear to be most powerful and
appealing when explicit a priori knowledge of underlying probability distributions is unknown, such that properly trained NN classifiers allow the nonparametric approximation
of the associated a posteriori class probabilities [101].
©2001 CRC Press LLC
• Neural networks offer a universal approximation capability, which allows accurate approximation of unknown systems based on sparse sets of noisy data. In this context,
some neural models have also effectively incorporated statistical signal processing and
optimization techniques.
• Temporal neural models, which are specifically designed to deal with temporal signals,
further expand the application domain in multimedia processing, particularly audio,
speech, and audiovisual integration and interactions.
• A hierarchical network of neural modules will be vital to facilitate search mechanisms
used in a voluminous, or Web-wide, database. Typically, in a tree network structure,
kernels that are common to all the models form the root of the tree. The leaves of the
tree correspond to the individual neural modules, whereas the paths from root to leaf
connect the modules to their respective kernels.
Consequently, NNs have recently received increasing attention in many multimedia applications. Here we list just a few examples: (1) human perception: facial expression and
emotion categorization [105], human color perception [109], and multimedia data visualization [3, 102]; (2) computer–human communication: face recognition [70], lipreading analysis [19, 20, 65, 95], and human–human and computer–human communication/interaction [89];
and (3) multimodal representation and information retrieval: hyperlinking of multimedia objects [64], queries and searches of multimedia information [75], 3D object representation and
motion tracking [118], and image sequence generation and animation [78]. More concrete
application examples will be discussed in the subsequent sections.
6.1.2
Focal Technical Issues Addressed in the Chapter
This chapter will focus on vital technical issues in the research frontier on information processing technology, particularly those closely related to IMP. More specifically, this chapter
will demonstrate why and how CI, with neural networks in particular, offers as a core technology for: efficient representations for audiovisual information (Section 6.2.1); detection
and classification techniques (Section 6.2.2); fusion of multimodal signals (Section 6.2.3);
and multimodal conversion and synchronization (Section 6.2.4). Here let us first offer some
motivations as well as a brief explanation on the key technical points.
Efficient Representations for Audiovisual Information
An efficient representation of the information can facilitate many useful multimedia functionalities, such as object-based indexing and access. To this end, it is vital to have sophisticated
preprocessing of the image or video data. For many multimedia applications, preprocessing
is usually carried out on the input signals to make the subsequent processing modeling and
classification tasks easier (e.g., segmentation of 2D or 3D images and video for content-based
coding and representation in the context of the MPEG or JPEG standards). The more sophisticated the representation obtained by preprocessing, the less sophisticated the classifier would
need to be. Hence, a synergistic balance (and eventually interaction) between representation
and indexing needs to be explored.
An efficient representation of vast amounts of multimedia data can often be achieved by adaptive data clustering or model representation mechanisms, which happen to be the most promising strength of many well-established unsupervised neural networks [e.g., self-organizing
feature map (SOFM) and principal component analysis (PCA) neural network]. The evolution
from conventional statistical clustering and/or contour and shape modeling to these unsupervised NNs will be highlighted in Section 6.2.1.
©2001 CRC Press LLC
Some of these NNs have been incorporated for various feature extraction, moving object
tracking, and segmentation applications. Illustrative samples for such preprocessing examples
are provided in Section 6.3.1.
Detection and Classification for Audiovisual Databases
As most digital text, audio, and visual archives exist on various servers throughout the world,
it becomes increasingly difficult to locate and access the information. It thus necessitates
automatic search tools for indexing and access. Detection and classification constitute a very
basic tool for most search and indexing mechanisms. Detection of a (deformable) pattern or
object has long been an important machine learning and computer vision problem. The task
involves finding a specific (but locally deformable) pattern in images (e.g., human faces). What
is critically needed are powerful search strategies to identify contents on speech or visual clues,
possibly without the benefit of textual information. These will have important commercial
applications, including automatic teller machine (ATM), access control, surveillance, and
video conferencing systems.
Several static supervised NNs (i.e., no feedback connections are used in the network),
which are useful for detection and classification, will be covered in Section 6.2.2. Built upon
these NNs, many NN content-based image search systems have been developed for various
applications. On the horizon are several promising tools which allow users to specify image
queries by giving examples, drawing sketches, selecting visual features (e.g., color, texture,
shape, and motion), and arranging the spatiotemporal structure of features. Some exemplar NN
systems will be presented in Section 6.3.4. They serve to demonstrate the fact that unsupervised
and supervised NN models are useful means for developing reliable search mechanisms.
Multimodal Media Fusion: Combine Multiple Sources
Multimedia signal processing is more than simply “putting together” text, audio, images,
and video. The correlation between audio and video can be utilized to achieve more efficient
coding and recognition. New application systems and thus new research opportunities arise in
the area of fusion and interaction among these media.
Humans perform most perception and recognition tasks based on joint processing of the
input multimodal data. The biological cognitive machines of humans handle multimodal
data through visual, auditory, and sensory mechanisms via some form of adaptive processing
(learning/retrieving) algorithms, which remain largely mysterious to us. Motivated by the
nature of biological information processing, fusion NN models which combine information
from multiple sensor and data sources are being pursued as a universal data processing engine
for multimodal signals. Linear fusion networks and nonlinear fusion networks are discussed
in Section 6.2.3.
Audio–video interaction can be used for personal authentication and verification. A visual/auditory fusion network for such an application is discussed in Section 6.3.2.
Multimodal Conversion and Synchronization
One of the most interesting interactions among different media is the one between audio and
video. In multimodal speech communication, audio–video interaction has a significant role,
as evidenced by the McGurk effect [74]. It shows that human perception of speech is bimodal
in that acoustic speech can be affected by visual cues from lip movements. For example, one
experiment showed that when a person sees a speaker saying /ga/, but hears the sound /ba/,
the person perceives neither /ga/ nor /ba/, but something close to /da/. In video conferencing
applications, it is conceivable that the video frame rate is severely limited by the bandwidth
and is by far very inadequate for lip synchronization perception. One solution is to warp the
©2001 CRC Press LLC
acoustic signal to synchronize it with the person’s mouth movements, which will be useful for
dubbing in a studio and other non-real-time applications.
There is a class of temporal neural models (i.e., feedback connections are used to keep
track of temporal correlation of signals) that can facilitate the conversion and synchronization
processes. Prominent temporal NN models and popular statistical approaches will be reviewed
in Section 6.2.4. Verbal communication has been efficiently achieved by combining speech
recognition and visual interpretation of lip movements (or even facial expressions or body language). As another example, an NN-based lipreading system via audio and visual integration
will be presented in Section 6.3.3. Other potential applications include dubbing of movies,
segmentation of video scenes, and human–computer interfaces.
6.1.3
Organization of the Chapter
Section 6.2 reviews some of the key NNs, then highlights their usefulness to IMP applications. Built upon these NN models, exemplar IMP applications will be illustrated in Section 6.3.
Some open technical issues and promising application trends will be suggested in Section 6.4.
6.2
Useful Neural Network Approaches to Multimedia Data
Representation, Classification, and Fusion
We will discuss in this section a variety of statistical learning techniques adopted by NNs.
Through these techniques, machines may be taught to automatically interpret and represent
possible variations of the same object or pattern. Some of these NNs (e.g., the self-organization
feature map) can be perceived as a natural evolution from traditional statistical clustering and
parameter estimation techniques (e.g., vector quantization (VQ) and expectation maximization). These NNs can also be incorporated into traditional pattern recognition techniques (e.g.,
active contour model) to enhance the performance.
6.2.1
Multimedia Data Representation
From the learning perspective, neural networks are grouped into unsupervised learning and
supervised learning networks. Static features extraction is often inadequate for an adaptive
environment where users may require adaptive and dynamic feature extraction tools. Unsupervised neural techniques are very amenable to dynamic feature extraction. The SOFM is
one representative of an unsupervised NN, which combines the advantages of statistical data
clustering (such as vector quantization and PCA) and local continuity constraint (as imposed
in the active contour model search).
Self-Organizing Feature Map (SOFM)
The basic idea of constructing an SOFM is to incorporate into the competitive learning
(clustering) rule some degree of sensitivity with respect to the neighborhood or history. This
provides a way to avoid totally uncommitted neurons, and it helps enhance certain topological
properties which should be preserved in the feature mapping (or data clustering).
Suppose that an input pattern has n features and is represented by a vector x in an ndimensional pattern space. The network maps the input patterns to an output space. The
output space in this case is assumed to be 1D or 2D arrays of output nodes, which possess
a certain topological ordering. The question is how to cluster these data so that the ordered
©2001 CRC Press LLC
relationship can be preserved. Kohonen proposed to allow the centroids (represented by output
nodes of an SOFM) to interact laterally, leading to the self-organizing feature map [52, 53],
which was originally inspired by a biological model.
The most prominent feature is the concept of excitatory learning within a neighborhood
around the winning neuron. The size of the neighborhood slowly decreases with each iteration.
A version of the training rule is described below:
1. First, a winning neuron is selected as the one with the shortest Euclidean distance (nearest
neighbor),
x − wi ,
between its weight vector and the input vector, where wi denotes the weight vector
corresponding to the ith output neuron.
2. Let i ∗ denote the index of the winner and let I ∗ denote a set of indices corresponding to
a defined neighborhood of winner i ∗ . Then the weights associated with the winner and
its neighboring neurons are updated by
wj = η x − wj ,
for all the indices j ∈ I ∗ , and η is a small positive learning rate. The amount of updating
may be weighted according to a preassigned “neighborhood function,” (j, i ∗ ).
wj = η j, i ∗ x − wj ,
(6.1)
for all j . For example, a neighborhood function (j, i ∗ ) may be chosen as
2
j, i ∗ = exp − rj − ri ∗ /2σ 2
(6.2)
where rj represents the position of the neuron j in the output space. The convergence of
the feature map depends on a proper choice of η. One plausible choice is that η = 1/t,
where t denotes the iteration number. The size of neighborhood (or σ ) should decrease
gradually.
3. The weight update should be immediately succeeded by the normalization of wi .
In the retrieving phase, all the output neurons calculate the Euclidean distance between the
weights and the input vector and the winning neuron is the one with the shortest distance.
By updating all the weights connecting to a neighborhood of the target neurons, the SOFM
enables the neighboring neurons to become more responsive to the same input pattern. Consequently, the correlation between neighboring nodes can be enhanced. Once such a correlation
is established, the size of a neighborhood can be decreased gradually, based on the desire of
having a stronger identity of individual nodes.
Application examples: There are many examples of successful applications of SOFMs. More
specifically, the SOFM network was used to evaluate the quality of a saw blade by analyzing
its vibration measurements, which ultimately determines the performance of a machining
process [7]. The major advantage of SOFMs is their unsupervised learning capability, which
makes them ideal for machine health monitoring situations (e.g., novelty detection in medical
images can then be performed online or classes can be labeled to give diagnosis [35]). A
good system configuration algorithm produces the required performance and reliability with
maximum economy. Actual design changes are frequently kept to a minimum to reduce the risk
of failure. As a result, it is important to analyze the configurations, components, and materials
of past designs so that good aspects may be reused and poor ones changed. A generic method
©2001 CRC Press LLC
of configuration evaluation based on an SOFM has been successfully reported [88]. The
SOFM architecture with activation retention and decay in order to create unique distributed
response patterns for different sequences has also been successfully proposed for mapping
between arbitrary sequences of binary and real numbers, as well as phonemic representations
of English words [45]. By using a selective learnable SOFM, which has the special property
of effectively creating spatially organized internal representations and nonlinear relations of
various input signals, a practical and generalized method was proposed in which effective
nonlinear shape restoration is possible regardless of the existence of distortion models [34].
There are many other examples of successful applications (e.g., [24, 62, 111]).
The Self-Organizing Tree Map (SOTM)
The motivation for the self-organizing tree map (SOTM) [54] — SOFM with a hierarchical
structure is different from Kohonen’s motivation for the original SOFM — is a nonparametric
regression model, but it is an effective tool for accurate clustering/classification leading to
segmentation and other image/multimedia processing applications.
The SOFM is a good clustering method, but it has some undesirable properties when an input
vector distribution has a prominent maximum. The results of the best-match computations tend
to be concentrated on a fraction of nodes in the map. Therefore, the reference vectors lying in
zero-density areas may be affected by input vectors from the surrounding nonzero distribution
areas. Such phenomena are largely due to the nonparametric regression nature of the SOFM.
In order to overcome the aforementioned problems, tree-structured SOFMs were proposed.
A typical example is the SOTM [54]. The main characteristic of the SOTM is that it exhibits
better fitting of the input data.
In the SOTM, the relationships between the output nodes are defined adaptively during
learning. Unlike the SOFM, which has a user-predefined and fixed number of nodes in the
network, the number of nodes is determined automatically by the learning process based on
the distribution of the input data. The clustering algorithm starts from an isolated node and
coalesces the nearest patterns or groups according to a hierarchy control function from the root
node to the leaf nodes to form the tree. The proposed approach has the advantage of K-means,
with their ability to accurately locate cluster centers, and the SOFM’s topology-preserving
property. The SOTM also provides a better and faster approximation of prominently structured
density functions.
Using the definitions of the input vector x(t) and the weight vector wj (t), the SOTM
algorithm is summarized as follows:
1. Select the winning node j ∗ with minimum Euclidean distance dj ,
dj ∗ x, wj ∗ = min dj x, wj
j
2. If dj ∗ (x, wj ∗ ) ≤ H (t) where H (t) is the hierarchy control function, which controls the
number of levels of the tree and decreases with time, then assign x to the j th cluster and
update the weight vector wj according to the following learning rule:
(6.3)
wj ∗ (t + 1) = wj ∗ (t) + η(t) x(t) − wj ∗ (t)
where η(t) = e(−t/T1 ) (with T1 determining the rate of convergence) is the learning rate,
which decreases with time and satisfies 0 < η(t) < 1.
Else form a new subnode with x as the weight vector.
3. Repeat by going back to step 1.
©2001 CRC Press LLC
The hierarchy control function H (t) = e(−t/T2 ) (with T2 being a constant which regulates
the rate of decrease) controls the number of levels of the tree. It adaptively partitions the input
vector space into subspaces.
With the decrease of the hierarchy control function H (t), a subnode forms a new branch.
The evolution process progresses recursively until it reaches the leaf node. The entire tree
structure preserves topological relations from the root node to the leaf nodes.
The SOTM is much better than the SOFM at preserving the topological relations of the input
dataset, as shown in the example. The learning of the tree map in Figure 6.1a is driven by
sample vectors uniformly distributed in the English letter “K.” The tree mapping starts from
the root node and gradually generates its subnodes as H (t) decreases. By properly controlling
the rate of decrease α(t), the final representation of the letter “K” is shown in Figure 6.1b. For
comparison, the SOFM is also used in this example, as shown in Figure 6.1c. The superiority
of the SOTM is apparent.
The other tree-structured SOFM models that share many similarities with the SOTM include
the self-generating neural networks [128], the hierarchical SOTM [51], and the self-partitioning
neural networks [104].
Application examples: The SOTM and the other tree-structured SOFMs have been used in
many image and multimedia applications. Self-generating neural networks have been applied
to visual communications [128], the hierarchical SOFM for range image segmentation [51], the
self-partitioning neural networks for target detection and recognition [104], and the SOTM for
quality cable TV transmission [54], image segmentation, and image/video compression [55].
Principal Component Analysis (PCA)
Principal component analysis (PCA) provides an effective way to find representative components of a large set of multivariate data. The basic learning rules for extracting principal
components follow the Hebbian rule and the Oja rule [57, 94]. PCA can be implemented using
an unsupervised learning network with traditional Hebbian-type learning. The basic network
is one where the neuron is a simple linear unit with output a(t) defined as follows:
a(t) = w(t)T x(t) .
(6.4)
To enhance the correlation between the input x(t) and the output a(t), it is natural to use a
Hebbian-type rule:
w(t + 1) = w(t) + βx(t)a(t) .
(6.5)
The above Hebbian rule is impractical for PCA, taking into account the finite-word-length
effect, since the training weights will eventually overflow (i.e., exceed the limit of dynamic
range) before the first component totally dominates and the other components sufficiently
diminish. An effective technique to overcome the overflow problem is to keep normalizing the
weight vectors after each update. This leads to the Oja learning rule or, simply, the Oja rule:
(6.6)
w(t + 1) = w(t) + β x(t)a(t) − w(t)a(t)2 .
In contrast to the Hebbian rule, the Oja rule is numerically stable.
For the extraction of multiple principal components, a lateral network structure was proposed [57]. The structure incorporates lateral connections into the network. The structure,
together with an orthogonalization learning rule, helps ensure the preservation of “orthogonality” between multiple principal components. A numerical analysis on their learning rates and
convergence properties has also been established.
©2001 CRC Press LLC
(a
(b
FIGURE 6.1
The SOTM for representation: (a) English letter “K;” (b) the representation of “K” by
the SOTM; (c) the representation of “K” by the SOFM. (Cont.).
©2001 CRC Press LLC
(c
FIGURE 6.1
(Cont.) The SOTM for representation: (a) English letter “K;” (b) the representation of
“K” by the SOTM; (c) the representation of “K” by the SOFM.
Application examples: The lipreading system of Bregler and Konig [8], an early attempt in
using both audio and visual features, used PCA to guide the snake search (the so-called active
shape models [23]) on gray-scale video for the visual front end. There are two ways to perform
PCA: (1) contour-based PCA is directly based on the located points from the snake search
(form feature vectors using the located points and projected onto a few principal components);
(2) area-based PCA is directly based on the gray-level matrix surrounding the lips. Instead of
reducing the dimensionality of the visual features, as performed by the contour-based KLT,
one can reduce the variation of mouth shapes by summing fewer principal components to
form the contours. It was concluded that gray-level matrices contain more information for
classifying visemes. Another attempt in PCA-based lip motion modeling is to express the
PCA coefficients as a function of a limited set of articulatory parameters which describe the
external appearance of the mouth [66]. These articulatory parameters have been directly
estimated from the speech waveform based on a bank of (time-delay) NNs. A PCA-based
Eigenface technique for a face recognition algorithm was studied in [6]. Its performance
was compared with a computationally compatible “Fisherface” method based on tests on the
Harvard and Yale Face Databases.
6.2.2
Multimedia Data Detection and Classification
In many application scenarios [e.g., optical character recognition (OCR), texture analysis,
face detection] several prior examples of a targeted class or object are available for training,
whereas the a priori class probability distribution is unknown. These training examples may
be best exploited as valuable teacher information in supervised learning models. In general,
detection and classification based on supervised learning models by far outperform those via
©2001 CRC Press LLC
unsupervised clustering techniques. That is why supervised neural networks are generally
adopted for detection and classification applications.
Multilayer Perceptron
Multilayer perceptron (MLP) is one of the most popular NN models. In this model, each
neuron performs a linear combination on its inputs. The result is then nonlinearly transformed
by a sigmoidal function. In terms of structure, the MLP consists of several layers of hidden
neuron units between the input and output neuron layers. The most commonly used learning
scheme for the MLP is the back-propagation algorithm [106]. The weight updating for the
hidden layers is performed based on a back-propagated corrective signal from the output
layer. It has been shown that the MLP, given its flexible network/neuron dimensions, offers a
universal approximation capability. It was demonstrated in [129] that two-layer perceptrons
(i.e., networks with one hidden layer only) should be adequate as universal approximators of
any nonlinear functions.
Let us assume an L-layer feed-forward neural network (with Nl units at the lth layer). Each
unit, say the ith unit at the (l +1)th layer, receives the weighted inputs from other units at the lth
layer to yield the net input ui (l +1). The net input value ui (l +1), along with the external input
θi (l + 1), will determine the new activation value ai (l + 1) by the nonlinear activation function
fi (l + 1). From an algorithmic point of view, the processing of this multilayer feed-forward
neural network can be divided into two phases: retrieving and learning.
Retrieving phase: Suppose that the weights of the network are known. In response to the input
(test pattern) {ai (0), i = 1, . . . , N0 }, the system dynamics in the retrieving phase of an L-layer
MLP network iterate through all the layers to generate the response {ai (L), i = 1, . . . , NL } at
the output layer.
ui (l + 1) =
Nl
wij (l + 1)aj (l) + θi (l + 1)
j =1
ai (l + 1) = fi (ui (l + 1)) = fi (l + 1)
(6.7)
where 1 ≤ i ≤ Nl+1 , 0 ≤ l ≤ L − 1, and fi is nondecreasing and differentiable (e.g., sigmoid
function [106]). For simplicity, the external inputs {θi (l + 1)} are often treated as special
modifiable synaptic weights {wi,0 (l + 1)} which have clamped inputs a0 (l) = 1.
Learning phase: The learning phase of this L-layer MLP network follows a simple gradient
descent approach. Given a pair of input/target training patterns, {ai (0), i = 1, . . . , N0 }, {tj ,
j = 1, . . . , NL }, the goal is to iteratively (by presenting a set of training pattern pairs many
times) choose a set of {wij (l), ∀l} for all layers so that the squared error function E can be
minimized:
E=
NL
1 (ti − ai (L))2
2
(6.8)
i=1
To be more specific, the iterative gradient descent formulation for updating each specific weight
wij (l) given a training pattern pair can be written as
wij (l) ⇐ wij (l) − η
∂E
∂wij (l)
(6.9)
where ∂w∂E
can be computed effectively through a numerical chain rule by back-propagating
ij (l)
the error signal from the output layer to the input layer.
©2001 CRC Press LLC
Other popular learning techniques of MLPs include discriminative learning [49], the support
vector machine [36], and learning by evolutionary computation [137].
Due to the popularity of MLPs, it is not possible to exhaust all the numerous IMP applications
using them. For example, Sung and Poggio [114] used MLP for face detection and Huang [40]
used it as preliminary channels in an overall fusion network. More details about using MLPs
for multimodal signal will be discussed in the audiovisual processing section.
RBF and OCON Networks
Another type of feed-forward network is the radial basis function (RBF) network. Each
neuron in the hidden layer employs an RBF (e.g., a Gaussian kernel) to serve as the activation
function. The weighting parameters in the RBF network are the centers, the widths, and the
heights of these kernels. The output functions are the linear combination (weighted by the
heights of the kernels) of these RBFs. It has been shown that the RBF network has the same
universal approximation power as an MLP [98].
The conventional MLP adopts an all-class-in-one-network (ACON) structure, in which
all the classes are lumped into one supernetwork. The supernet has the burden of having
to simultaneously satisfy all the teachers, so the number of hidden units tends to be large.
Empirical results confirm that the convergence rate of ACON degrades drastically with respect
to the network size because the training of hidden units is influenced by (potentially conflicting)
signals from different teachers [57].
In contrast, it is natural for the RBF to adopt another type of network structure — the oneclass-in-one-network (OCON) structure — where one subnet is designated to one class only.
The difference between these two structures is depicted in Figure 6.2. Each subnet in the
OCON network specializes in distinguishing its own class from the others, so the number of
hidden units is usually small. In addition, OCON structures have the following features:
FIGURE 6.2
(a) An ACON structure; (b) an OCON structure.
©2001 CRC Press LLC
• Locally, unsupervised learning may be applied to determine the initial weights for individual subnets. The initial clusters can be trained by VQ or K-mean clustering techniques. If the cluster probabilities are desired, the EM algorithm can be applied to
achieve maximum likelihood estimation for each class conditional likelihood density.
• The OCON structure is suitable for incremental training (i.e., network upgrading through
the addition/removal of memberships [57, 58]).
• The OCON network structure supports the notion of distributed processing. It is appealing to smart card biometric systems. An OCON-type classifier can store personal
discriminant codes in individual class subnets, so the magnet strip in the card needs to
store only the network parameters in the subnet that have been designated to the card
holder.
Application examples: In [11], Brunelli and Poggio proposed a special type of RBF network called the “hyperBF” network for successful face recognition applications. In [72], the
associated audio information is exploited for video scene classification. Several audio features have been found to be effective in distinguishing audio characteristics of different scene
classes. Based on these features, a neural net classifier can successfully separate audio clips
from different TV programs.
Decision-Based Neural Network
A decision-based neural network (DBNN) [58] has two variants: one is a hard-decision
model and the other is a probabilistic model. A DBNN has a modular OCON network structure:
one subnet is designated to represent one object class. For multiclass classification problems,
the outputs of the subnets (the discriminant functions) will compete with each other, and the
subnet with the largest output value will claim the identity of the input pattern.
Decision-Based Learning Rule The learning scheme of the DBNN is decoupled into two
phases: locally unsupervised and globally supervised learning. The purpose is to simplify the
difficult estimation problem by dividing it into several localized subproblems and, thereafter,
the fine-tuning process would involve minimal resources.
• Locally Unsupervised Learning: VQ or EM Clustering Method
Several approaches can be used to estimate the number of hidden nodes, or the initial
clustering can be determined based on VQ or EM clustering methods.
– In the hard-decision DBNN, the VQ-type clustering (e.g., K-mean) algorithm can
be applied to obtain initial locations of the centroids.
– For the probabilistic DBNN, called PDBNN, the EM algorithm can be applied
to achieve the maximum likelihood estimation for each class conditional likelihood density. (Note that once the likelihood densities are available, the posterior
probabilities can be easily obtained.)
• Globally Supervised Learning
Based on this initial condition, the decision-based learning rule can be applied to further
fine-tune the decision boundaries. In the second phase of the DBNN learning scheme,
the objective of the learning process changes from maximum likelihood estimation
to minimum classification error. Interclass mutual information is used to fine-tune the
decision boundaries (i.e., the globally supervised learning). In this phase, DBNN applies
the reinforced–antireinforced learning rule [58], or discriminative learning rule [49], to
©2001 CRC Press LLC
adjust network parameters. Only misclassified patterns are involved in this training
phase.
• Reinforced–Antireinforced Learning Rules
Suppose that the mth training pattern x(m) is known to belong to class #i , and that the
leading challenger is denoted as j = arg maxj =i φ(x(m) , wj ). The learning rule is
(m+1)
(m)
= wi + η∇φ x(m) , wi ,
Reinforced learning:
wi
(m+1)
(m)
= wj − η∇φ x(m) , wj .
Antireinforced learning: wj
Application examples: DBNN is an efficient neural network for many pattern classification
problems, such as OCR and texture classification [57] and face and palm recognition problems [68, 71]. A modular neural network based on DBNN and a model-based neural network
have recently been proposed for interactive human–computer vision.
Mixture of Experts
Mixture of experts (MOE) learning [44] has been shown to provide better performance due
to its ability to effectively solve a large complicated task by smaller and modularized trainable
networks (i.e., experts), whose solutions are dynamically integrated into a coherent one using
the trainable gating network. For a given input x, the posterior probability of generating class
y given x using K experts is computed by
P (y|x, φ) =
K
gi P (y|x, θi ) ,
(6.10)
i=1
where y is a binary vector, φ is a parameter vector [v, θi ], gi is the probability for weighting
the expert outputs, v is a vector of the parameters for the gating network, θi is a vector of the
parameters for the ith expert network (i = 1, . . . , K), and P (y|x, θi ) is the output of the ith
expert network.
The gating network can be a nonlinear neural network or a linear neural network. To obtain
the linear gating network output, the softmax function is utilized [10]:
gi = exp (bi ) /
K
exp bj
(6.11)
j =1
where bi = viT x, with vi denoting the weights of the ith neuron of the gating network.
The learning algorithm for the MOE is based on the maximum likelihood principle to
estimate the parameters (i.e., choose parameters for which the probability of the training set
given the parameters is the largest). The gradient ascent algorithm can be used to estimate the
parameters.
Assume that the training dataset is {x(t) , y(t) }, t = 1, . . . , N. First, we take the logarithm
of the product of N densities of P (y|x, φ):
(t)
l(y, x, φ) =
.
(6.12)
log gi P y(t) |x(t) , θi
t
i
Then, we maximize the log likelihood by gradient ascent. The learning rule for the weight
vector vi in a linear gating network is obtained as follows:
(t)
(t)
vi = ρ
(6.13)
hi − gi x(t) ,
t
©2001 CRC Press LLC
where ρ is a learning rate and hi = gi P (y|x, θi )/ t gj P (y|x, θj ).
The MOE [44] is a modular architecture in which the outputs of a number of “experts,”
each performing classification tasks in a particular portion of the input space, are combined
in a probabilistic way by a “gating” network which models the probability that each portion
of the input space generates the final network output. Each local expert network performs
multi-way classification over K classes by using either a K-independent binomial model, with
each model representing only one class, or one multinomial model for all classes.
Application example: The MOE model was applied to a time series analysis with wellunderstood temporal dynamics, and produced significantly better results than single networks.
It also discovered the regimes correctly. In addition, it allowed the users to characterize the
subprocesses through their variances and avoid overfitting in the training process [76]. A
Bayesian framework for inferring the parameters of an MOE model based on ensemble learning by variational free energy minimization was successfully applied to sunspot time series
prediction [126]. By integrating pretrained expert networks with constant sensitivity into an
MOE configuration, the trained experts are able to divide the input space into specific subregions with minimum ambiguity, which produces better performance in automated cytology
screening applications [42]. By applying a likelihood splitting criterion to each expert in the
HME, Waterhouse and Robinson [125] first grew the HME tree adaptively during training;
then, by considering only the most probable path through the tree, they pruned branches away,
either temporarily or permanently, in case of redundancies. This improved HME showed significant speedups and more efficient use of parameters over the standard fixed HME structure
for both simulation problems and real-world applications, as in the prediction of parameterized speech over short time segments [127]. The HME architecture has also been applied to
text-dependent speaker identification [16].
A Network of Networks
A network of networks (NoN) is a multilevel neural network consisting of nested clusters
of neurons capable of hierarchical memory and learning tasks. The architecture has a fractallike structure, in that each level of organization consists of interconnected arrangements of
neural clusters. Individual elements in the model form level zero cluster organization. Local
groupings among the elements via certain types of connections produce level one clusters.
Other connections link level one clusters to form level two clusters, while the coalescence of
level two clusters yields level three clusters, and so on [115]. A typical NoN is schematically
depicted in Figure 6.3. The structure of the NoN makes it a natural choice for massive parallel
processing and a hierarchical search engine.
Training of the NoN is very flexible. Mean field theory [116] and Hebbian learning algorithms [2] were among the first to be used in the NoN.
Recently, EP was proposed to discover clusters in the NoN in the context of adaptive segmentation/image regularization [131]. First a population of potential processing strategies is
k , of data quality which is
generated and allowed to compete under a k-pdf error measure Epdf
defined as the following weighted probability density error measure:
k
Epdf
=
N
0
2
w(k) pkm (k) − pk (k) dk
(6.14)
where the variable k is defined in [131], E is a factor which characterizes the correlation of
each item in the dataset with a prescribed neighboring subset, pk (k) is the probability density
function of k within the dataset to be processed, pkm (k) characterizes the density function of a
©2001 CRC Press LLC
FIGURE 6.3
Schematic representation of a biologically inspired network: (a) the overall network;
(b) a simplified connection model within one part of the network in (a) (the black dot at
the top left corner, for example), which itself is a three-level NoN.
©2001 CRC Press LLC
model dataset with certain desired properties, and w(k) is the weighting coefficient defined as
w(k) =
1
max (pkm (k), pk (k))2
(6.15)
to compensate for the generally smaller contribution of the tail region to the total probability.
In the context of image processing, it was shown in [131] that a small k represents a smooth
image region, a medium k represents an area with one or two dominant edges, and a large
k represents a texture area. Optimization is carried out on (6.14) in order to identify the
clusters in terms of the following regularization parameter assignment function λ(σ ), which
is a decreasing sigmoid function of the local data standard deviation σ :
λ(σ ) =
λmax − λmin
+ λmin
1 + eβ(σ −α)
(6.16)
where λmin and λmax are the minimum and the maximum regularization parameters used, α
represents the offset of the sigmoidal function from the origin, and β controls the steepness
of the function. Apart from the assignment of the local regularization parameters, this function indirectly achieves segmentation if we identify image pixels with similar associated λ
values as a single cluster. Concatenating them with their respective strategy parameters [29]
σλmin , σλmax , σα , σβ into an eight-tuple, we define the following regularization strategy Sp as
the pth potential optimizer in the population:
(6.17)
Sp = λmin,p , λmax,p , αp , βp , σλmin ,p , σλmax ,p , σα,p , σβ,p
We generate a population P consisting of µ instances of Sp in the first generation and apply
the mutation operation [29] to each of these µ parents to generate µ descendants. In this and
subsequent generations, the potential optimizers undergo a competition process from which
the emerged winners are incorporated into the new population in the next generation.
Application example: The first engineering application of the NoN was in signal categorization by Anderson et al. [1]. Guan studied the NoN and proposed a hierarchical adaptive
image processing based on it [32]. He later developed a low-level vision model to recursively
perform segmentation and edge extraction [33].
6.2.3
Hierarchical Fuzzy Neural Networks as Linear Fusion Networks
In many multimedia applications, it is useful to have a versatile multimedia fusion subsystem,
where information from various sensors are laterally combined to yield improved classification.
Neural networks offer a natural solution for sensor or media fusion. This is because of their
capability for nonlinear and nonparametric estimation in the absence of complete knowledge
on the underlying models or sensor noises.
The problem of combining the classification power of several classifiers is of great importance to various applications. First, for several recognition problems, numerous types of
media could be used to represent and recognize patterns. In addition, for those applications
that deal with high-dimensional feature data, it makes sense to divide the feature vector into
several lower-dimensional vectors before integrating them for a final decision (i.e., divide and
conquer).
Most of the current information fusion models are based on a linear combination of outputs
weighted by some proper confidence parameters. This is largely motivated by the following
statistical and computational reasons:
• It can make use of the popular Bayesian formulation.
©2001 CRC Press LLC
• It can facilitate adoption of EM training of the confidence parameters.
Channel Fusion
Two channel fusion models were proposed to deal with information from different media
sources: class-dependent channel fusion and data-dependent channel fusion.
• The class-dependent channel fusion scheme deploys one PDBNN for each sensor channel. Each PDBNN receives only the patterns from its corresponding sensor. The class
and channel conditional likelihood densities (p(x|ωi , Cj )) are estimated. The outputs
from different channels are combined in the weighted sum fashion. The weighting parameters, P (Cj |ωi ), represent the confidence of the channel Cj producing the correct
answer for the object class ωi . P (Cj |ωi ) can be trained by the EM algorithm; after that,
its value is fixed during the identification process (recall that the values of the weighting
parameters in the HME are functions of the input pattern). Figure 6.4a illustrates the
p(xl ω )
i
P(Cl ω )
1 i
p(xl ω C )
i 1
*
*
PDBNN 1
x = [ x1
P(Cl ω )
2 i
p(xl ω C )
i 2
PDBNN 2
x2 ]
FIGURE 6.4
A media fusion network: linear fusion of probabilistic DBNN classifiers. (a) For the
applications where there are several sensor sources, the class-dependent channel fusion
scheme can be applied for classification. P (Cj |ωi ) is a trainable parameter. Its value is
fixed during the retrieving phase.
structure of the class-dependent channel fusion scheme. The class-dependent channel
fusion scheme considers the data distribution as the mixture of the likelihood densities
from various sensor channels. This is a simplified density model. If the feature dimension is very large and the number of training examples is relatively small, the direct
estimation approaches can hardly obtain good performance due to the curse of dimen-
©2001 CRC Press LLC
P(ω l x)
i
gating
network
P(Clx)
2
*
P(Clx)
1
*
P(ω l x C)
i
1
P(ω l x C)
i
2
softmax
softmax
PDBNN 1
PDBNN 2
x = [ x1
x2 ]
FIGURE 6.4
(Cont.) A media fusion network: linear fusion of probabilistic DBNN classifiers. (b) Datadependent channel fusion scheme. In this scheme, the channel weighting parameters are
functions of the input pattern x (P (Cj |x)).
sionality. For this kind of problem, since the class-dependent fusion scheme greatly
reduces the number of parameters, it could achieve better estimation results.
• Another fusion scheme is the data-dependent channel fusion. Figure 6.4b shows the
structure of this scheme. Like the class-dependent fusion method, each sensor channel
has a PDBNN classifier. The outputs of the PDBNNs are transformed into the posterior
probabilities by the softmax functions [10]. In this fusion scheme, the channel weighting
P (Cj |x) is a function of the input pattern x. Therefore, the importance of an individual
channel may vary if the input pattern is different.
Fuzzy Systems and Modular Neural Networks
The basic idea behind a fuzzy inference system is to incorporate the human “expert’s experience” into system design. The input–output relationship is described by a collection of fuzzy
inference rules involving linguistic variables. The typical architecture of a fuzzy system is
composed of four components:
• A fuzzifier, which maps crisp numbers into suitable linguistic values
• A fuzzy rule base, which stores the knowledge of the human experts and the empirical
observations
• An inference engine, which deduces the desired output by performing approximate
reasoning
©2001 CRC Press LLC
• A defuzzifier, which extracts a crisp value from a fuzzy set as a representative value.
Fuzzy logic systems, in contrast to neural networks, offer a structural framework with highlevel fuzzy rule thinking and reasoning. Fuzzy systems base their decisions on inputs in the
form of linguistic variables defined by membership functions, which are formulas used to
determine the fuzzy set to which a value belongs and the degree of membership in that set.
The variables are then matched with the preconditions of the linguistic rules to calculate the
firing strengths of the rules, and the response of each rule is obtained through fuzzy implication.
Following a compositional rule of inference, the response of each rule is weighted according
to the rule firing strength.
It has recently become popular for a fuzzy system to utilize Gaussian membership functions
and a centroid defuzzification scheme to calculate the output. This is in part due to the
capability of this combination to approximate any real continuous functions on a compact set
to an arbitrary accuracy, provided sufficient fuzzy logic rules are available [56, 123]. In the
neural network literature, it has also been established that neural networks with normalized
RBFs as the hidden node functions are also universal approximators [98]. Therefore, neural
networks, especially those with modular structures, and fuzzy systems are similar in terms of
approximation capabilities.
They also bear very sharp structural resemblance. A good example is to compare the fuzzy
inference engine and the MOE modular neural networks. Stretching the similarity further, the
intersection of fuzzy systems and neural networks actually defines a large family of learning
networks. In the following it can be shown that this family of models can be built upon a
common mathematical formulation and system architecture. In terms of learning capabilities,
neural networks with RBFs as hidden nodes are basically equivalent to fuzzy systems using
Gaussian membership function, product inference, and fuzzy rules with singleton consequents.
It has been shown that an RBF MOE network and a fuzzy inference system are essentially
equivalent as long as the gating network of the MOE generates the fuzzy membership values
according to the membership function and the And operation in the fuzzy If-Then rule.
Bearing the above analysis in mind, Kung et al. [60] demonstrated that a hierarchical fuzzy
neural network designed by combining the expert-level partitioning strategies of the MOE and
the class-level partitioning of the DBNN offers an attractive processing structure for linear
channel fusion. In particular, they proposed to adopt expert-in-class hierarchical structure
(ECHS) for class-dependent channel fusion and class-in-expert hierarchical structure (CEHS)
for data-dependent channel fusion.
Hierarchical Fuzzy Neural Networks for Class-Dependent Channel Fusion
The architecture of the ECHS is exactly the same as for the class-dependent channel fusion
model illustrated in Figure 6.4a. The inner blocks comprise expert-level modules, whereas
the outer blocks are on the class level. A typical example of this type of network is the
hierarchical DBNN [59], which describes the class discriminant function as a mixture of
multiple probabilistic distribution. That is, the discriminant function of the class ωc in the
hierarchical DBNN is a class conditional likelihood density which can be described as follows:
p (x(t)|ωi ) =
K
P (Ck |ωi ) p (x(t)|ωi , Ck ) ,
k=1
where p(x(t)|ωi , Ck ) is the discriminant function of subnet i in channel k, and p(x(t)|ωi )
is the combined discriminant function for class ωi . The channel confidence P (Ck |ωi ) can
be learned by the following procedure. Define αk = P (Ck |ωi ). At the beginning, assign
©2001 CRC Press LLC
αk = 1/K, ∀k = 1, . . . , K. At step j ,
(j )
α p(x(t)|ωi , Ck )
(j )
hk (t) = k (j )
,
l αl p(x(t)|ωi , Cl )
(j +1)
αk
=
N
1 (j )
hk (t) .
N
(6.18)
t=1
In an ECHS, each expert processes only the local features from its corresponding class. The
outputs from different experts are linearly combined. The weighting parameters, P (Ck |ωi ),
represent the confidence of expert Ek producing the correct answer for the object class ωi .
Once
K they are trained, their values remain constant during the retrieving phase. By definition,
k=1 P (Ck |ωi ) = 1, where K is the number of experts in the subnet ωi . So it has the
property of a probability function. Note that, within this expert-level (or rule-level) hierarchy,
each hidden node in one class must be used to model a certain local expert with a varying
degree of confidence, which reflects its ability to interpret a given input vector. The locally
unsupervised and globally supervised schemes described in the previous section can be adopted
to train the OCON network.
Hierarchical Fuzzy Neural Networks for Data-Dependent Channel Fusion
The architecture of the CEHS is exactly the same as for the data-dependent channel fusion
model illustrated in Figure 6.4b. The inner blocks comprise class modules, whereas the outer
blocks are the expert modules. Each expert has its own hierarchical DBNN classifier. The
outputs of the hierarchical DBNNs are transformed into the posterior probabilities by softmax
functions. In this fusion scheme, the expert weighting P (Ej |x) is a function of input pattern
x. Therefore, the importance of an individual expert may vary with different input patterns
observed.
The network adopts the posterior probabilities of electing a class given x(t) (i.e., P (ωi |x(t),
Ck )), instead of the likelihood of observing x(t) given a class (i.e., p(x(t)|ωi , Ck )), to model the
discriminant function of each cluster. For this version of hierarchical fuzzy neural networks,
a new confidence P (Ck |x(t)) is assigned, which stands for the confidence on expert k when
the input pattern is x(t). Accordingly, the probability model is modified to become
P (ωi |x(t)) =
K
P (Ck |x(t)) P (ωi |x(t), Ck ) ,
k=1
where P (ωi |x(t), Ck ) = P (ωi |Ck )p(x(t)|ωi , Ck )/p(x(t)|Ck ), and the confidence P (Ck |x)
can be obtained by the following equation:
P (Ck )p(x|Ck )
,
l P (Cl )p(x(t)|Cl )
where p(x(t)|Ck ) can be computed as p(x(t)|Ck ) = i P (ωi |Ck )p(x(t)|ωi , Ck ) and P (Ck )
can be learned by equation (6.18) with p(x(t)|ωi , Ck ) replaced by p(x(t)|Ck ). The term
P (Ck ) can be interpreted as “the general confidence” we have in channel k. Unlike in the
class-dependent approach, the fusion weights need to be computed for each testing pattern
during the retrieving phase. Notice that this data-dependent fusion scheme can be considered
a combination of PDBNN and MOE [44].
P (Ck |x(t)) = Application example: The class-dependent channel fusion scheme has been observed to
have very good classification performance on vehicle recognition and face recognition problems [67]. The experiment in [67] used six car models from different view angles to create the
training and testing database. Approximately 30 images (each 256 × 256 pixels) were taken
for each car model from various viewing directions. There were 172 examples in the dataset.
©2001 CRC Press LLC
Two classifier channels were built from two different feature extraction methods: one used
intensity information and the other edge information. With the fusion of these two channels
(with 94% and 85% recognition rates), the recognition rate reached 100%.
The fusion model was compared with a single network classifier. The input vectors of
these two networks were formed by concatenating the intensity vector with the edge vector.
Therefore, the input vector dimension became 144 × 2 = 288. The RBF-typed DBNN was
used as the classifier. The experimental result showed that the performance was worse than
for the fusion network (about 95.5% recognition rate).
6.2.4
Temporal Models for Multimodal Conversion and Synchronization
The class of neural networks that are most suitable for applications in multimodal conversion
and synchronization is the so-called temporal neural network. Unlike the feed-forward type
of artificial neural network, temporal networks allow bidirectional connections between a pair
of neuron units, and sometimes feedback connections from a unit to itself. Let us elaborate
further on this difference. From the perspective of connection patterns, neural networks can be
grouped into two categories: feed-forward networks, in which the associated network graphs
have no loops, and recurrent networks, where loops occur because of the existence of the
feedback connections. Feed-forward networks are static; that is, a given input can produce
only one set of output values rather than a sequence of data. Thus, they carry no memory. In
contrast, many temporal neural networks employ some kind of recurrent network structure.
Such an architectural attribute enables temporal information to be stored in the networks.
A simple extension to the existing feed-forward structure to deal with temporal sequence
data is the partially recurrent network (sometimes called the simple recurrent network). The
connection in a simple recurrent network (SRN) is mainly of the feed-forward type, but a
carefully chosen set of feedback connections is also included. In most cases the feedback
connections are fixed and not trainable. This judicious incorporation of recurrence allows the
network to remember cues from the past without appreciably complicating the overall training
procedure. The most widely used SRNs are Elman’s network and Jordan’s network [28,
47]. The time-delay neural network (TDNN) is a further extension to cope with the shiftinvariance property required in speech recognition. It is achieved by making time-shifted
copies of the hidden units and linking them to the output layer [122]. Several fully recurrent
neural network architectures with the corresponding learning algorithms are real-time recurrent
learning (RTRL) networks [130] and back-propagation through time (BPTT) networks [39,
106]. The computational requirements of these and several variants are very high. Among
all the recurrent networks, BPTT’s performance is the best unless online learning is required,
in which case the RTRL is required instead. But for many applications involving temporal
sequence data, an SRN or a TDNN may suffice and is much less costly than RTRL or BPTT.
Time-Delay Neural Network
Figure 6.5 shows the TDNN architecture [122] for a three-class temporal sequence recognition task. A TDNN is basically a feed-forward multilayer (four layers) neural network with
time-delay connections at the hidden layer to capture varying amounts of contexts. The basic
unit in each layer computes the weighted sum of its inputs and then passes this sum through a
nonlinear sigmoid function to the higher layer. The TDNN classifier shown in Figure 6.5 has
an input layer with 12 units, a hidden layer with 8 units, and an output layer with 3 units (one
output unit represents one class).
When a TDNN is used for speech recognition, the speech utterance is partitioned frame by
frame (e.g., 30 ms frame with 15 ms advance). Each frame is transformed into 12 coefficients,
and every three frames with successive time delay 0, 1, and 2 are used as inputs to the 8 time
©2001 CRC Press LLC
B D G
3 units
Output Layer
8 units
Hidden Layer
12 LPC coefficient
k=1,2,3
Integration
Input Layer
j=1,2,..,8
Wjkl
l=0,1,..,4
i=1,2,..,12
Wijd
d=0,1,2
FIGURE 6.5
The architecture of a time-delay neural network (TDNN).
delay hidden units [i.e., each neuron in the first hidden layer now receives input (via 3 × 12
weighted connections) from the coefficients in the 3-frame window]. The 8-unit hidden layer
is delayed 5 times to form a 40-unit layer. At the second hidden layer, each unit looks at all
5 copies of the delayed 8-unit hidden blocks of the first hidden layer. Finally, the output is
obtained by integrating the information from the second hidden layer over time. This procedure
can be formalized using the following equations:
T
1 (t)
bc
T −6
t=7


12 2
4
7 (t−l−d)
H
I
= S
wcj
wij
+ θjI + θcH  ,
lS
d xi
yc =
bc(t)
©2001 CRC Press LLC
j =0 l=0
i=1 d=0
(6.19)
(6.20)
(t)
where T is the total number of frames, x is the input, {bc } are the outputs of the c class at the
second layer at different time instances, and S(·) is the sigmoid function. The tapped delay
line structure of the input layer implies the adoption of the shift invariant assumption (i.e., the
absolute time of a particular event is not important).
Like an MLP, a TDNN is also trained by the back-propagation learning rule [122]. Suppose
the input to the TDNN is a vector x; then the updating of the weights, w, can be described by
w ⇐ w − η
∂E
∂w
where
C
E = E({w}, {x}) =
1
(tc − yc (x))2 .
2
c=1
Therefore, through this training procedure, the local short duration features in speech signal
can be formed at the lower layer and more complex longer duration features formed at the
higher layer. The learning procedure ensures that each of the units in each layer has its weights
adjusted in a way that improves the network’s overall performance [41].
After the TDNN has learned its internal representation, it performs recognition by passing
input speech over the TDNN neurons and selecting the class that has the highest output value.
Section 6.3.3 presents an example employing such a TDNN model to audiovisual synchronization in the lipreading application.
6.3
Neural Networks for IMP Applications
Neural networks have played a very important role in the development of multimedia application systems [17, 43, 92, 124]. Their usefulness ranges from low-level preprocessing to
high-level analysis and classification. A complete multimedia system consists of many of the
following information processing stages, for which neural processing offers an efficient and
unified core technology:
• Visualization, Tracking, and Segmentation
– Neural networks have been found useful for some visualization applications, such
as optimal image display [61] and color constancy and induction [25].
– Feature-based tracking is crucial to motion analysis and the motion/shape reconstruction problem. Neural networks can be applied to motion tracking schemes
for feature- and object-level tracking [18].
– Segmentation is a very critical task for both image and video processing. Object
boundary detection methods can use a hierarchical technique by adopting pyramid
representation of images for computation efficiency [14, 73]. Active contour (e.g.,
snake) could also take advantage of the NN’s adaptive learning capability for
continuous and fast tracking of the region of interest (ROI) [21]. Both unsupervised
and supervised neural networks may be adopted for object boundary detection
methods, based on a variety of cues including motion, intensity, edge, color, and
texture.
©2001 CRC Press LLC
• Detection and Recognition
– Neural networks can be applied to machine learning and computer vision problems
with applications to detection and recognition of a specific object class. Examples
are online OCR applications [12, 30], signature verification [5], currency recognition [117], and structure from motion [63].
– Neural networks can facilitate detection or recognition of high-level features such
as human faces in pictures or a certain object shape under inspection.
– Multimodality recognition and authentication will have useful applications in network security and access control.
• Multimodal Coding, Conversion, and Synchronization
– Multimodal coding, conversion, and synchronization will remain a challenging
research task. Static MLP networks for multimodal facial image coding driven by
speech and phonemes were already studied in [82].
– Temporal NN models (e.g., TDNN) for multimodality synchronization, integrating
audio and visual signals for lipreading, will be elaborated in Section 6.3.3.
• Video and Image Content Indexing and Browsing
It is important for a system to possess the ability to fast access audiovisual objects,
manipulate them, and present them in a highly flexible way. For video content selection,
the ability to extract and utilize proper information content inherent in video clips may
lead to efficient search schemes for many disciplines:
– object-based and subject-based video indexing and databases
– video skimming and browsing
– content-based retrieval
Again, neural processing presents a promising approach for these tasks.
• Interactive Human–Computer Communications
Teaching a computer to understand human behavior and imitate human action can have
profound impact on successful multimedia systems. The process needs investigation in
the following two areas:
– multimodal human–computer interaction
– interactive human–computer vision
In this area, neural networks also offer attractive solutions.
6.3.1
Image Visualization and Segmentation
The task of feature extraction is critical to search schemes, because an efficient representation
of the information can facilitate many subsequent multimedia functionalities, such as featurebased or object-based indexing and access. Efficient representation of multimedia data can be
achieved by neural clustering mechanisms. The general objectives are (1) to extract the most
salient features to make classification tasks easier, and (2) to extract representation of media
information needed at various levels of abstraction.
©2001 CRC Press LLC
Although perfect segmentation and tracking of 3D video objects may not always be required,
it is desirable to have such capability in telemedicine and biomedically related applications.
Using the local energy surface as a principal feature, an SOFM can provide sufficient D
resolution of surface details of specific objects through the process of 3D segmentation. The
technique has been applied to the segmentation and visualization of specimen chromosomes
in microscopy images and the CAT images of human brains [93, 103].
6.3.2
Personal Authentication and Recognition
Neural networks have been recognized as an established and mature tool for many pattern
classification problems. In particular, they have been successfully applied to face recognition
applications. By combining face information with other biometric features such as speech,
this feature fusion approach offers improved accuracy as well as some degree of fault tolerance
(i.e., it could tolerate temporary failure of one of the bimodal channels).
Face Detection and Recognition
For many visual monitoring and surveillance applications, it is important to determine human
eye positions from an image or an image sequence containing a human face. Once the human
eye positions are determined, all of the other important facial features, such as positions of
nose and mouth, can easily be determined. The basic facial geometry information, such as the
distance between two eyes, nose and mouth size, etc., can further be extracted. This geometry
information can then be used for a variety of tasks, such as the recognition of a face from a
given face database.
There are many successful neural network examples for face detection and recognition.
Brunelli and Poggio have adopted an RBF network for face recognition [11]. Pentland et
al. [80, 96, 120] used eigenface subspace to determine the classes of face patterns. Eigenface
and Fisherface recognition algorithms were studied and compared in [6]. Cox et al. [26]
proposed a mixture–distance VQ network for face recognition and reached a 95% rate in a
large (685 persons) database. In [67, 69], neural networks were successfully applied to the
detection of human faces and the location of eyes on the face.
6.3.3
Audio-to-Visual Conversion and Synchronization
There already exist a few application examples that apply temporal neural models to conversion and/or synchronization. Included in this subsection is an example using TDNN for
lipreading applications.
Audio and Visual Integration for Lipreading Applications
Although the theory of automatic speech recognition (ASR) is well advanced, it is still not
widely adopted in practical applications due to the contamination of the speech signals with
background noise in adverse environments such as offices, automobiles, aircraft, and factories.
To improve the performance of the speech recognition system, the following approaches can be
used: (1) compensate for the noise in the acoustic speech signals prior to or during the recognition process [81], or (2) use multimodal information sources, such as semantic knowledge
and visual features, to assist acoustic speech recognition. The latter approach is supported by
the evidence that humans rely on other knowledge sources, such as visual information, to help
constrain the set of possible interpretations [133].
Due to the maturity of digital video technology, it is now feasible to incorporate visual information in the speech understanding process (lipreading). These new approaches offer effective
©2001 CRC Press LLC
integration of visually derived information into the state-of-the-art speech recognition systems
so as to gain an improved performance in noise without suffering degraded performance on
clean speech [108]. Other important evidence to support the use of lipreading in human speech
perception is offered by the auditory–visual blend illusion or the McGurk effect [74].
Three mechanisms concerning the means by which the two disparate (audio and visual)
streams of information are integrated have been proposed [113]. First, vision is used to direct
the attention, which commonly occurs in situations such as crowded rooms where several
people are talking at once. Second, visual information provides redundancy to the audio
information. Finally, visual information complements the audio information, especially when
listening conditions are poor. Most current research efforts concentrate on the third mechanism
of integration. A complete audiovisual lipreading system can be decomposed into the following
three major components [108]:
1. Audiovisual information preprocessing: explicit feature extraction from audio and visual
data
2. Pattern recognition strategy: hidden Markov modeling, pattern matching with dynamic
or linear time warping, and various forms of neural networks
3. Integration strategy: decision from audio and visual signal recognition
Audiovisual Information Preprocessing
Audio information processing has been well documented in speech recognition literature [99]. Briefly, digitized speech is commonly sampled at 8 KHz. The sampled speech
is pre-emphasized, then partitioned into frames with a fixed time interval (say, 32 ms long)
and with some overlap (say, 16 ms). For each frame, an N -dimensional feature vector is
extracted (e.g., 12-order LPC cepstral coefficients, 12-order delta cepstral coefficients, 12order delta–delta coefficients, a log–energy coefficient, a delta–log–energy coefficient, and a
delta–delta–log–energy coefficient).
There are two major types of visual features useful for lipreading: contour-based and areabased features. The active contour model [50] is a good example of an approach based on
contour-based features, which have been applied to locating object contours in many image
analysis problems [21, 22]. PCA of a gray-level image matrix, a typical area-based method,
has been successfully used for principal feature extraction in pattern recognition problems [77,
120]. Most early systems used explicit contour feature extraction. Petajan [97] extracted
contour features from binary thresholded mouth images. This approach was also used by
Goldschen [31]. Deformable template approaches to obtain contour features, such as snake,
have been the dominant method for contour feature extraction [8, 37, 100]. Chiou and Hwang
made the first attempt in using neural networks to guide the search of the deformable template
for lipreading applications [20]. These methods attempt to directly measure physical aspects
of the mouth that are invariant to changes in lighting, camera distance, and orientation. Areabased techniques have primarily been based on neural networks [112, 136]. These area-based
features are directly derived from the gray-level matrix surrounding the lips and allow the
extraction of more detailed information in the vicinity of the mouth, including the cheek and
chin. However, purely area-based approaches tend to be very sensitive to changes in position,
camera distance, rotation, and the identity of the speaker.
Pattern Recognition Strategies
Most lipreading systems have used similar pattern recognition strategies as adopted in traditional speech recognition approaches, such as dynamic time warping [97] and hidden Markov
models [20, 107]. Neural network architectures have also been extensively explored, such as
©2001 CRC Press LLC
the static feed-forward back-propagation networks used by Yuhas et al. [136], the TDNNs
used by Stork et al. [112], the multistage TDNNs used in [27], and the HMM recognizer,
which uses neural networks for performing the observation probabilities calculation [9].
The speech data used in Yuhas’ experiments were captured from a male speaker under a
well-lit condition. It is based on an NTSC video with 30 frames per second. Nine different
phonemes were recognized. A reduced subimage (20 × 25) centered around the mouth was
automatically identified for visual features, which were then converted into the corresponding
“clean” audio short-term cepstrum magnitude envelope (STSAE) by a feed-forward backpropagation network. The resulting cepstrum were weight averaged, with the noisy cepstrum
directly derived from the audio signals. The weighting between the visual converted STSAE
and the audio STSAE was determined based on the environment’s SNR. Another feed-forward
neural network collected the sequence of the combined STSAE as the inputs and performed
the recognition of vowels.
The work presented by Stork et al. [112] used a TDNN for recognizing the combined audio
and video speech data for five speakers. In their experiments, a video-only (VO) TDNN was
used to recognize the visual speech inputs, which were acquired every 10 ms. From the 10ms visual frame, five features (noise–chin separation, vertical separation of mouth opening,
horizontal separations estimated from upper and lower lips, and horizontal separation of mouth
opening) were estimated and combined by the VO TDNN to produce the classification posterior
probabilities P (C|V ), where C represents one of the 10 spoken letters. Similarly, an audioonly (AO) TDNN was used to recognize the audio speech inputs, which again were acquired
every 10 ms. From the 10-ms audio frame, 14 mel-scale coefficients (from 0 to 5 KHz)
were estimated and used by the AO TDNN to produce the classification posterior probabilities
P (C|A). The resulting classification posterior probability P (C|V , A) is approximated as
P (C|V , A) ∝ P (C|V )P (C|A) .
It was shown in [112] that this combined VO and AO TDNN network, a single video–audio
(VA) TDNN, receives the concatenated video and audio features (19 dimensions) as inputs,
thus illustrating the importance of adopting separate modules for different media types.
The See Me, Hear Me project [27] developed at Carnegie Mellon University extended the
idea of using two separate (VO and AO) TDNNs in performing continuous letter recognition
encountered in the continuous spelling tasks. The audio features consist of 16 mel-scale Fourier
coefficients obtained at a 10-ms frame rate. The visual features were formed from the PCA
transform with reduced dimensionality (only 32 out of 24 × 16 smoothed eigenlips). The two
TDNNs were used for recognizing the phoneme (out of 62) and viseme (out of 42), which
were then combined statistically for recognition of the continuous letter sequence based on the
dynamic time warping algorithm.
The project presented in [8] also combined acoustic and visual features for effective lipreading. Instead of using neural networks as the temporal sequence classifier, this project adopted
the HMMs and used an MLP to calculate the observation probabilities {P (phoneme|audio,
visual)}. The system combined the 10-order PCA transform coefficients (and/or the delta features) from the gray-level eigenlip matrix (instead of the PCA from the snake points) from the
video data and nine of the acoustic features from audio data [38]. They used a discriminatively
trained MLP to compute the observation probabilities (the likelihood of the input speech data
given the state of a subword model) needed by the Viterbi algorithm. Theoretically, the MLP
provides the posterior probabilities, instead of the likelihood, which can be easily converted
to likelihood according to Bayes’ rule using the prior probability information. This bimodal
hybrid speech recognition system has already been applied to a multispeaker spelling task,
and work is in progress to apply it to a speaker-independent spontaneous speech recognition
system, the “Berkeley Restaurant Project (BeRP).”
©2001 CRC Press LLC
Decision Integration
As discussed in the previous subsection, audio and visual features can be combined into
one vector before pattern recognition; then the decision is solely based on the result of the
pattern recognizer. In the case of some lipreading systems, which perform independent visual
and audio evaluation, some rule is required to combine the two evaluation scores into a single
one. Typical examples have included the use of heuristic rules to incorporate knowledge of
the relative confusability of phonemes in the evaluation of two modalities [97]; others have
used a multiplicative combination of independent evaluation scores for each modality. These
postintegration methods possess the advantages of conceptual and implementational simplicity
as well as giving the user the flexibility to use just one of the subsystems if desired.
6.3.4
Image and Video Retrieval, Browsing, and Content-Based Indexing
Digital video processing has recently become an important core information processing technology. The MPEG-4 audiovisual coding standards tend to allow content-based interactivity,
universal accessibility, and a high degree of flexibility and extensibility. To accommodate
voluminous multimedia data, researchers have long suggested the content-based indexing and
retrieval paradigm. Content-based intelligent processing is so critical because it encompasses
various application domains including video coding, compaction, object-oriented representation of video, content-based retrieval in the digital library, video mosaicing, video composition
(a combination of natural and synthetic scenes), and so forth [15].
Subject-Based Retrieval for Image and Video Databases
A neural network–based tagging algorithm has been proposed for subject-based retrieval for
image and video databases [135]. Object classification for tagging is performed offline using
DBNN. A hierarchical multiresolution approach is used which helps cut down the search space
of looking for a feature in an image. The classification is performed in two phases, first using
color, and then texture features are applied to refine the classification (both via DBNN). The
general indexing scheme and tagging procedure are depicted in Figure 6.6. The system [135]
allows the customer to search the image database by supplying the semantic subject. The
images are not manipulated directly in the online phase. Each image is classified into a series
of predefined subjects offline using color and texture features and neural network techniques.
Queries are answered by searching the tag database. Unlike previous approaches, which
directly manipulate images online using templates or low-level image parameters, this system
tags the images offline, which greatly enhances performance.
Compared to most of the other existing content-based retrieval systems, which only support
similarity-based retrieval, this system supports subject-based retrieval by using descriptions
of visual objects as search keys. The difference between subject-based and similarity-based
retrieval lies in the necessity for identifying visual objects in the images. Therefore, previous
low-level models are not suitable for subject-based retrieval. Novel models are needed for
subject-based retrieval that could be utilized in film- and TV program-oriented digital video
databases. Neural networks provide a natural effective technology for intelligent information
processing.
The tagging procedure includes four steps. In the first step, each image is cut into 25 equal
size blocks. Each block may contain single or multiple objects. In the second step, color
information is employed for an initial classification where each block is classified into one
of the following families: black family, gray family, white family, red family, yellow family,
green family, cyan family, blue family, or magenta family in the HSV color space. In the next
step, texture features are applied to refine the classification using DBNN if the result of color
©2001 CRC Press LLC
FIGURE 6.6
A subject-based indexing system: (a) visual search methodology; (b) tagging procedure;
(c) tagging illustration.
classification is a non-singleton set of subject categories. Each block may be further classified
into one of the following categories: sky, foliage, fleshtone, blacktop, white object, ground,
light, wood, unknown, and unsure. Finally, an image tag generated from the lookup table
using the object recognition results is saved in the tag database. The experimental results of
the Web-based implementation shows that this model is very efficient for a large film- or TV
program-oriented digital video database.
©2001 CRC Press LLC
Transform Domain–Based Retrieval for Digital Image and
Video Library (DIVL)
Transform domain–based retrieval offers an attractive alternative to content-based retrieval.
With the increasing popularity of the use of compressed images and videos, an intuitive approach for lowering computational complexity and increasing the efficiency of image and video
retrieval systems is to perform retrieval directly in the compressed domain. The advantages
of this approach are that no extra time is required to calculate features and no extra space is
required to store them. Chapter 14 of this book presents a method of using energy histograms
of the low-frequency DCT coefficients as features for the retrieval of images and videos compressed in the DCT domain. One of the attractive features of this approach is that the DCT
coefficients obtained from coding are representative features of the images, and there is no
need to process the images to obtain features as required by most other content-based methods.
It is observed that the method is sufficient for performing high-level retrieval on medium-size
DIVLs, and it represents a promising solution to efficient retrieval. However, when the size
of the DIVLs gets larger (i.e., when the number of images are in the range of millions), any of
the current retrieval methods based on matching, including those in the compressed domain,
would inevitably slow down considerably. Real-time processing becomes a critical issue. The
intuitive solution is to introduce a preprocessing scheme to cut down the amount of matching
performed. Neural networks offer attractive solutions to this problem.
One proposal consists of the following four basic steps for the preprocessing stage:
1. Average the corresponding DCT coefficients in all the 8 × 8 DCT transformed blocks.
This operation results in an 8 × 8 feature matrix representing the image.
2. Cluster the images in a DIVL into categories by the SOFM or the SOTM, to ensure more
precise clustering by using the most significant coefficients in the feature matrix, which
are normally the low-frequency coefficients.
3. A general regression neural network (GRNN) [110] or a PCA network is then used to
identify the coefficients that are most effective to distinguish between the categories.
4. The features selected in step 3 are used to train a classification machine.
When the DIVL receives a query, the classification machine first determines the specific
category to which the query belongs, prior to matching.
Further considerations must be taken into account to ensure reliable performance. Averaging
the DCT coefficients in a large image may result in too much loss of information. One way
to preserve information is to adopt a divide-and-conquer strategy. In particular, each image is
partitioned into N subimages, and then the GRNN is used to identify the most effective features
for categorizing the subimages at the same geographical location in the various images. Then
we can regard the problem as a multisensor fusion problem where we consider those features
extracted from the same subimage as arising from the same sensor. Afterward, the modular
structured fusion network or the FNN can be applied to this task. In this model, there are N
experts, with each of them specializing in representing one particular subimage. Each expert
is a DBNN consisting of M neurons, which measures the similarity of that subimage to a
particular category.
The matching process will only be performed with those images in a particular category as
predicted by the FNN. To minimize the possibility of matching images in a wrong category
due to misclassification by the FNN, the ranking of the similarity should be checked. If the
differences between the top two or three categories are small, matching should be carried out
in all of these categories, instead of the top category only.
©2001 CRC Press LLC
A prototype system has been built based on steps 1, 3, and 4 of the above principle and
tested on a small database (3000 images). Ten categories were identified in the database. It
was observed that a correct matching rate of 95% was achieved if only the top-ranked categories
were searched. The rate was further increased to 99.5% if the three top-ranked categories were
searched [121].
The aforementioned DIVL architecture is hierarchical and clustered. Such architecture is
well adapted for searching, but it would be difficult to encode new information in this hierarchy
due to the following facts: (1) the strictly top-down links in the architectures make it hard to
merge and split clusters or change the borders of the clusters when new data is entered into
the database, and (2) a global training has to be performed to accommodate new information.
The hierarchically structured NoN and SOTM offer potential solutions to this problem due to
the coexistence of both top-down and lateral links in these networks.
For these two networks, each data cluster (class of images) is represented by a particular
subnetwork. It has been shown in both the NoN [2] and the SOTM [54] that the clusters
are not isolated from one another, but are sparsely connected. Therefore, the structure of the
networks dynamically changes according to the availability of new information. Split and
merge, or change of borders, is executed smoothly and continuously. In addition, because
of the modularized architecture, retraining is restricted to some limited subarchitecture of the
network (e.g., the cluster directly affected and a number of surrounding clusters).
Face-Based Video Indexing and Browsing
A video indexing and browsing scheme based on human faces has been proposed by S.H. Lin
et al. [69]. The scheme is implemented by applying face detection and recognition techniques.
In many video applications, browsing through a large amount of video material to find the relevant clips is an extremely important task. The video database indexed by human faces provides
users with the facility to efficiently acquire video clips featuring the person of interest. For example, a film-study student may conveniently extract the clips of his/her favorite actor/actress
from a movie archive to study a performance, and a TV news reporter may quickly find in a
news database the clips containing images of some politician in order to edit the evening news.
The scheme contains three steps. The first step of the face-based video browser is to segment
the video sequence by applying a scene change detection algorithm. Scene change detection
gives an indication of when a new shot starts and ends. Each segment created by scene
change detection can be considered as a story unit of this sequence. After video sequence
segmentation, a probabilistic DBNN face detector [69] is invoked to find the segments (shots)
that most possibly contain human faces. From every video shot, we take its representative
frame and feed it into a face detector. Those representative frames from which the detector
gives high face detection confidence scores are annotated and serve as the indices for browsing.
This scheme can also be very helpful to algorithms for constructing hierarchies of video
shots for video browsing purposes. One such algorithm [134], for example, proposes using
global color and luminance information as similarity measures to cluster video shots in an
attempt to build video shot hierarchies. Their similarity metrics enable very fast processing of
videos. However, in their demonstration, some shots featuring the same anchorman fail to be
grouped together due to insufficient image content understanding. For this type of application,
we believe that the existence of similar objects, and human objects in particular, should provide
a good similarity measure. As reported in [13], this scheme successfully classifies these shots
to the same group.
©2001 CRC Press LLC
6.3.5
Interactive Human–Computer Vision
The importance of interaction between humans and computers in multimedia systems can
never be underestimated. We would like computers to be capable of understanding human
intention and expression from audio, visual gestures, body movements, and so forth, as well as
to imitate these actions. The multimodality research described in the previous sections is useful
to tackle the understanding problem. For imitating human action, interactive human–computer
vision (IHCV) may provide the solution.
Developing vision algorithms that can adapt processes designed to track, predict, and describe specific human–computer interactions in ways that are useful to the specific user in a
given task is very important in multimedia systems. It enables augmented behaviors, such as
augmented reality as an aid to human performance, by taking over tasks or making them easier.
In doing so, we do not necessarily require that these computational algorithms exactly correspond to how the brain enables perceptual and cognitive processes. Rather, these algorithms
are designed to be useful to the user insofar as they reflect the actions or behavior, or provide
important information to the user over the course of execution. These algorithms dynamically
adapt to the behaviors of individual users to evolve into ever more useful and reliable systems.
IHCV is a learning problem. We develop learning algorithms that are expressive enough
to track, predict, and describe how humans extract features and interpret images in different
tasks. For example, we could have an iconic description of structures such as edges, textures,
contours, etc. We can also have a symbolic description of structures such as mathematical
formulas.
Two different examples of scene annotation that involve the IHCV approach are
• Tracking/prediction of human edge/feature labeling: Different tasks and image properties require the recognition of different types of edges/features.
• Learning to recognize human symbol drawing (e.g., equations): The recognition performance is invariant to size, orientation, position, and specific distortions.
To summarize, the objective of using MNNs in the task is to track what humans do and predict
new cases.
In [132], a new approach to extract iconic structures was proposed. The iconic structures
in images are those referred to as edges, textures, contours, etc. In IHCV, the issue that
must be addressed properly is the adaptive extraction of those structures considered important
for human perception. Typically, the factors to be considered are the varying illumination
conditions of the background and the prototypes of the features representing the structures
under a particular level of background illumination.
The DBNNs proposed by Kung and Taur [58] are particularly suited for such tasks. The
motivation for using this architecture is that, in feature extraction, it would be more natural
to adopt multiple sets of decision parameters and apply the appropriate set of parameters as a
function of the local context, instead of adopting a single set of parameters across the whole
image as in the traditional approaches.
The modular decision-based architecture thus constitutes a natural representation of the
above adaptive decision process if we designate each subnetwork to represent a different
background illumination level, and each unit in the subnetwork to represent different prototypes
of features under the corresponding illumination level. When analyzing an input feature vector,
a two-stage decision procedure is performed by a DBNN:
• Within a subnetwork, the units representing different prototypes under the corresponding
illumination condition compete with one another. The unit giving the strongest output
claims the identity of the input feature vector.
©2001 CRC Press LLC
• The subnets then compete with one another, and the one with the largest output value
will claim the identity of the input feature vector.
One very attractive feature of the DBNN is its robustness against noise and interference.
Since the DBNN learns the average background information, the noise and interference are
filtered out as random signals. Such robustness has been clearly demonstrated in edge detection [132].
6.4
Open Issues, Future Research Directions, and Conclusions
In this chapter, we have focused on the main attributes of neural networks relevant to their
application to intelligent multimedia applications. Space limitations prohibit more exhaustive
coverage of the subjects. More illustrative examples can be found in [92, 124] and numerous
signal processing journals.
Although NNs have been quite successful in many applications of IMP, critical research
topics remain to be solved. From the commercial system perspective, there are many promising
application-driven research problems. These include analysis of multimodal scene change
detection, facial expressions and gestures, fusion of gesture/emotion and speech/audio signals,
automatic captioning for the hearing-impaired or second-language TV audiences, multimedia
telephone, and interactive multimedia services for audio, speech, image, and video contents.
From a long-term research perspective, there is a need to establish a fundamental and coherent theoretical ground for intelligent multimedia technologies. A powerful preprocessing
technique, capable of yielding salient object-based video representation, would provide a
healthy footing for online object-oriented visual indexing. This suggests that a synergistic
balance and interaction between representation and indexing must be carefully investigated.
Another fundamental research subject requiring immediate attention is the modeling and evaluation of perceptual quality in multimodal human communication. For content-based visual
query, incorporating user feedback in the interactive search process will also be a challenging
but rewarding topic.
At the beginning of the chapter, we pointed out that integrating the three branches of computational intelligence may offer excellent design strategies for multimedia systems due to their
synergistic power. The hierarchical FNN is one good example. However, such synergies have
not been extensively explored in intelligent multimedia research. Investigation into this field
will bring about new methodologies and techniques for future multimedia systems.
In conclusion, future telecommunication will place a major emphasis on media integration
for human communication. Multimedia systems can achieve their potential only when they are
truly integrated in three key ways: integration of content, integration with human users, and
integration with other media systems [91]. Therefore, the following technologies will emerge
to lead the future multimedia research [90]:
1. Technologies for generating any kind of cyberspace
2. Technologies for warping into cyberspace
3. Technologies for manipulating objects in cyberspace
4. Technologies for communicating with residents of cyberspace
To sum up, the research and application opportunities in intelligent multimedia processing
are truly boundless. We must now explore further their vast benefits and enormous potential.
©2001 CRC Press LLC
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retrieval in image databases,” Proceedings of the First International Workshop on Image
Databases and Multi-Media Search, pp. 146–153, Amsterdam, The Netherlands, 1996.
[120] M. Turk and A. Pentland, “Eigenfaces for recognition,” Journal of Cognitive Neuroscience, vol. 3, pp. 71–86, 1991.
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using time-delay neural networks,” IEEE Transactions on Acoustics, Speech, and Signal
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IEEE Workshops on Multimedia Signal Processing, IEEE Press, Princeton, NJ, 1997.
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of experts,” Advances in Neural Information Processing, vol. 8, pp. 584–590, Nov. 1995.
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experts,” Advances in Neural Information Processing, vol. 8, pp. 351–357, Nov. 1995.
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©2001 CRC Press LLC
Chapter 7
On Independent Component Analysis for
Multimedia Signals
Lars Kai Hansen, Jan Larsen, and Thomas Kolenda
7.1
Background
Blind reconstruction of statistically independent source signals from linear mixtures is relevant to many signal processing contexts [1, 6, 8, 9, 22, 24, 36]. With reference to principal
component analysis (PCA), the problem is often referred to as independent component analysis
(ICA).1
The source separation problem can be formulated as a likelihood formulation (see, e.g.,
[7, 32, 35, 37]). The likelihood formulation is attractive for several reasons. First, it allows
a principled discussion of the inevitable priors implicit in any separation scheme. The prior
distribution of the source signals can take many forms and factorizes in the source index
expressing the fact that we look for independent sources. Second, the likelihood approach
allows for direct adaptation of the plethora of powerful schemes for parameter optimization,
regularization, and evaluation of supervised learning algorithms. Finally, for the case of linear
mixtures without noise, the likelihood approach is equivalent to another popular approach
based on information maximization [1, 6, 27].
The source separation problem can be analyzed under the assumption that the sources either
are time independent or possess a more general time-dependence structure. The separation
problem for autocorrelated sequences was studied by Molgedey and Schuster [33]. They
proposed a source separation scheme based on assumed nonvanishing temporal autocorrelation
functions of the independent source sequences evaluated at a specific time lag. Their analysis
was developed for sources mixed by square, nonsingular matrices. Attias and Schreiner derived
a likelihood-based algorithm for separation of correlated sequences with a frequency domain
implementation [2]–[4]. The approach of Molgedey and Schuster is particularly interesting as
regards computational complexity because it forms a noniterative, constructive solution.
Belouchrani and Cardoso presented a general likelihood approach allowing for additive
noise and nonsquare mixing matrices. They applied the method to separation of sources taking
discrete values [7], estimating the mixing matrix using an estimate–maximize (EM) approach
with both a deterministic and a stochastic formulation. Moulines et al. generalized the EM
approach to separation of autocorrelated sequences in the presence of noise, and they explored
a family of flexible source priors based on Gaussian mixtures [34]. The difficult problem
1 There are a number of very useful ICA Web pages providing links to theoretical analysis, implementations, and
applications. Follow links from the page http://eivind.imm.dtu.dk/staff/lkhansen/ica.html.
© 2001 CRC Press LLC
of noisy, overcomplete source models (i.e., more sources than acquired mixture signals) was
recently analyzed by Lewicki and Sejnowski within the likelihood framework [28, 31].
In this chapter we study the likelihood approach and entertain two different approaches
to the problem: a modified version of the Molgedey–Schuster scheme [15], based on time
correlations, and a novel iterative scheme generalizing the mixing problem to separation of
noisy mixtures of time-independent white sources [16]. The Molgedey–Schuster scheme
is extended to the undercomplete case (i.e., more acquired mixture signals than sources),
and further inherent erroneous complex number results are alleviated. In the noisy mixture
problem we find a maximum posterior estimate for the sources that, interestingly, turns out
to be nonlinear in the observed signal. The specific model investigated here is a special case
of the general framework proposed by Belouchrani and Cardoso [7]; however, we formulate
the parameter estimation problem in terms of the Boltzmann learning rule, which allows for a
particular transparent derivation of the mixing matrix estimate.
The methods are applied within several multimedia applications: separation of sound, image
sequences, and text.
7.2
Principal and Independent Component Analysis
PCA is a very popular tool for analysis of correlated data, such as temporal correlated image
databases. With PCA the image database is decomposed in terms of “eigenimages” that often
lend themselves to direct interpretation. A most striking example is face recognition, where
so-called eigenfaces are used as orthogonal preprocessing projection directions for pattern
recognition. The principal components (the sequence of projections of the image data onto the
eigenimages) are also uncorrelated and, hence, perhaps the simplest example of independent
components [9]. The basic tool for PCA is singular value decomposition (SVD).
Define the observed M × N signal matrix, representing a multichannel signal, by
X = {Xm,n } = {xm (n)} = [x(1), x(2), . . . , x(N )]
(7.1)
where M is the number of measurements and N is the number of samples. xm (n), n =
1, 2, . . . , N is the mth signal and x(n) = [x1 (n), x2 (n), · · · , xM (n)] . In the case of image
sequences, M is the number of pixels.
For the fixed choice of P ≤ M, the SVD of X reads2
X = U DV =
P
i=1
ui Di,i vi ,
Xm,n =
P
Um,i Di,i Vn,i
(7.2)
i=1
where M × P matrix U = {Um,i } = [u1 , u2 , . . . , uP ] and N × P matrix V = {Vn,i } =
[v1 , v2 , . . . , vP ] represent the orthonormal basis vectors (i.e., eigenvectors of the symmetric
matrices XX and X X, respectively). D = {Di,i } is a P × P diagonal matrix of singular
values. In terms of independent sources, SVD can identify a set of uncorrelated time sequences,
the principal components: Di,i vi , enumerated by the source index i = 1, 2, . . . , P . That is,
we can write the observed signal as a weighted sum of fixed eigenvectors (eigenimages) ui .
However, considering the likelihood for the time-correlated source density, we are often
interested in a slightly more general separation of image sources that are independent in time
2 Usually, SVD expresses X = U
D
V
where U
is M × M, D
is M × N , and V
is N × N. U is the first P columns
, D is the P × P upper-left submatrix of D,
and V is the first P columns of V
.
of U
© 2001 CRC Press LLC
but not necessarily orthogonal in space (i.e., we would like to be able to perform a more general
decomposition of the signal matrix),
X = AS,
Xm,n =
P
Am,i Si,n
(7.3)
i=1
where A is a general mixing matrix of dimension M × P and S is a source data matrix with
dimension P × N consisting of P ≤ M independent sources. Finding A, S is often referred
to as ICA (see, e.g., [6, 9]).
7.3
Likelihood Framework for Independent Component Analysis
Reconstruction of statistically independent components/sources from linear mixtures is relevant to many information processing contexts (see, e.g., [27] for an introduction and a recent
review). We will derive a solution to the source separation based on the likelihood formulation
(see, e.g., [7, 32, 37]). An additional benefit from working in the likelihood framework is
that it is possible to discuss the generalizability of the ICA representation; in particular, we
use the generalization error as a tool for optimizing the complexity of the representation (see
also [14, 17]).
The noisy mixing model takes the form
X = AS + E
(7.4)
where E is the M × N noise signal matrix. The noise is supposed to obey a specific zero
mean, parameterized stationary probability distribution. The source signals are assumed to be
stationary and mutually independent — that is, p(si (k)sj (n)) = p(si (k))p(sj (n)), ∀ i, j ∈
[1; M], ∀ n, k ∈ [1; N ]. The properties of the source signals are introduced by a parameterized
prior probability density p(S|ψ), where ψ is the parameter vector. The likelihood of the
parameters of the noise distribution, the parameters of the source distribution, and those of the
mixing matrix is given by
L(A, θ , ψ) = p(X|A, θ , ψ) = p(X − AS|θ )p(S|ψ)dS
(7.5)
where p(X − AS|θ ) = p(E|θ) is the noise distribution parameterized by the vector θ . We will
assume that the noise can be modeled by i.i.d. Gaussian sequences with a common variance
θ = σ 2,
M N
1
1 2
2
exp − 2
εm (n) .
(7.6)
p(E|σ ) =
(2π σ 2 )MN/2
2σ
m=1 n=1
We will consider two different assumptions about the independent source distributions leading
to different algorithms.
For the time-independent white source problem, the parameter-free source distribution
of [32] is deployed:
N P
P
1
p(S) =
p(si ) = N P exp −
log cosh si (n)
(7.7)
π
i=1
© 2001 CRC Press LLC
n=1 i=1
where S = {s1 , s2 , · · · , sP } and si = [si (1), si (2), · · · si (N )] . In the time-correlated case,
it is assumed that the sources are stationary, independent, possess time autocorrelation, have
zero mean, and are Gaussian distributed:3
p(S|ψ) =
P
i=1
p(si |ψi ) =
P
i=1
(2π )N/2
1
1
exp − si s−1
s
i
i
2
det(si )
(7.8)
where ψ = [ψ1 , · · · , ψP ] and si = E[si si ] = Toeplitz([γsi (0), . . . , γsi (N − 1)]) 4 is the
N × N Toeplitz autocorrelation matrix consisting of autocorrelation function values, γsi (m) =
E[si (n)si (n + m)], m = 0, 1, . . . , N − 1. The autocorrelation matrix si is supposed to be
parameterized by ψi .
7.3.1
Generalization and the Bias-Variance Dilemma
The parameters of our blind separation model are estimated from a finite random sample,
and therefore they also are random variables which inherit noise from the dataset on which
they were trained. Within the likelihood formulation, the generalization error of a specific set
of parameters is given by the average negative log-likelihood5
G(A, θ , ψ) = − log L(A, θ , ψ) · p∗ (X) dX
= [− log p(X − AS|θ )p(S|ψ) dS] · p∗ (X) dX
(7.9)
where p∗ (X) is the true distribution of data. The generalization error is a principled tool for
model selection. In the context of blind separation, the optimal number of sources retained
in the model is of crucial interest. We face a typical bias-variance dilemma [13]. If too few
components are used, a structured part of the signal will be lumped with the noise, hence
leading to a high generalization error because of “lack of fit.” On the other hand, if too many
sources are used, we expect “overfit” because the model will use the additional degrees of
freedom to fit nongeneric details into the training data. The generalization error in (7.9) can
be estimated using a test set of data independent of the training set.6
7.3.2
Noisy Mixing of White Sources
The specific model investigated here is a special case of the general framework proposed
by Belouchrani and Cardoso [7]; however, we formulate the parameter estimation problem in
terms of the Boltzmann learning rule, which allows for a particular transparent derivation of
the mixing matrix estimate.
Let us first address the problem of estimating the sources if the mixing parameters are known
(i.e., for given A and σ 2 ). Note that MacKay [32] showed that the gradient descent scheme
3 By assuming stationarity, we implicitly neglect transient behavior due to initial conditions.
4 Toeplitz(·) transforms a vector into a Toeplitz matrix.
5 Note the close connection between generalization error and the Kullback–Leibler information (KL), as
p∗ (X)
p∗ (X) dX
p(X|A, θ , ψ)
= G(A, θ , ψ) + log(p∗ (X))p∗ (X) dX
KL( p∗ (X) : p(X|A, θ , ψ) ) =
log
6 That is, we evaluate (7.9) on the test data by using p (X) = δ(X − X ) where δ is the Dirac delta function and
∗
test
Xtest are the test data.
© 2001 CRC Press LLC
for the likelihood problem, for vanishing noise variance, is equivalent to the Bell–Sejnowski
rule [6]. Here we want to consider the more general noisy case. We use Bayes’ formula
p(S|X) ∝ p(X|S)p(S) to obtain the posterior distribution of the sources
N
P M N
1 2
p(S|X, A, σ ) ∝ exp − 2
εm (n) −
log cosh si (n)
2σ
m=1 n=1
i=1 n=1
M N
N
P 1 2
= exp − 2
(X − AS)m,n −
log cosh Si,n . (7.10)
2σ
2
m=1 n=1
i=1 n=1
The maximum a posteriori (MAP) source estimate is found by maximizing this expression
w.r.t. S 7 , leading to the following nonlinear equation to solve iteratively for the MAP estimate
S,
+ A X − σ 2 tanh S
= 0 .
−A AS
(7.11)
There are two problems with equation (7.11). First, the equation is nonlinear — although only
weakly nonlinear for low noise levels.8 Second, A A may be ill conditioned or even singular.
A useful rewriting that takes care of potential ill-conditioning of the system matrix leads to the
iterative scheme,
−1 (j +1) = A A + σ 2 I
(j ) − tanh S
(j +1)
S
A X + σ 2 S
(7.12)
where j denotes the iteration number and I is the identity matrix. This form suggests an
approximate solution for low noise levels
(1) = S (0) + σ 2 H −1 S (0) − tanh S (0) ,
S
S (0) = H −1 A X, H = A A + σ 2 I ,
(7.13)
exposing the fact that the presence of additive noise turns the otherwise linear separation
problem into a nonlinear one. A nonlinear source estimate is also found in Lewicki and
Sejnowski’s analysis of the overcomplete problem [31].
Since the likelihood is of the hidden Gibbs form we can use a generalized Boltzmann learning
rule to find the gradients of the likelihood of the parameters A, σ 2 . These averages can be
estimated in a mean field approximation [16, 38] leading to recursive rules for A and σ 2 ,
−1
S
+ βI
= XS
S
A
,
σ2 =
1
S
S
X−A
Tr X − A
MN
(7.14)
(7.15)
where β is a regularization constant representing the lumped effect of neglected fluctuations
in the mean field approach. β is estimated by
7 Note in the case of zero noise, the posterior expression leads to the expression given in [32], and the solution is
obtained by the Bell–Sejnowski algorithm [6].
8 This expression is the gradient of the exponent of the posterior distribution. A globally convergent iterative solution
can be assured if solving by gradient ascent ∇S = η · ∂ log p(S|X, A, σ 2 )/∂S, with a sufficiently small step size, η.
Here, however, we aim for a fast approximate solution for S.
© 2001 CRC Press LLC
β =
σ
2
P N
1 1−
tanh2 Si,n
PN
.
(7.16)
i=1 n=1
(See [16].)
Fluctuation corrections (hence the magnitude of β) can be derived in the low noise limit,
based on a Gaussian approximation of the likelihood [16].
The overall algorithm then consists of iterating (7.13), (7.14)–(7.16), (7.12), (7.14)–(7.16),
etc. Convergence of the algorithm is discussed in [16].
7.3.3
Separation Based on Time Correlation
Molgedey and Schuster [33] have proposed a simple noniterative source separation scheme
based on assumed nonvanishing (time) autocorrelation functions of the independent sources
that can be Gaussian distributed.9 Their idea was developed for sources mixed by square,
nonsingular A matrices. Here we generalize their approach in three ways:
• Handling the undercomplete case of more mixture signals than sources (i.e., P ≤ M).
In particular, the algorithm is well suited for cases where P M.
• Alleviating inherent erroneous complex valued results.
• Allowing for simultaneous use of more cross-correlation matrix function values maintaining the simple noniterative solution.
Define the M × M cross-correlation function matrix for the mixture signals
Cx (τ ) = E x(n)x (n + τ ) = i, j ∈ [1; M] : xi (n)xj (n + τ )
(7.17)
where τ = 0, ±1, ±2, · · · is a time lag and E{·} is the expectation operator. Note for τ = 0 we
get the usual cross-correlation matrix, Cx (0) = E{x(n)x (n)}, which is positive semidefinite.
Assume the noise-free model (7.3), x(n) = As(n), where s(n) = [s1 (n), . . . , sP (n)] , x(n) =
[x1 (n), . . . , xM (n)] and further that the M × P mixing matrix has rank(A) = P ≤ M. Since
Cx (0) = ACs (0)A where Cs (0) is the P × P cross-correlation matrix for the source signals,
and rank(A) = P , then rank(Cx (0)) = P . An eigenvalue decomposition of Cx (0) reads
Cx (0) = QLQ
(7.18)
where Q = [q1 , q2 , . . . , qM ] is the orthogonal matrix (Q Q = I ) of eigenvectors qi and
L = diag(l1 , . . . , lM ) is the diagonal matrix of eigenvalues l1 ≤ l2 ≤ · · · ≤ lP ≤ 0 and
lP +1 = lP +2 = · · · = lM = 0. Consider projection onto the P -dimensional full rank
subspace,
x
x=Q
(7.19)
= [q1 , q2 , . . . , qP ] is the M × P projection matrix and x is the P × 1 projected
where Q
mixture signal vector. Now define quotient matrix
−1
K = C
x (τ )C
x (0).
9 At most, one source is allowed to be white.
© 2001 CRC Press LLC
(7.20)
ACs (τ )A Q,
the quotient matrix can be expressed as10
Since C
x (τ ) = Q
−1 −1
A Cs τ C s 0 Q
A
K= Q
(7.21)
According to Appendix A, the quotient matrix has the eigenvalue decomposition K = −1
where is a diagonal matrix of real eigenvalues and are the associated real eigenvectors.
Define a permutation matrix11 P = [ej1 , . . . , ejP ] where ej = {δij , i ∈ [1; P ]} are P dimensional unit column vectors and [j1 , j2 , . . . , jP ] is a permutation of the numbers [1; P ].
Note that P P = I . Further, define a diagonal scaling matrix = diag([ξ1 , . . . , ξP ]) with
ξi = 0. Comparing with (7.21) shows that eigenvalue decomposition of K can be used to
identify the mixing matrix A, as shown by:
−1 −1
A
A Cs τ C s 0 Q
= P P −1 −1 P P −1
K= Q
(7.22)
where P is a permutation matrix and a diagonal scaling matrix as defined in Section 7.2.
Consequently,
A = P ,
Q
Cs (τ )Cs−1 (0)
= P
−1
(7.23)
−1
P.
(7.24)
Here we use the fact that Cs (τ ) is diagonal due to independence of the source signals.
Consider measurements of the cross-correlation function matrix for T different τ ’s and
define the extended quotient matrix:
Kext =
T
j =1
−1
αj · C
x (τj )C
x (0)
(7.25)
where αj are scalar weights. Then eigenvalue decomposition of Kext = −1 leads to
A = P ,
Q
T
j =1
αj · Cs (τj )Cs−1 (0) = P −1 −1 P .
(7.26)
(7.27)
The generalized Molgedey–Schuster algorithm for identification of mixing and source signals
up to scaling and permutations is thus summarized in the following steps:
1. Perform eigenvalue decomposition: Cx (0) = QLQ .
x.
2. Compute projected mixing signals, x=Q
3. Choose αj and τj for j = 1, 2, . . . , T and compute the extended quotient matrix Kext .
4. Perform eigenvalue decomposition: Kext = −1 .
5. Up to scaling and permutations, the mixing matrix and sources are identified as:
A = Q
−1
X .
S = A A
A X = −1 Q
10 Note that Q
A has a full rank equal to P .
11 W P gives a permutation of W ’s columns, whereas P W gives a permutation of the rows.
© 2001 CRC Press LLC
(7.28)
(7.29)
Estimation of Mixing Matrix and Source Signals
The procedure described above is based on true cross-correlation function matrices which
in practice are estimated from available data. Consider the estimate
x (τ ) = 1 Xτ X + XXτ
C
2N
(7.30)
where Xτ = {xm (n+τ )} is the time-shifted data matrix. Here we consider a cyclic permutation
by τ time steps (i.e., Xτ = {xm ((n + τ )N )} where (·)N denotes the argument modulo N ).
Equation (7.30) respects the fact that the true correlation matrix function Cx (τ ) is symmetric.
Consider the SVD of X = U DV in (7.2) with P selected so that D consists of positive
singular values only. When Xτ is formed by cyclic permutation, XX = Xτ Xτ ; hence,
Xτ = U DVτ where Vτ is the cyclic permutation of V . The P × N projected mixture signal
The estimated
τ = DVτ as U is an estimate of Q.
= U X = DV and X
matrix is X
quotient matrix is according to (7.20), given by
−1
=C
K
x (τ )C
x (0)
1 −1
=
XX
Xτ X + XXτ
2
−1
1
= D Vτ V + V Vτ D DV V D
2
1 = D Vτ V + V Vτ D −1 .
2
(7.31)
The generalized Molgedey–Schuster ICA algorithm can be summarized in the following steps:
1. Perform SVD: X = U DV with P selected so that all singular values in D are positive.
There is an option for regularization by discarding some of the smallest singular values,
causing a reduction of P .
2. Perform eigenvalue decomposition of the estimated quotient matrix12
= 1 D Vτ V + V Vτ D −1
K
2
−1 .
=
(7.32)
3. Estimate the mixing matrix and source signals:
= U ,
A
−1
= DV .
S
(7.33)
(7.34)
4. Cross-correlation matrix functions of the source signals are estimated as
S
= N −1 · s (0) = N −1 S
−1 D 2 − ,
C
s (τ ) = C
s (0) .
C
(7.35)
(7.36)
s (0) and C
s (τ ) are not diagonal.
is nonorthogonal in general implies that C
The fact that That is, finite sequence source signals cannot be expected to be uncorrelated. Unlike PCA, this
scheme and other ICA schemes do not automatically produce a set of uncorrelated features.
12 When T > 1 the term (V V + V V ) is replaced by T
τ
τ
j =1 αj (Vτ V + V Vτj ).
j
© 2001 CRC Press LLC
7.3.4
Likelihood
The major advantage of the Molgedey–Schuster algorithm is its noniterative nature; however,
it is not directly guaranteed to minimize the likelihood. Still, the likelihood is a convenient
tool for understanding the nature of the modeling. Deploying one τ (T = 1) is consistent with
parameterizing the source distribution p(S|ψ) in (7.8) using one parameter per source. As
more τ ’s are deployed, a more flexible parameterization of the likelihood applies.
The likelihood can be computed in a simple way using Fourier techniques. This also enables
computation of validation/generalization error, and consequently a principled way to select
optimal τ ’s aiming at achieving minimum generalization error. However, that discussion is
beyond the scope of this chapter.
7.4
Separation of Sound Signals
In this example the aim is to demonstrate how ICA is applied to separation of sound signals.
This could be thought of as a special case of blind signal separation in connection with the
cocktail party problem illustrated in Figure 7.1.
FIGURE 7.1
In the cocktail party problem, speech from a group of people is recorded by a number
of microphones. Without prior knowledge of the dynamics in the voices, how they are
mixed, or presence of additional noise sources, the goal is to separate the voices of the
individual speakers into different output channels.
The present example deals with speech from three persons that are assumed statistically independent. The sampling frequency of the signals is 11,025 Hz and they consist of 50,000 samples
each. A linear instantaneous mixing with a fixed known 3 × 3 mixing matrix is deployed and
enables a quantitative evaluation of the ICA separation. The source and mixing signals are
shown in Figure 7.2. In general these assumptions would not hold in real-world applications
due to echo, noise, delay, and various nonlinear effects. In such cases more elaborate source
separation is needed, as described, for example, in [2]–[4], [10]. In order to evaluate the results
of the separation, we consider the so-called system matrix defined as
−1
C
s (0)1/2
SM = A
PA
(7.37)
is the estimated mixing matrix, P is a permutation matrix, and C
s (0) is the crosswhere A
correlation matrix of the estimated source signals. If the separation is successful, the system
matrix equals the identity matrix.
© 2001 CRC Press LLC
FIGURE 7.2
The original source sound signals s1 (n), s2 (n), and s3 (n) consist of 50,000 samples and
are assumed to be statistically independent. The mixture signals x1 (n), x2 (n), and x3 (n)
are linear instantaneous combinations of the source signals.
7.4.1
Sound Separation using PCA
The PCA described in Section 7.2 is often used because it is simple and relatively fast.
Moreover, it offers the possibility of reducing the number of sources by ranking sources
according to power (variance). The result of the PCA separation is shown in Figure 7.3 and
the corresponding system matrix in Table 7.1. Obviously the result is poor when comparing
estimated sources to the original sources in Figure 7.2. This is also confirmed by inspecting
the system matrix in Table 7.1.
FIGURE 7.3
Separated sound source signals using PCA. Right panels show error signals, ei (n) =
si (n) −
si (n).
© 2001 CRC Press LLC
Table 7.1 System Matrix for the PCA
Separation of Sound
Signals


0.56
0.98
0.62




SM =  0.28 0.72 0.23 


0.18 0.50 0.06
7.4.2
Sound Separation using Molgedey–Schuster ICA
The main advantage of the Molgedey–Schuster ICA algorithm is that it is noniterative and
consequently very fast. A standard T = 1 ICA was employed, and the choice τ = 1 gave the
best performance. In Figure 7.4 the estimated sound signals from the separation are shown.
Comparison with original source signals in Figure 7.2 indicates very good separation. The
system matrix in Table 7.2 and an additional listening test also confirm this result.
FIGURE 7.4
Separated sound source signals using Molgedey–Schuster ICA. Right panels show error
si (n).
signals, ei (n) = si (n) −
Table 7.2 System Matrix for the
Molgedey–Schuster ICA Separation of Sound
Signals


 1.00 0.02 0.03 


SM =  0.02 1.00 −0.01 


−0.03 −0.03 −1.00
© 2001 CRC Press LLC
7.4.3
Sound Separation using Bell–Sejnowski ICA
The very commonly used Bell-Sejnowski ICA [6] is equivalent to maximum likelihood with
assumptions like those presented in Section 7.3.2 in the case of zero noise. Bell–Sejnowski
ICA iteratively computes an estimate of the mixing matrix by updating proportionally to the
natural gradient of the likelihood. The step size (gradient parameter) was initially 10−4 and a
line search was employed using bisection. The algorithm was terminated when the negative
log-likelihood was below 10−12 . Due to the iterative nature, this algorithm is much more time
consuming than the Molgedey–Schuster algorithm.
In Figure 7.5 and Table 7.3 the results of the separation are shown. Clearly, the system matrix
is closer to the identity matrix than that of Molgedey–Shuster, at the expense of increased
computational burden.
FIGURE 7.5
Separated sound source signals using Bell–Sejnowski ICA. Right panels show error signals, ei (n) = si (n) −
si (n).
Table 7.3 System Matrix for the Bell–Sejnowski
ICA Separationof Sound Signals

 1.00 −0.01 0.01 


SM =  0.00 1.00 −0.01 


0.01 0.01 1.00
7.4.4
Comparison
Table 7.4 lists the norm of the system matrix deviation from the identity matrix as well as
computation time.
Obviously, PCA was outperformed by both ICA algorithms due to very restricted separation
capabilities. Both ICA algorithms performed very well. The major difference is computation
time; MS-ICA was more than 200 times faster than BS-ICA. The advantage of the BS-ICA
© 2001 CRC Press LLC
Table 7.4 Norm of the System Matrix’s Deviation from the
Identity Matrix and Computation Time in Seconds
|SM − I |
PCA
MS-ICAa
BS-ICAb , 22 iterations
BS-ICAb , 56 iterations
1.21
0.05
0.05
0.01
Computation Time (s)
0.25
0.25
56.10
152.18
a MS-ICA, Molgedey–Schuster ICA.
b BS-ICA, Bell–Sejnowksi ICA for 22 and 56 iterations, respectively.
algorithm is that the system matrix can be significantly closer to unity provided sufficient
computation time. A hybrid of MS-ICA and BS-ICA in which MS-ICA is used to initialize
BS-ICA seems obvious.
Listening to the separated signals, it was hardly impossible to tell the difference between
the ICA results.
7.5
Separation of Image Mixtures
Applying ICA to images has been carried out in a number of applications ranging from face
recognition to localizing activated areas in the brain (see, e.g., [5, 16, 19, 20, 29, 30]).
In this section we illustrate some of the basic features using ICA in contrast to or in combination with PCA for image segmentation. From a sequence of images, the objective is to
extract sequence images where common features have been separated into different images. In
the present case ICA is based on raw images; however, in principle, the segmentation can also
be done from features extracted from the images. The simple dataset as shown in Figure 7.6 is
used in this example. There are P = 4 original source images of N = 9100 (91 by 100) pixels
rearranged into the P × N source matrix S so that each row represents an image. The M × N
signal matrix X with M = 6 is generated by using the following M × P mixing matrix


1 1 0 1
 −1 1 0 1 


 1 1 −2 1 

 .
A=
(7.38)

 −1 −1 −2 1 
 1 −1 0 1 
−1 −1 0 1
7.5.1
Image Segmentation using PCA
The result of applying PCA to the face dataset is shown in Figure 7.7. The number of
nonzero eigenvalues is correctly determined to be 4. Notice that the eyebrow and mouth
positions operate in pairs; when the mouth is “smiling” it cannot be “sad” and likewise for the
eyebrows. PCA is able to detect this behavior but mixes both eyebrows and mouth pieces in
sources 2 and 3. Further, only the nose is present in source 1. This is a typical effect in PCA
because its decomposition is based on finding the directions with the most variance, which is
not always well suited for the data.
© 2001 CRC Press LLC
FIGURE 7.6
The artificial face dataset used for image segmentation. The top row shows the P = 4
sources of N = 9100 pixels, which is multiplied with the mixing A in the middle row to
generate the signal matrix X with M = 6 components in the bottom row.
FIGURE 7.7
Applying PCA to the artificial face data. The sources V are shown in the top row and
the corresponding mixing matrix estimate is shown in the bottom row. Unfortunately,
PCA mixes the eyebrows and mouth pieces in sources 2 and 3. Further, only the nose is
present in source 1.
7.5.2
Image Segmentation using Molgedey–Schuster ICA
ICA on images can be performed either to the signal matrix X or the transpose X . In the
first case N = number of pixels and M = number of images in sequence corresponding to
assuming independence of pixels in the sources. In this case the sources are images and the
mixing matrix is the time sequence. In the second case N = number of images in sequence
and M = number of pixels corresponding to assuming independence driving time sequence
sources. Thus, the mixing matrix corresponds to (eigen)images. This is summarized in
Table 7.5.
© 2001 CRC Press LLC
Table 7.5 Two Ways of Performing ICA on Image Sequences
Signal Matrix
M
N
S
A
Assumption
X
X
No. of images in sequence
No. of pixels
Images
Time sequence
Pixel independence
No. of pixels
No. of images in sequence
Time sequence
Images
Time independence
The result when assuming pixel independence (i.e., using X as signal matrix) is shown in
Figure 7.8. The result when assuming time independence (i.e., using X as signal matrix) is
shown in Figure 7.9.
FIGURE 7.8
MS-ICA on the artificial face data with the pixel-independence assumption (i.e., X is the
signal matrix). The estimated sources (eigenimages) are shown in the top row and the
associated mixing matrix (time sequences) in the bottom row. Unlike PCA in Figure 7.7,
MS-ICA does not mix eyebrows and mouths (i.e., the sources are almost perfect except
for a small problem with the nose component in source 1). Also, the mixing matrix A is
almost perfect in comparison with Figure 7.6.
7.5.3
Discussion
Real image applications often show preference toward ICA over PCA. This is mainly because
ICA is able to produce a nonorthogonal basis and is not constrained by the variance ranking
inherent in PCA. Using PCA as preprocessing to ICA in order to determine the number of
sources has proven successful [6]. Also, the PCA estimate of the mixing matrix can be used
as initialization for an iterative ICA scheme such as Bell–Sejnowski [6] and the algorithm of
Section 7.3.2. Performing ICA using the Molgedey–Schuster algorithm gives better results
than PCA, at comparable computational cost.
The choice of pixel independence vs. time independence is related to the problem at hand.
In the image segmentation problem above, pixel independence gave the best result; however,
other cases have shown preference to time independence (see, e.g., [15, 16]).
© 2001 CRC Press LLC
FIGURE 7.9
MS-ICA on the artificial face data with the time-independence assumption (i.e., X as
signal matrix). The estimated sources (time sequences) are shown in the bottom row and
associated mixing matrix (eigenimages) in the top row. The mouth is present in both
eigenimages 2 and 3, thus producing a slightly worse result than that in Figure 7.8.
7.6
7.6.1
ICA for Text Representation
Text Analysis
The field of text analysis aims at searching for specific information and structure in text
data, which has emerged rapidly in recent years due to the Internet and other massive text
databases. The general ways of searching and grouping are usually Boolean13 search and
query14 subset selection. These methods are straightforward but are not, however, based on
statistical modeling. Due to the large amount of data, any statistical approach has been very
difficult, and only in recent years has a serious effort been carried out.
The general idea behind many text analysis algorithms is the so-called N -gram histogram.
The N -gram histogram is based on counting the simultaneous occurrence of N words or terms.
We consider merely 1-gram histograms as higher order histograms that often have large areas
of infinitesimal probability mass due to the infrequent occurrence of many word combinations.
In Figure 7.10 a 1-gram histogram is shown and is referred to as the term/document matrix.
The term/document matrix can contain features extracted from the documents and be used
as a signal matrix X for PCA and ICA. Recently PCA and ICA have been applied to text
analysis [21, 23, 25], and in the following we shall apply both PCA and ICA to the 1-gram
histogram using the MED dataset [11]. The MED dataset is a commonly studied collection
of medical abstracts. It consists of 1033 abstracts, of which 30 labels have been assigned to
696 of the documents. The goal is not to compare the performance of ICA to other unsupervised
methods, but rather to demonstrate its capability in text analysis. Consequently, we restrict
the study to 124 abstracts — that is, the first five groups/classes in the MED dataset that can
be characterized by the following verbal descriptions:
1. The crystalline lens in vertebrates, including humans.
13 A Boolean search operates from AND and OR operators.
14 When a query is made, a subset of the data is selected. This can be done, for example, by a Boolean search — often
found by SQL statements.
© 2001 CRC Press LLC
FIGURE 7.10
The term/document matrix X is a 1-gram histogram. The rows represent different
words/terms appearing in a collection of text documents. In the present study we use
M = 1159 terms. Each column represents the histogram for a specific document or text
group. In the present example, N = 124 documents were used.
2. The relationship of blood and cerebrospinal fluid oxygen concentrations or partial pressures. A method of interest is polarography.
3. Electron microscopy of lung or bronchi.
4. Tissue culture of lung or bronchial neoplasms.
5. The crossing of fatty acids through the placental barrier. Normal fatty acid levels in
placenta and fetus.
When constructing the histogram term/document matrix, words that occur in more than one
abstract were chosen as term words. In order to facilitate the analysis, commonly used words15
were removed; 1159 terms remained in the matrix. In summary, the term/document matrix
X is M = 1159 by N = 124. The ICA algorithm used in this example is the noisy mixing
algorithm described in Section 7.3.2.
15 A stop word list was defined.
© 2001 CRC Press LLC
7.6.2
Latent Semantic Analysis — PCA
A classical method for both search and grouping (clustering) is latent semantic analysis
(LSA), introduced by [11]. The principle of LSA is to build the term/document matrix and
find a better basis representation using PCA. Consider the SVD X = U DV where U contains
the eigenvectors of the term covariance matrix XX . Likewise, V contains the eigenvectors of
the document covariance matrix X X. D is the diagonal matrix of increasing singular values
equal to the square root of the eigenvalues. Paraphrased, U provides relative coordinates for
the covariance between different terms and, likewise, V relative coordinates for the documents.
In Figure 7.11 the documents are represented by a 3D PCA basis. A clear data cluster structure
is noticed.
FIGURE 7.11
PCA on the term/document matrix. The documents are plotted with different signatures
corresponding to the prelabeling into five classes. A clear cluster structure is noticed.
Using clustering techniques, the documents can now be clustered into groups of similar
meaning. This also enables the characterization of a new document by projecting onto the
identified PCA basis.
7.6.3
Latent Semantic Analysis — ICA
The objective of ICA in LSA is that it should serve as a clustering algorithm so that different
semantic groups are represented by separate independent components. The ICA algorithm
produces the mixing matrix A in which each column represents a histogram associated with a
specific semantic cluster. The source matrix S expresses how the documents contribute to the
semantic clusters.
Since we typically face problems with thousands of words in the terms list and possibly much
fewer documents, this is a so-called extremely ill-posed learning problem, which can be remedied without loss of generality by PCA projection. The PCA decomposes the term/document
matrix on eigen-histograms. These eigen-histograms are subject to an orthogonality constraint,
being eigenvectors to a symmetric real matrix. We are interested in a slightly more general
separation of sources that are independent as sequences, but not necessarily orthogonal in the
word histogram; that is, we would like to be able to perform a more general decomposition
of the data matrix, corresponding to the model in equation (7.4). Before performing the ICA
we can make use of the PCA for simplification of the ICA problem. The approach here is
similar to the so-called “cure for extremely ill-posed learning” [26] problem used to simplify
supervised learning in short image sequences. We first note that the likelihood, considered as
a function of the columns of A (histograms), can be split in two parts: part A1 , orthogonal
to the subspace spanned by the M rows of X, and part A2 , situated in the subspace spanned
© 2001 CRC Press LLC
by the N columns of X. The first part is trivially minimized for any nonzero configuration of
sources by putting A1 = 0. It simply does not “couple” to data. The remaining part A2 can
be projected onto an N -dimensional hyperplane spanned by the documents. In this way we
reduce the high-dimensional separation problem to the separation of a square (projected) data
matrix of size N × N . We note that it often may be possible to further limit the dimensionality
of the PCA subspace, hence further reducing the histogram dimensionality M of the remaining
problem. Using the “cure for extremely ill-posed learning” method, the problem is reduced to
an M = 124 by N = 124 problem without loss of generality. However, we expect that even
fewer components are needed for creating a generalizable model. In Figure 7.12 we show the
test and training set errors evaluated on training sets of 104 patterns randomly chosen among
the set of 124. The test set consists of the remaining 20 documents in each resample. The
FIGURE 7.12
ICA analysis of the MED dataset. Training and test error as a function of the number
of sources, or number of components P . The training set consists of 104 documents
randomly chosen among the set of 124 possible, and the remaining 20 are used for test.
The test curve shows a shallow minimum for P = 4 components, reflecting the biasvariance trade-off discussed in Section 7.3.1.
generalization error shows a shallow minimum for P = 4 independent components, reflecting
the bias-variance trade-off (Section 7.3.1) as a function of the complexity of the estimated mixing matrix. In Figure 7.13 we show scatterplots in the most variant independent components.
Although the distribution of documents forms a rather well-defined group structure in the PCA
scatterplots, clearly the ICA scatterplots are much better axis aligned. We conclude that the
nonorthogonal basis found by ICA better “explains” the group structure. To further illustrate
this finding we have converted the ICA solution to a pattern recognition device by a simple
heuristic. We assign a group label based on the magnitude of the recovered source signal. In
Tables 7.6 and 7.7 we show that this device is quite successful in recognizing the group structure, although the ICA training procedure is completely unsupervised. For an ICA with three
independent components, two are recognized perfectly and three classes are lumped together.
The four-component ICA, which is the generalization optimal model, “recognizes” three of the
© 2001 CRC Press LLC
FIGURE 7.13
ICA applied on the term/document matrix. The documents are plotted with different
signatures corresponding to the prelabeling into five classes. ICA projects the natural
clusters along the basic vectors, making them easy to separate.
five classes almost perfectly and confuses the two classes 3 and 4. Inspecting the groups, we
found that the two classes indeed are on very similar topics,16 and investigating classifications
for five or more ICA components did not resolve the ambiguity between them. The ability of
the ICA classifier to identify the topic structure is further illustrated in Figure 7.14, where we
show scatterplots coded according to ICA classifications. This shows that the ICA is better
than PCA-based LSI in identifying relevant latent semantic structure. Finally, we inspect the
histograms produced by ICA by back-projection using the PCA basis. Thresholding the ICA
histograms, we find the salient terms for the given component. These terms are keywords for
the given topic, as shown in Tables 7.6 and 7.7, and follow nicely the behavior of the confusion
matrices.
Table 7.6 Confusion Matrix for a Simple Classifier Constructed from
the Three-Component ICA
1
2
Class
3
4
5
Keywords
IC1 37 0 0 0 0
Lens protein
IC2 0 16 1 1 0
Arterial blood cerebral oxygen rise
IC3 0 0 21 22 26
Acid blood cell fatty free glucose insulin
Note: Two of the five MED classes are recovered, whereas the last independent
component contains a mixture of the remaining three classes.
7.7
Conclusion
This chapter discussed the use of ICA for multimedia applications. In particular, we applied
ICA to the separation of speech signals, segmentation of images, and text analysis/clustering.
16 They both concern medical documents on diseases of the human lungs.
© 2001 CRC Press LLC
FIGURE 7.14
ICA analysis of the MED dataset. The dataset consists of 124 documents in five topics.
The source signals recovered in the ICA have been converted to a simple classifier, and we
have coded these classes by different shades. From top to bottom we show scatterplots
in the principal component representation 1 vs. 2 and 3 vs. 2, with shading signifying
the classification proposed by the ICA with 2, 3, 4, and 5 independent components,
respectively.
A likelihood framework for ICA was presented and enables a unified view of different
algorithms. Furthermore, this enables formulation of the generalization error, defined as the
expected negative log-likelihood on independent examples. The generalization error is a
principled tool for model optimization (e.g., number of sources retained in the model).
We focused on two ICA algorithms: separation based on time correlation and noisy mixing
of white sources. In the first case we presented a generalized version of the Molgedey–
Schuster algorithm, allowing for handling of undercomplete problems, alleviating inherent
erroneous complex valued results, and allowing for simultaneous use of more cross-correlation
measurements while maintaining the simple noniterative nature of the algorithm. In the noisy
mixing case, a maximum a posteriori estimate for source estimation was employed, and the
mixing matrix and noise variance were estimated via Boltzmann learning.
© 2001 CRC Press LLC
Table 7.7 Confusion Matrix for a Simple Classifier Constructed from the
Four-Component ICA
1
2
Class
3
4
5
Keywords
IC1 31 0 0 0 0
Lens protein
IC2 0 16 0 1 0
Arterial blood cerebral oxygen rise
IC3 6 0 22 21 2
Alveolar cell lens lung
IC4 0 0 0 1 24
Acid blood fatty free glucose insulin
Note: Three of the five MED classes are recovered, whereas the remaining two classes are mixed.
The two unresolved classes are related because both make reference to the lung physiology.
Acknowledgment
This work was funded by the Danish Research Councils through the Distributed Multimedia
Technologies and Applications within the Center for Multimedia and the THOR Center for
Neuroinformatics. Andrew Back is acknowledged for valuable discussions concerning the
Molgedey-Schuster algorithm.
Appendix A: Property of the Quotient Matrix
THEOREM 7.1
−1
The quotient matrix K = C
x (τ )C
x (0) has real eigenvalues and eigenvectors, and obtains
the eigenvalue decomposition K = −1 .
ACs (τ )A Q.
PROOF
C
x (τ ) is symmetric since it can be expressed as C
x (τ ) = Q
Further, C
x (0) is positive definite, as Cs (0) is positive definite. A similarity transform of K is
given by
−1/2
Ksim = C
x
1/2
−1/2
(0)KC
x (0) = C
x
−1/2
(0)C
x (τ )C
x
(0)
(7.39)
Ksim is thus symmetric with real eigenvalues and eigenvectors [18, Theorem 4.1.5], and obtains
the eigenvalue decomposition EE where E is the orthogonal (E E = I ) matrix of
eigenvectors and is a diagonal matrix of eigenvalues. Since K and Ksim are similar, they
have the same eigenvalues, counting multiplicity [18, Corollary 1.3.4]. Finally, using the
1/2
−1/2
similarity transform K = C
(0), K obtains the eigenvalue decomposition
x (0)Ksim C
x
1/2
K = −1 where = C
(0)E.
x
© 2001 CRC Press LLC
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[32] D. MacKay, “Maximum Likelihood and Covariant Algorithms for
dependent Components Analysis,”
Draft 3.7,
1996. Available
ftp://mroa.cam.ac.uk/hello.ps.gz.
Invia
[33] L. Molgedey and H. Schuster, “Separation of Independent Signals Using Time-Delayed
Correlations,” Physical Review Letters, vol. 72, no. 23, pp. 3634–3637, 1994.
[34] E. Moulines, J.-F. Cardoso, and E. Gassiat, “Maximum Likelihood for Blind Separation
and Deconvolution of Noisy Signals Using Mixture Models,” Proceedings of ICASSP’97,
vol. 5, pp. 3617–3620, 1997.
[35] B.A. Olshausen, Learning Linear, Sparse, Factorial Codes, A.I. Memo 1580, MIT Press,
Cambridge, MA, 1996.
[36] E. Oja, “PCA, ICA, and Nonlinear Hebbian Learning,” Proceedings of the International
Conference on Artificial Neural Networks ICANN-95, pp. 89–94, 1995.
[37] B.A. Pearlmutter and L.C. Parra, “Maximum Likelihood Blind Source Separation, A
Context-Sensitive Generalization of ICA,” in Advances in Neural Information Processing
Systems 9, M.C. Mozer et al. (eds.), MIT Press, Cambridge, MA, pp. 613–619, 1997.
[38] C. Peterson and J.R. Anderson, “Mean Field Theory Learning Algorithm for Neural
Networks,” Complex Systems, vol. 1, pp. 995–1019, 1987.
© 2001 CRC Press LLC
Chapter 8
Image Analysis and Graphics for Multimedia
Presentation
Tülay Adali and Yue Wang
8.1
Introduction
The success of multimedia applications is highly dependent on the effective representation of
the information of interest from data that now come in a variety of forms. For the effective use
of computer-reconstructed images, two steps are key: analysis of images through extraction
of the key features of the image and the visualization of these features in a way that is suitable
for the application at hand.
Model-based image analysis aims at capturing the intrinsic character of images with few
parameters and is also instrumental in helping to understand the nature of the imaging process.
Key issues in image analysis include model selection, parameter estimation, imaging physics,
and the relationship of the image to the task (how the image is going to be utilized) [11, 28].
Stochastic model-based image analysis has been the most popular among the model-based
image analysis methods because, most often, imaging physics can be modeled effectively with
a stochastic model. For example, the suitability of standard finite normal mixture models has
been verified for a number of medical imaging modalities [33, 73, 77]. In the first part of
the chapter, we discuss a complete treatment of the stochastic model-based image analysis
that includes model and model order selection, parameter estimation, and final segmentation.
We focus on models that use finite normal mixtures and show examples in medical image
segmentation and computer-aided diagnosis.
Computer graphics can play a central role in helping multimedia meet its challenges. Representing images in a form that matches our perceptual capabilities (mainly visual) and a
problem’s particular needs makes the process of getting information and digesting it easier
and more effective. More specifically, good use of visualization and computer graphics in the
multimedia environment can make a number of important tasks easier and more effective, such
as,
1. Analyzing information on the images
2. Monitoring image content and changes
3. Interacting with image databases
4. Collaborating with other sites/groups
5. Handling video e-mail or browsing on the Web
© 2001 CRC Press LLC
In the second part of this chapter, we discuss how to use graphics modeling and visualization technologies to achieve this task. We address methods for graphical modeling and
reconstruction and introduce deformable surface–spine models. We discuss applications in
reconstruction of synthetic and range datasets and of 3D surgical prostate models.
8.2
Image Analysis
Stochastic model-based image analysis is a technique for partitioning an image into distinctive meaningful regions based on the statistical properties of both the gray-level and the
context images. A good segmentation result depends on suitable model selection for the given
image. For medical images, such as magnetic resonance (MR), positron emission tomography
(PET), and radiographic images, model selection can be justified in terms of imaging physics,
or alternatively, a better understanding of the imaging physics can be used to select a suitable
model for a given imaging modality [33, 77]. Model selection refers to the determination of
both the local statistical distributions of each region and the number of image regions.
In image analysis, we can treat pixel and context modeling separately, assuming that each
pixel can be decomposed into a pixel image and a context image. Pixel image is defined as the
observed gray level associated with the pixel, and finite mixture models have been the most
popular pixel image models. In particular, standard finite normal mixtures (SFNMs) have
been very widely used in statistical image analysis, and efficient algorithms are available for
calculating the parameters of the model. Furthermore, by incorporating statistical properties
of context images, where context image is defined as the membership of the pixel associated
with different regions, a localized SFNM formulation can be used to impose local consistency
constraints on context images in terms of a stochastic regularization scheme [74]. The next
section describes the finite mixtures model and addresses identification of the model (i.e.,
estimation of the parameters of the model and the model order selection). In Section 8.2.2,
we discuss approaches to modeling context. Also, it is important to note that, even though
texture is an important property in the perception of images by humans, it is typically difficult
to describe. It can be identified in terms of five perceptual dimensions: coarseness, contrast,
directionality, line-likeness regularity, and roughness [60], and can be incorporated into the
graphical representation discussed in Section 8.3.
8.2.1
Pixel Modeling
Given a digital image consisting of N ≡ N1 × N2 pixels, assume that this image contains
K regions and that each pixel is decomposed into a pixel image x and a context image l.
By ignoring information regarding the spatial ordering of pixels, we can treat context images
(i.e., pixel labels) as random variables and describe them using a multinomial distribution with
unknown parameter πk . Since this parameter reflects the distribution of the total number of
pixels in each region, πk can be interpreted as a prior probability of pixel labels determined
by the global context information. Thus, the relevant (sufficient) statistics are the pixel image
statistics for each component mixture and the number of pixels of each component. The
marginal probability measure for any pixel image (i.e., the finite mixtures distribution) can be
obtained by writing the joint probability density of x and l and
then summing the joint density
over all possible outcomes of l (i.e., by computing p(xi ) l p(xi , l)), resulting in a sum of
© 2001 CRC Press LLC
the following general form:
p (xi ) =
K
πk pk (xi ) , i = 1, . . . , N
(8.1)
k=1
where xi is the gray level of pixel i. pk (xi )’s are conditional region
probability density functions
(pdfs) with the weighting factor πk , satisfying πk > 0, and K
k=1 πk = 1. The generalized
Gaussian pdf given region k is defined by [89]
αβk
1
exp − |βk (xi − µk )|α , α > 0, βk =
pk (xi ) =
2(1/α)
σk
(3/α)
(1/α)
1/2
(8.2)
where µk is the mean, (·) is the gamma function, and βk is a parameter related to the variance
σk by
βk =
1
σk
(3/α)
(1/α)
1/2
.
(8.3)
When α 1, the distribution tends to be a uniform pdf; for α < 1, the pdf becomes sharper;
for α = 2.0, one has the Gaussian pdf; and for α = 1.0, the Laplacian pdf exists. Therefore, the
generalized Gaussian model is a suitable model to fit the histogram distribution of those images
whose statistical properties are unknown since the kernel shape can be controlled by selecting
different α values. The finite Gaussian mixture model (FGGM) for α = 2 is commonly
referred to as the standard finite normal mixture model and has been the most frequently used
form. It can be written as
pk (xi ) =
K
k=1
with
πk g xi µk , σk2
i = 1, 2, . . . , N
(8.4)
2
−
µ
(x
1
)
i
k
g xi µk , σk2 = √
exp −
2σk2
2π σk
where µk and σk2 are the mean and variance of the kth Gaussian kernel and K is the number
of Gaussian components.
The whole image can be well approximated by an independent and identically distributed
random field X. The corresponding joint pdf is
P (x) =
K
N πk pk (xi )
(8.5)
i=1 k=1
where x = [x1 , x2 , . . . , xN ] and x ∈ X. Based on the joint probability measure of pixel
images,
the likelihood function under finite mixture modeling can be expressed as L(r) = N
i=1 pr (xi )
where r : {K, α, πk , µk , σk , k = 1, . . . , K} denotes the model parameter set.
8.2.2
Model Identification
Once the model is chosen, identification addresses the estimation of the local region parameters (πk , µk , σk , k = 1, . . . , K) and the structural parameters (K, α). In particular the
estimation of the order parameter, K, is referred to as model order selection.
© 2001 CRC Press LLC
Parameter Estimation
With an appropriate system likelihood function, the objective of model identification is to
estimate the model parameters by maximizing the likelihood function, or equivalently minimizing the relative entropy between the image histogram px (u) and the estimated pdf pr (u),
where u is the gray level [2, 69]. There are a number of approaches to perform the maximum
likelihood (ML) estimation of finite mixture distributions [66]. The most popular method is
the expectation–maximization (EM) algorithm [18, 53]. The EM algorithm first calculates
the posterior Bayesian probabilities of the data through the observations, obtains the current
parameter estimates (E step), and then updates parameter estimates using generalized mean
ergodic theorems (M step). The procedure cycles back and forth between these two steps.
The successive iterations increase the likelihood of the model parameters. A neural network
interpretation of this procedure is given in [49].
We can use relative entropy (the Kullback–Leibler distance) [31] for parameter estimation
[i.e., we can measure the information theoretic distance between the histogram of the pixel
images, denoted by px , and the estimated distribution pr (u), which we define as the global
relative entropy (GRE)]:
D (px ||pr ) =
px (u) log
u
px (u)
.
pr (u)
(8.6)
It can be shown that, when relative entropy is used as the distance measure, distance minimization is equivalent to the ML estimation of the model parameters [2, 69].
For the case of the FGGM model, the EM algorithm can be applied to the joint estimation
of the parameter vector and the structural parameter α as follows [18]:
EM Algorithm
1. For α = αmin , . . . , αmax
• m = 0, given initialized r(0)
• E step: for i = 1, . . . , N, k = 1, . . . , K, compute the probabilistic membership
(m)
π pk (xi )
(m)
zik = K k (m)
k=1 πk pk (xi )
• M step: for k = 1, . . . , K, compute the updated parameter estimates

N1 N2

1 
(m+1)
(m)

 πk
=
zik


N


i=1


N

1
(m+1)
(m)
µk
=
zik xi
(m+1)


N πk

i=1


N


1

2(m+1)
(m)
(m+1) 2

σ
=
zik (xi − µk
)

 k
(m+1)
N πk
i=1
• When |GRE(m) (px ||pr ) − GRE(m+1) (px ||pr )| ≤ is satisfied, go to step 2.
Otherwise, m = m + 1 so go to E step.
2. Compute GRE, and go to step 1.
© 2001 CRC Press LLC
(8.7)
(8.8)
3. Choose the optimal r̂ that corresponds to the minimum GRE.
The EM algorithm, however, in general, has the reputation of being slow, because it has
a first-order convergence in which new information acquired in the expectation step is not
used immediately [84]. Recently, a number of online versions of the EM algorithm have
been proposed for large-scale sequential learning (e.g., see [41, 47, 66, 69, 81]). Such a
procedure obviates the need to store all the incoming observations, changing the parameters
immediately after each data point, allowing for high data rates. Titterington [66] has developed
a stochastic approximation procedure that is closely related to the probabilistic self-organizing
mixture (PSOM) algorithm we are going to introduce here, and shows that the solution can be
made consistent. Other similar formulations have been proposed by Marroquin et al. [41] and
Weinstein et al. [81].
For the adaptive estimation of the SFNM model parameters, we can derive an incremental
learning algorithm by the simple stochastic gradient descent minimization of D(px ||pr ) [69,
73] given in (8.6) with the pr given by (8.4):
(t)
(t)
(t)
(8.9)
µk (t+1) = µk + a(t) xt+1 − µk z(t+1)k ,
2
2(t+1)
2(t)
(t)
2(t)
(t)
z(t+1)k ,
σk
= σk + b(t) xt+1 − µk
− σk
k = 1, . . . , K
(8.10)
where a(t) and b(t) are introduced as the learning rates, two sequences converging to zero,
ensuring unbiased estimates after convergence. For details about derivation and the approximations, see [69, 70]. Based on generalized mean ergodic theorem [17], updates can also be
obtained for the constrained regularization parameters, πk , in the SFNM model. For simplicity,
given an asymptotically convergent sequence, the corresponding mean ergodic theorem (i.e.,
the recursive version of the sample mean calculation) should hold asymptotically. Thus, we
define the interim estimate of πk by [71]:
(t+1)
πk
=
t
1 (t)
(t)
π +
z
.
t +1 k
t + 1 (t+1)k
(8.11)
Hence the updates given by (8.9), (8.10), and (8.11) together with evaluation of (8.7) using (8.4) provide the incremental procedure for computing the SFNM component parameters.
Their practical use, however, requires strongly mixing conditions and a decaying annealing
procedure (learning rate decay) [17, 25, 51]. In finite mixtures parameter estimation, algorithm
initialization must be chosen carefully and appropriately. In [71], an adaptive Lloyd–Max histogram quantization (ALMHQ) algorithm is introduced for threshold selection which is also
well suited to initialization in ML estimation. It can be used for initializing the network
parameters, µk , σk2 , and πk , k, 1, 2, . . . , K.
Model Order Selection
Determination of the region parameter K directly affects the quality of the resulting model
parameter estimation and, in turn, affects the results of segmentation. In a statistical problem
formulation such as the one introduced in the previous section, the use of information theoretic criteria for the problem of model determination arises as a natural choice. Two popular
approaches are Akaike’s information criterion (AIC) [4] and Rissanen’s minimum description
length (MDL) [55]. Akaike proposed to select the model that gives the minimum AIC, which
is defined by
(8.12)
AIC (Ka ) = −2 log L r̂ML + 2Ka
© 2001 CRC Press LLC
where r̂ML is the maximum likelihood estimate of the model parameter set r, and K is the
number of free adjustable parameters in the model [4, 33]. AIC selects the correct number of
image regions K0 when
K0 = arg
min AIC(K) .
(8.13)
1≤K≤Kmax
Rissanen addressed the problem from a quite different point of view. He reformulated the
problem explicitly as an information coding problem in which the best model fitness was
measured such that it assigned high probabilities to the observed data while at the same time
the model itself was not too complex to describe [55]. The model is selected by minimizing
the total description length defined by
MDL (Ka ) = − log L r̂ML + 0.5Ka log(N ) .
(8.14)
Similarly, the correct number of distinctive image regions K0 can be estimated as
K0 = arg
min MDL(K) .
1≤K≤Kmax
(8.15)
A more recent formulation of information theoretic criterion, the minimum conditional bias
and variance (MCBV) criterion [69, 75], selects a minimum conditional bias and variance
model (i.e., if two models are about equally likely, MCBV selects the one whose parameters
can be estimated with the smallest variance). The formulation is based on the fundamental
argument that the value of the structural parameter cannot be arbitrary or infinite, because
although such an estimate might be said to have low “bias,” the price to be paid is high
“variance” [23].
Since the joint maximum entropy is a function of Ka and r̂, by taking the advantage of
the fact that model estimation is separable in components and structure, we define the MCBV
criterion as
Ka
MCBV(K) = − log L x|r̂ML +
H r̂kML
(8.16)
k=1
where − log(L(x|r̂ML )) is the conditional bias (a form of information theoretic distance) [17,
a
54] and K
k=1 H (r̂kML ) is the conditional variance (a measure of model uncertainty) [51, 54]
of the model. Because both of these terms represent natural estimation errors about their true
models, they can be treated on an equal basis. A minimization of the expression in (8.16) leads
to the following characterization of the optimum estimation:
K0 = arg
min MCBV(K) .
(8.17)
1≤K≤Kmax
That is, if the cost of model variance is defined as the entropy of parameter estimates, the cost
of adding new parameters to the model must be balanced by the reduction they permit in the
ideal code length for the reconstruction error. A practical MCBV formulation with code-length
expression is further given by [17, 75]
Ka
1
MCBV(K) = − log L x|r̂ML +
log 2π eVar r̂kML
2
(8.18)
k=1
where the calculation of H (r̂kML ) requires the estimation of the true ML model parameter
values. It is shown that, for a sufficiently large number of observations, the accuracy of the
© 2001 CRC Press LLC
ML estimation tends quickly to be the best possible accuracy determined by the Cramer–
Rao lower bounds (CRLBs) [51]. Thus, the CRLBs of the parameter estimates are used in
the actual calculation to represent the “conditional” bias and variance [50]. We have found
that, experimentally, the MCBV formulation for determining the value of K0 exhibits very
good performance consistent with both the AIC and the MDL criteria. It should be noted,
however, that these are not the only plausible approaches to the problem of order selection; other
approaches such as cross-validation techniques may also be quite useful [20, 36, 42, 48, 80].
8.2.3
Context Modeling
Once the pixel model is estimated, the segmentation problem is the assignment of labels to
each pixel in the image. A straightforward way is to label pixels into different regions by maximizing the individual likelihood function pk (x) (i.e., to perform ML classification). Usually,
this method may not achieve good performance because it does not use local neighborhood
information in the decision. The CBRL algorithm [27] is one approach that can incorporate the
local neighborhood information into the labeling procedure and thus improve the segmentation performance. The CBRL algorithm to perform/refine pixel labeling based on the localized
FGGM model can be defined as follows [37]:
Let ∂i be the neighborhood of pixel i with an m×m template centered at pixel i. An indicator
function is used to represent the local neighborhood constraints Rij (li , lj ) = I (li , lj ), where
li and lj are labels of pixels i and j , respectively. Note that pairs of labels are now either
compatible or incompatible. Similar to the procedure in [27], one can compute the frequency
of neighbors of pixel i that have the same label values k as at pixel i
(i)
πk = p li = k l∂i =
m2
1
−1
I k, lj
(8.19)
j ∈∂i,j =i
(i)
where l∂i denotes the labels of the neighbors of pixel i. Since πk is a conditional probability
of a region, the localized FGGM pdf of gray-level xi at pixel i is given by
K
(i)
πk pk (xi )
p xi l∂i =
(8.20)
k=1
where pk (xi ) is given in (8.2). Assuming gray values of the image are conditional independent,
the joint pdf of x, given the context labels l, is
P (x|l) =
N K
i=1 k=1
(i)
πk pk (xi )
(8.21)
where l = (li : i = 1, . . . , N ).
It is important to note that the CBRL algorithm can obtain a consistent labeling solution based
on the localized FGGM model (8.20). Since l represents the labeled image, it is consistent if
Si (li ) ≥ Si (k), for all k = 1, . . . , K and for i = 1, . . . , N [27], where
(i)
Si (k) = πk pk (xi ) .
(8.22)
Now we can define
A(l) =
N
i=1
© 2001 CRC Press LLC
k
I (li , k)Si (k)
(8.23)
as the average measure of local consistency and
I (li , k) Si (k), i = 1, . . . , N
LCi =
(8.24)
k
represents the local consistency based on l. The goal is to find a consistent labeling l that
can maximize (8.23). In the real application, each local consistency measure LCi can be
maximized independently. In [27], it has been shown that when Rij (li , lj ) = Rj i (lj , li ), if
A(l) attains a local maximum at l, then l is a consistent labeling.
(0)
Based on the localized FGGM model, li can be initialized by an ML classifier,
(0)
(8.25)
li = arg max pk (xi ) , k = 1, . . . , K .
k
Then, the order of pixels is randomly permutated and each label li is updated to maximize LCi
— that is, classify pixel i into kth region if
(i)
(8.26)
li = arg max πk pk (xi ) , k = 1, . . . , K
k
(i)
where pk (xi ) is given in (8.2) and πk is given in (8.19). By considering (8.25) and (8.26),
we can give a modified CBRL algorithm as follows [37]:
CBRL Algorithm
1. Given l(0) , m=0
2. Update pixel labels
• Randomly visit each pixel for i = 1, . . . , N
• Update its label li according to
(m)
li
3. When
8.2.4
(l(m+1) ⊕l(m) )
N1 N2
= arg
(i)(m)
max πk
pk
k
(xi )
≤ 1%, stop; otherwise, m = m + 1, and repeat step 2.
Applications
Simulated Data
To verify the steps of the statistical image analysis framework we discussed, let us first
consider a simulated image. Our example is the image shown in Figure 8.1a, which is made up
of four overlapping normal components. Each component represents one local region. Noise
levels are set to keep the same signal-to-noise ratio (SNR) between regions, where the SNR is
defined by
SNR = 10 log10
(*µ)2
σ2
(8.27)
where *µ is the mean difference between regions and σ 2 is the noise power. The AIC, MDL,
and MCBV curves as a function of the number of local clusters K are shown in Figure 8.1b.
According to the information theoretic criteria, the minima of these curves indicate the correct number of local regions. From this experimental figure, it is clear that the number of
© 2001 CRC Press LLC
FIGURE 8.1
Experimental results of model selection, algorithm initialization, and final quantification on the simulated image: (a) original image with four components; (b) curves of the
AIC/MDL/MCBV criteria where the minimum corresponds to K0 = 4; (c) initial histogram learning by the ALMHQ algorithm; (d) final histogram learning by the PSOM
algorithm.
Table 8.1 True Parameter Values and the Estimates for the Simulated Image of
Figure 8.1
k
π
µ
σ2
1
0.25
86
400
True
2
3
0.125 0.5
126
166
400
400
4
0.125
206
400
1
0.234
81
235
Initial
2
3
0.234 0.364
131
167
158
157
4
0.185
205
177
1
0.23
84
354
Final
2
3
0.135 0.48
121
164
365
373
4
0.157
201
463
local regions suggested by these criteria are all correct. After the algorithm initialization by
ALMHQ [71], network parameters are finalized by the PSOM algorithm given in (8.9)–(8.11).
The GRE value (8.6) is used as an objective measure to evaluate the accuracy of quantification. The results of the distribution learning by PSOM are shown in Figures 8.1c and d.
The GRE in the initial stage achieves a value of 0.0399 nats, and after the final quantification
by PSOM, is down to 0.008 nats. The numerical results are given in Table 8.1, where the
units of µ and σ 2 simply represent the observed gray levels of the pixel images, whereas π
is the probability measure. To simplify the representation, we omit their units as in [38, 56].
References [69, 70, 73] discuss these examples in more detail and present comparative results
for parameter estimation using EM and PSOM, noting the advantages of PSOM due to its
incremental nature.
Figure 8.2 shows the results of final image segmentation using the CBRL algorithm. We use
the ML classifier to initialize the image segmentation (i.e., to initialize the quantified image by
selecting the pixel label with the largest likelihood at each pixel) using equation (8.25). This
gives a suitable starting point for relaxation labeling [74]. CBRL is then used to fine tune the
image segmentation. Since the true scene is known in this experiment, the percentage of total
classification error is used as the criterion for evaluating the performance of the segmentation
technique. In Figure 8.2, the initial segmentation by the ML classification and the stepwise
results of three iterations in PCRN are presented. In this experiment, algorithm initialization
results in an average misclassification of 30%. It can be clearly seen that a dramatic improvement is obtained after several iterations of the CBRL by using local constraints determined by
the context information. Also, note that the convergence is fast because after the first iteration
most misclassifications are removed. The final percentage of classification errors for Figure 8.2
is about 0.7935%.
© 2001 CRC Press LLC
FIGURE 8.2
Image segmentation by PCRN on simulated image (with initialization by ML classification).
Brain MR Analysis
Quantitative analysis of brain tissues refers to the problem of estimating tissue quantities
from a given image and segmentation of the image into contiguous regions of interest to
describe the anatomical structures. The problem has recently received much attention largely
due to the improved fidelity and resolution of medical imaging systems. Because of its ability
to deliver high resolution and contrast, MR imaging (MRI) has been the dominant modality
for research on this problem [14, 16, 38, 56, 83]. Based on the statistical properties of MR
pixel images, use of an SFNM distribution is justified to model the image histogram, and it
is shown that the SFNM model converges to the true distribution when the pixel images are
asymptotically independent [73].
For this study, we use data consisting of three adjacent, T1-weighted MR images parallel to
the AC–PC line. Since the skull, scalp, and fat in the original brain images do not contribute
to the brain tissue, we edit the MR images to exclude nonbrain structures prior to tissue
quantification and segmentation, as explained in [70, 74]. This also helps us to achieve better
quantification and segmentation of brain tissues by delineation of other tissue types that are
not clinically significant [38, 56, 83]. The extracted brain tissues are shown in Figure 8.3.
Evaluation of different image analysis techniques is a particularly difficult task, and dependability of evaluations by simple mathematical measures such as squared error performance is
questionable. Therefore, most of the time, the quality of the quantified and segmented image usually depends heavily on subjective and qualitative judgments. Besides the evaluation
performed by radiologists, we use the GRE value to reflect the quality of tissue quantification.
Based on the pre-edited MR brain image, the procedure for analysis of tissue types in a slice
© 2001 CRC Press LLC
FIGURE 8.3
Three sample MR brain tissues.
is summarized as follows:
1. For each value of K (number of tissue types), K = Kmin , . . . , Kmax , ML tissue quantification is performed by the PSOM algorithm [equations (8.9)–(8.11)].
2. Scan the values of K = Kmin , . . . , Kmax , and use MCBV (8.16) to determine the suitable
number of tissue types.
3. Select the result of tissue quantification corresponding to the value of K0 determined in
step 2.
4. Initialize tissue segmentation by ML classification (8.25).
5. Finalize tissue segmentation by CBRL [implementing (8.26)].
The performance of tissue quantification and segmentation is then evaluated in terms of the
GRE value, convergence rate, computational complexity, and visual judgment.
The brain is generally composed of three principal tissue types: white matter (WM), gray
matter (GM), and cerebrospinal fluid (CSF), plus their combinations, called the partial volume
effect. We consider the pairwise combinations as well as the triple mixture tissue, defined as
CSF–white–gray (CWG). More important, since the MRI scans clearly show the distinctive
intensities at local brain areas, the functional areas within a tissue type need to be considered.
In particular, the caudate nucleus and putamen are two important local brain functional areas
because, in our complete image analysis framework, we allow the number of tissue types to vary
from slice to slice (i.e., we do consider adaptability to different MR images). We let Kmin = 2
and Kmax = 9 and calculate AIC(K) [eq. (8.12)], MDL(K) [eq. (8.14)], and MCBV(K)
[eq. (8.16)] for K = Kmin , . . . , Kmax . The results with these three criteria are shown in
Figure 8.4, which suggests that the three sample brain images chosen contain 6, 8, and 6 tissue
types, respectively. According to the model fitting procedure using information theoretic
criteria, the minima of these criteria indicate the most appropriate number of tissue types,
which is also the number of hidden nodes in the corresponding PSOM (mixture components
in SFNM). In the calculation of MCBV using (8.18), as discussed, one can use the CRLBs to
represent the conditional variances of the parameter estimates, given by [50]:
πk (1 − πk )
Var π̂kML =
,
N
σ2
Var µ̂kML = k , and
N πk
2σ 4 (N πk − 1)
2
.
= k
Var σ̂kML
N 2 πk2
© 2001 CRC Press LLC
(8.28)
(8.29)
(8.30)
FIGURE 8.4
Results of model selection for slices 1–3 (K0 = 6, 8, 6, left to right).
Note that since the true parameter values in the above equations are not available, their ML
estimates are used to obtain the approximate CRLBs. From Figure 8.4, it is clear that, with
real MR brain images, the overall performance of the three information theoretic criteria is
fairly consistent. However, it is noted that AIC has a tendency to overestimate while MDL has
a tendency to underestimate the number of tissue types [68], and MCBV provides a solution
between those of AIC and MDL, which can be a desirable choice in terms of providing a
balance between the bias and variance of the parameter estimates.
When performing the computation of the information theoretic criteria, we use PSOM to
iteratively quantify different tissue types for each fixed K. The PSOM algorithm is initialized
by the ALMHQ [71]. For slice 2, the results of final tissue quantification with K0 = 7, 8, 9
are shown in Figure 8.5. Table 8.2 gives the numerical result of final tissue quantification for
FIGURE 8.5
Histogram learning for slice 2 (K = 7, 8, 9 from left to right).
slice 2 corresponding to K0 = 8, where a GRE value of 0.02 to 0.04 nats is achieved. These
quantified tissue types agree with those of a physician’s qualitative analysis results [69].
Table 8.2 Result of Parameter Estimation for Slice 2
Tissue Type
π
µ
σ2
1
0.0251
38.848
78.5747
2
0.0373
58.718
42.282
3
0.0512
74.400
56.5608
4
0.071
88.500
34.362
5
0.1046
97.864
24.1167
6
0.1257
105.706
23.8848
7
0.2098
116.642
49.7323
8
0.3752
140.294
96.7227
The CBRL tissue segmentation for slice 2 is performed with K0 = 7, 8, 9, and the algorithm
is initialized by ML classification [eq. (8.25)] [66]. CBRL updates are terminated after 5 to 10
© 2001 CRC Press LLC
iterations since further iterations produced almost identical results. The segmentation results
are shown in Figure 8.6. It is seen that the boundaries of WM, GM, and CSF are successfully
FIGURE 8.6
Results of tissue segmentation for slice 2 with K0 = 7, 8, 9 (from left to right).
delineated. To see the benefit of using information theoretic criteria in determining the number
of tissue types, the decomposed tissue type segments are given in Figure 8.7 with K0 = 8. As
can be observed in Figures 8.6 and 8.7, the segmentation with eight tissue types provides a very
meaningful result. The regions with different gray levels are satisfactorily segmented, and the
major brain tissues are clearly identified. If the number of tissue types were “underestimated”
by one, tissue mixtures located within the putamen and caudate areas would be lumped into
one component, but the results would still be meaningful. When the number of tissue types
is “overestimated” by one, there is no significant difference in the quantification result, but
the white matter would be divided into two components. For K0 = 8, the segmented regions
represent eight types of brain tissues: CSF, CG, CWG, GW, GM, putamen area, caudate area,
and WM, as shown in Figure 8.7. These segmented tissue types again agree with the results
of a radiologist’s evaluation [69].
Mammogram Analysis
Another example application area for the image analysis framework we have introduced is
in segmentation and extraction of suspicious mass areas from mammographic images. With an
appropriate statistical description of various discriminate characteristics of both true and false
candidates from the localized areas, an improved mass detection may be achieved in computeraided diagnosis (Figure 8.8). Preprocessing can be an important tool for analysis depending
on the application. In this example, one type of morphological operation is derived to enhance
disease patterns of suspected masses by cleaning up unrelated background clutters, and then
image segmentation is performed to localize the suspected mass areas using the stochastic
relaxation labeling scheme [35, 37]. The mammograms for this study were selected from
the Mammographic Image Analysis Society (MIAS) database and the Brook Army Medical
Center (BAMC) database created by the Department of Radiology at Georgetown University
Medical Center. The areas of suspicious masses were identified by an expert radiologist based
on visual criteria and biopsy-proven results. The BAMC films were digitized with a laser film
digitizer (Lumiscan 150) at a pixel size of 100 × 100µm and 4096 gray levels (12 bits). Before
the method was applied, the digital mammograms were smoothed by averaging 4 × 4 pixels
into 1 pixel. According to radiologists, the size of small masses is 3 to 15 mm in effective
diameter. A 3-mm object in an original mammogram occupies 30 pixels in a digitized image
with a 100-µm resolution. After the image size is reduced by four times, the object will occupy
the range of about 7 to 8 pixels. An object the size of 7 pixels is expected to be detectable by
© 2001 CRC Press LLC
FIGURE 8.7
Result of tissue type decomposition for slice 2 that represents eight types of brain tissues:
CSF, CG, CWG, GW, GM, putamen area, caudate area, and WM (left to right, top to
bottom).
any computer algorithm. Therefore, the shrinking step is applicable for mass cases and can
save computation time.
Consider the use of the FGGM model and the two information criteria — AIC and MDL
— to determine the mixture number K. Tables 8.3 and 8.4 show the AIC and MDL values
with different K and α of the FGGM model based on one original mammogram. As can be
seen, although with different α, all AIC and MDL values achieve the minimum when K = 8.
This indicates that AIC and MDL are relatively insensitive to the change of α. With this
observation, we can decouple the relation between K and α and choose the appropriate value
of one while fixing the value of the other. Figure 8.9a and b are two examples of AIC and MDL
curves with different K and fixed α = 3.0. Figure 8.9a is based on the original mammogram
and Figure 8.9b is based on the enhanced mammogram. As we can see in Figure 8.9a, both
criteria achieved the minimum when K = 8. It should be noted that although no ground truth
is available in this case, our extensive numerical experiments have shown a very consistent
performance of the model selection procedure and all the conclusions were strongly supported
by the previous independent work reported by [5]. Figure 8.9b indicates that K = 4 is the
appropriate choice for the mammogram enhanced by a dual morphological operation. This is
believed to be reasonable since the number of regions decreases after background correction.
© 2001 CRC Press LLC
FIGURE 8.8
Examples of mass enhancement: (a) original mammogram; (b) enhanced mammogram;
(c) different original mammogram; (d) enhanced result of (c).
The order is then fixed at K = 8, and the value of α is changed for estimating the FGGM
model parameters using the EM algorithm given in Section 8.2.2 with the original mammogram.
The GRE value between the histogram and the estimated FGGM distribution is used as a
measure of the estimation bias, and it is noted that the GRE achieved a minimum distance
when the FGGM parameter α = 3.0, as shown in Figure 8.10. A similar result was shown
when the EM algorithm was applied to the enhanced mammogram with K = 4 (Figure 8.11).
This indicated that the FGGM model might be better than the SFNM model (α = 2.0) for
© 2001 CRC Press LLC
Table 8.3 Computed AICs for the FGGM Model with
Different α
K
α = 1.0
α = 2.0
α = 3.0
α = 4.0
2
3
4
5
6
7
8
9
10
651250
646220
645760
645760
645740
645640
645550(min)
645580
645620
650570
644770
644720
644700
644670
644600
644570(min)
644590
644600
650600
645280
645260
645120
645110
645090
645030(min)
645080
645100
650630
646200
646060
646040
645990
645900
645850(min)
645880
645910
Table 8.4 Computed MDLs for the FGGM Model with
Different α
K
α = 1.0
α = 2.0
α = 3.0
α = 4.0
2
3
4
5
6
7
8
9
10
651270
646260
645860
645850
645790
645720
645680(min)
645710
645790
650590
644810
644770
644770
644750
644700
644690(min)
644710
644750
650630
645360
645280
645280
645150
645120
645100(min)
645140
645180
650660
646350
646150
646100
646090
645930
645900(min)
645930
645960
Table 8.5 Comparison of Segmentation Error Resulting
from Noncontextual and Contextual Methods
GRE value
Soft classification
0.0067
Method
Bayesian classification
0.4406
CBRL
0.1578
mammographic images when the true statistical properties of the mammograms are generally
unknown, although the SFNM has been successfully used in a large number of applications
as well as in our previous example. Hence, the choice of the best model to describe the data
depends on the nature of the data for the given problem.
After the determination of all model parameters, every pixel of the image is labeled to a
different region (from 1 to K) based on the CBRL algorithm. Then, the brightest region,
corresponding to label K, plus a criterion of closed isolated area, is chosen as the candidate
region of suspicious masses. These results are noted to be highly satisfactory when compared
to outlines of the lesions [37]. Also, similar to the previous example, GRE values can be used
to assess the performance of the final segmentation. Table 8.5 shows our evaluation data from
three different segmentation methods when applied to these real images.
© 2001 CRC Press LLC
FIGURE 8.9
The AIC and MDL curves with different number of regions K. (a) The results based
on the original mammogram, the optimal K = 8; (b) the results based on the enhanced
mammogram, the optimal K = 4.
8.3
Graphics Modeling
Reconstruction of a 3D surface from a set of processed images is an important problem in the
presentation and understanding of multimedia data. The data generated by imaging modalities
such as 2D/3D camera, medical imaging, and other imaging devices provide a series of image
slices of the object. The problem is then to infer a 3D representation of the object which will
allow visualization as well as analysis of the geometry parameters of the object. In general,
the 3D reconstruction process consists of three steps:
1. Extracting object contours from 2D cross-sectional images
2. Interpolating the intermediate contours between successive slices or among data points
3. Reconstructing surfaces or volumes from serial cross-sectional contours
Based on various image analysis algorithms, step 1 may be achieved through image segmentation or edge detection, which was discussed in the first part of this chapter and earlier in
this book. In this section, we focus our discussions on steps 2 and 3.
Surface reconstruction is to form surfaces between contours of successive contours. If the
interslice distances between the successive contours is small, the 3D structure of the object can
be captured well by using surface reconstruction methods. However, if the contours are not
closely spaced, the empty space between contours should be filled before surface reconstruction
methods are applied. This procedure is usually referred to as contour interpolation. Many
interpolation methods have been developed for various applications. For example, a linear
interpolation algorithm is proposed in [82] to reconstruct prostatectomy specimens together
with an enhanced extrapolation algorithm to overcome the difficulties in branching shapes and
concave surfaces. This method is similar to the shape interpolation method described in [52]
but has limitations in working with hemispherical shapes or round objects, primarily because
of its linear characteristics. The elastic interpolation method given in [10] performs nonlinear
contour interpolation by generating a series of intermediate contours filling the gap between the
start and the goal contours. This method is based on Burr’s dynamic elastic contour model [8]
and can handle the branching situation very well by using union and/or intersection operators.
© 2001 CRC Press LLC
FIGURE 8.10
The comparison of learning curves and histogram of the original mammogram with
different α; K = 8. The optimal α = 3.0. (a) α = 1.0, GRE = 0.0783; (b) α = 2.0, GRE
= 0.0369; (c) α = 3.0, GRE = 0.0251; (d) α = 4.0, GRE = 0.0282.
Many researchers have proposed using the elastic contour interpolation method for interpolating intermediate contours from the initial contours. The interpolation method assigns
contours with the elastic property, and then, by applying forces onto them, deforms the start
contour to conform to the goal contour. For example, a deformable surface–spine model has
been proposed in [86, 87] to reconstruct the surface model from the interpolated contours. The
deformable surface–spine model is a coupled dynamic system, where the surface and spine
are confined in the following way: a deformable spine (axis) is determined from its contours,
then all the surface patches are contracted to the spine through expansion/compression forces
radiating from the spine while the spine itself is also confined to the surfaces. The surface
refinement is governed by a second-order partial differential equation from Lagrangian mechanics, and the refining process is accomplished when the energy of this dynamic deformable
surface–spine model reaches its minimum. A finite-element method is further used to solve the
dynamic Lagrangian equation by constructing 9-degree-of-freedom (dof) triangular elements
and 4-dof spine elements. In sum, both the elastic interpolation method and the deformable
surface–spine model can be jointly used for building 3D graphics models for visualization and
animation.
A contour on a plane z = zk can be defined as a linked list of vertices: C = {(xi , yi ), 1 ≤ i ≤
N }, or equivalently, defined as a concatenation of linked line segments where a line segment is
represented by its two end vertices (xi , yi ) and (xi+1 , yi+1 ). Note that we drop z coordinates
in the above expressions to simplify the notations. Given a start contour C1 = {(x1i , y1i ), 1 ≤
i ≤ N1 } and a goal contour C2 = {(x2i , y2i ), 1 ≤ i ≤ N2 }, to interpolate between the start
and goal contours, one must find a particular “force field” acting on the start contour and try to
© 2001 CRC Press LLC
FIGURE 8.11
The comparison of learning curves and histogram of the enhanced mammogram with
different α; K = 4. The optimal α = 3.0. (a) α = 1.0, GRE = 0.0493; (b) α = 2.0, GRE
= 0.0126; (c) α = 3.0, GRE = 0.0105; (d) α = 4.0, GRE = 0.0676.
deform it to conform to the goal contour. Thus, a three-step procedure is designed to achieve
this task as described below [39].
1. Finding the Closest Line Segment
Let P1i and P1(i+1) denote a line segment on the first contour C1 , and P2j and P2(j +1) a line
segment on the second contour C2 . In order to find the closest line segment of each vertex,
a distance measure including the Euclidean distance and orientation property (the directional
incompatibility) is used.
The directional incompatibility φ(i, j ) between a vertex P1i of C1 and a line segment
between the vertices P2j and P2(j +1) of C2 is defined as
φ(i, j ) =
|(P1(i+1) − P1i ) × (P2(j +1) − P2j )|
.
|P1(i+1) − P1i | |P2(j +1) − P2j |
(8.31)
The above equation tells us that φ(i, j ) = sin θ, where θ is the angle between two vectors
P1(i+1) − P1i and P2(j +1) − P2j . The Euclidean distance from a vertex P1i to a line segment
B|,
where 0 ≤ θ ≤ π, A = P1i − P2j , and B =
of P2(j +1) − P2j is η(i, j ) = |A × B|/|
B|
a term |R·
P2(j +1) − P2j . Additionally, if A cos θ < 0 or A cos θ > |B|,
has to be included
|B|
, and P is either the point P or P
in η(i, j ), where R = P1i − P2j
2j
2(j +1) depending on
2j
which one is closer to P1i . Then, the total distance between a vertex P1i and a line segment of
P2(j +1) − P2j is defined as the weighted sum, d(i, j ) = φ(i, j ) + ω η(i, j ), where ω is the
© 2001 CRC Press LLC
FIGURE 8.12
(a) The suspected mass segmentation results based on the original mammogram, (b) the
results based on the enhanced mammogram, K = 4, α = 3.0. (c) and (d) are the results
based on another original mammogram and its enhanced image.
weight that in practice can be set to 1 in most cases. The closest line segment can be determined
by finding the point index Ji giving minimum distance d(i, j ) — that is, minj (i, j ) = d(i, Ji ).
2. Determining Displacement and Force Field
The displacement vector associated with a vertex P1i and a line segment between the vertices
P2Ji and P2(Ji +1) is defined as
© 2001 CRC Press LLC
1 (i, Ji ) =
D

− P ,
 P2J
1i
i
or A cos θ ≤ 0
if A cos θ ≥ |B|
 A sin θ
,
if 0 < A cos θ < |B|
By î−Bx jˆ
,
|B|
(8.32)
is the point P
where P2J
2Ji or P2(Ji +1) depending on which is closer to P1i . Similarly, by
i
2 (j, Ij ) can be
reversing the roles of the start and the goal contours, the displacement vector D
determined at each vertex P2 of C2 .
A force field is then defined as a function of the “pushing” and “pulling” forces:
N
N2
1
j =1 G2j D2 (j, Ij )
i=1 G1i D1 (i, Ji )
−1
,
(8.33)
−
F (x, y) = γ
N1
N2
i=1 G1i
j =1 G2j
where G1i and G2j are designed to provide the effect that close neighbors have more influence
than that of far neighbors, and they can be defined as Gaussian functions with covariance σk
defined as σk = σ0 f −k , where f is a constant 1 ≤ f ≤ 2, and γ can be regarded as a damping
coefficient. For a discussion of how these parameters affect the dynamic behavior of the elastic
contour model, see [39].
3. Generating Intermediate Contours
Consider a start contour C1 and a goal contour C2 . One can compute the initial force field F 0
according to the method described in step 2. Using F 0 , one defines the contour Ik+1 from Ik
interactively by providing I0 = C1 ; that is,
Ik+1 = Ik + F k (xki , yki ) .
8.3.1
(8.34)
Surface Reconstruction
Surface reconstruction is usually achieved by forming triangular patches between successive
pairs of contours, which is often referred to as the tiling problem or triangulation problem in
the literature. Solutions to the tiling problem can be categorized into two groups: (1) optimal
approaches in some given criterion and (2) primarily heuristic approaches. Optimal methods
provide the best triangulation in the sense of the given criterion and are often based on a graph
description where a path in the graph defines a possible solution. A cost function (criterion)
is assigned to each arc of the graph, and the optimal solution is obtained by finding the path
with minimum or maximum cost function in the graph. For example, one can use maximizing
volume as a cost function or use minimizing area instead. These two methods produce good
results in practice, although the second method is preferred over the first because there is no
need to deal separately with the convex and concave parts of objects. Heuristic approaches, on
the other hand, are computationally less expensive and they usually define triangular patches
one by one using only a local decision criterion. For instance, the triangular patches can be
sequentially determined by choosing the shorter edge of two possible edges defining a patch.
Most heuristic methods suffice when contours are similar in shape and orientation and are
mutually centered. However, if contours are very different in shape, orientation, and position,
heuristic methods can produce incorrect results.
In contrast to linear methods (tiling triangular patches), nonlinear surface reconstruction
methods have been intensively proposed and studied. For example, a uniform B-spline approach has been developed to represent sectional contours and to further interpolate the surface
between slices. A Hermite interpolation function with curvature sampling and a fast nearest
mapping algorithm between two cross-sections is also proposed to perform nonlinear surface
© 2001 CRC Press LLC
reconstruction using physically based deformable modeling. A more elegant approach to
surface reconstruction using physical deformable models has been recently developed in the
computer vision community and is now widely used in many areas such as computer graphics
and animation, dynamics simulation, and modeling. We discuss this approach next.
8.3.2
Physical Deformable Models
Deformable models are based on variational principles of continuum mechanics. These dynamic principles are usually expressed in the form of dynamic differential equations. Elastic
models [63] simulate nonlinear elastic materials. They incorporate deformation energies that
are invariant with respect to rigid-body motions, impart no deformation, and grow monotonically with the magnitude of the deformation. The energy functionals are expressed as integral
measures of the instantaneous deformation of a model away from its prescribed reference shape.
The deformation is quantified in a convenient way using the fundamental forms of differential
geometry (metrics, curvatures, etc.). Lagrange equations of motion balance the resulting elastic forces against inertial forces due to the mass distribution of the model, frictional damping
forces, and externally applied forces. Elastically deformable models can efficiently model a
variety of smooth objects with different shapes. They can also dynamically respond to external
forces, which is very important in modeling human organs for the purpose of surgical planning
and simulation in particular. Several deformable models (e.g., controlled-continuity splines
under tension [61], symmetry-seeking models [64], and deformable superquadrics [61]), have
been developed and applied to surface reconstruction [64], shape and motion recovery [44],
and object recognition [57]. A dynamic finite-element surface model was proposed by Terzopoulos to track moving anatomical structures (e.g., the left ventricle) in 4D cardiac images
for functional deformation analysis [59].
Inelastic models [91] are a powerful model-building medium. Unlike elastic models, which
immediately regain their natural, undeformed shapes, inelastic models are commonly associated with high-polymer solids such as modeling clay or silicon putty. Consequently, inelastic
models serve as a sort of freely sculptable computational plasticine. Free-form shapes may be
created by interactively applying simulated forces on the inelastic model to stretch, squash, and
mold it. Inelastic models tractably simulate three canonical inelastic behaviors — viscoelasticity, plasticity, and fracture. These behaviors may be incorporated into any of the elastic
models described above by introducing internal processes that dynamically control resilience
and fragility as a function of deformation.
Stochastic models combine deterministic deformable behaviors with random processes.
This leads to the marriage of two well-known modeling techniques: splines and fractals. On
one hand, spline shapes are easily constrained and are suitable for modeling smooth, manmade objects such as teapots, whereas fractals, although difficult to constrain, are suitable
for synthesizing the various irregular shapes found in nature, such as a mountainous terrain.
Constrained fractals are a class of deformable models that combine these seemingly opposed
features by exploiting the remarkable relationship between fractals and generalized energyminimizing splines, which may be derived through Fourier analysis. Constrained fractals
are generated by a stochastic relaxation algorithm that bombards a spline subject to shape
constraints with modulated white noise, letting the spline diffuse the noise into the desired
fractal spectrum as it settles into equilibrium. In general, elastically deformable models are
suitable to model relatively smooth objects, whereas inelastic models have the potential to
model complex (moderately irregular) objects. On the other hand, stochastic deformable
models are extremely important to model the various irregular shapes found in nature, such as
mountainous terrain.
© 2001 CRC Press LLC
8.3.3
Deformable Surface–Spine Models
The surface and spine can be defined as geometric mappings from material (parametric)
coordinate domains into 3D Euclidean space 3 . The surface can be defined by the following
mapping M:
M : (u, v) → x(u, v, t) = (x(u, v, t), y(u, v, t), z(u, v, t)) ,
(8.35)
where (u, v) ∈ [0, 1]2 are the bivariate material coordinates; x(u, v, t), y(u, v, t), and
z(u, v, t) are the coordinates of a point on the surface in 3 ; and t denotes the time-varying
property of the deformable surface. Similarly, the spine can be defined by the mapping m:
m : s → x(s, t) = (x(s, t), y(s, t), z(s, t)) ,
(8.36)
where s ∈ [0, 1] is the univariate material coordinate and x(s, t), y(s, t), and z(s, t) are the
coordinates of a point on the spine in 3 .
The strain energy E can be found to characterize the deformable material of either the surface
or the spine, which will be discussed in the next section as an instance of the spline function.
Then the continuum mechanical equation
µ
∂ 2x
∂x δE(x)
+γ
+
= f(x)
2
∂t
δx
∂t
(8.37)
governs the nonrigid motion of the surface (spine) in response to an extrinsic force f(x), where
µ is the mass density function of the deformable surface (spine) and γ is the viscosity function
of the ambient medium. The third term on the left-hand side of the equation is the variational
derivative of the strain energy functional E, the internal elastic force of the surface (spine).
The deformable energy of surface x(u, v, t) can be defined by
2
2
1 1
∂x ∂x ∂x ∂x Esurface (u, v, t) =
w10 + 2w11 × + w01 ∂u
∂u
∂v
∂v
0
0
2 2
2 2
2 2 ∂ x
∂ x + w02 ∂ x du dv , (8.38)
+ w20 2 + 2w22 2
∂u∂v
∂u
∂v where the weights w10 , w11 , and w01 control the tensions of the surface and w20 , w22 , and w02
control its rigidities (bending energy). The deformable energy of spine x(u, t) is given by
2 2 1 2
dx d x
Espine (s, t) =
w1 + w2 2 ds .
(8.39)
ds
ds
0
The weight w1 controls the tension along the spine (stretching energy), while w2 controls its
rigidity (bending energy).
To couple the surface with the spine, one should enforce v ≡ s, which maps the spine
coordinate into the coordinate along the length of the surface. Then connect the spine with the
surface by introducing the following forces on the surface and spine, respectively [64]:
a
fsurface
(8.40)
(u, s, t) = −(a/ l) x̄surface − xspine
a
fspine (s, t) = a x̄surface − xspine
(8.41)
where a controls the strength of the forces; x̄surface is the centroid of the coordinate
curve (s
∂xsurface 1 1
= constant) circling the surface and defined as x̄surface = l 0 xsurface ∂u du, where l is
© 2001 CRC Press LLC
1 the length given by l = 0 ∂xsurface
∂u du. In general, the above forces coerce the spine staying
on an axial position of the surface. Further, if necessary, we can encourage the surface to be
radially symmetric around the spine by introducing the following force:
b
fsurface
= b (r̄ − |r|) r̂ ,
(8.42)
where b controls the strength of the force; r is the radial vector of the surface with respect
to the spine as r(u, s) = xsurface − xspine ; the unit radial vector r̂(u, s) = r/|r|; and r̄(s) =
∂xsurface
1 1
du, as the mean radius of the coordinate curve s = constant. Also, it is possible
l 0 |r| ∂u
to provide control over expansion and contraction of the surface around the spine. This can be
realized by introducing the following force:
c
= cr̂ ,
fsurface
(8.43)
where c controls the strength of the expansion or contraction force. The surface will inflate if
c > 0 and deflate if c < 0.
Summing the above coupling forces in the motion equation associated with surface and spine,
we obtain the following dynamic system describing the motion of the deformable surface–spine
model:
µ
µ
∂xsurface
δEsurface
∂ 2 xsurface
ext
+γ
+
= fsurface
2
∂t
δx
∂t
a
b
c
+ fsurface
+ fsurface
,
+ fsurface
∂ 2 xspine
∂xspine
δEspine
ext
a
+γ
+ fspine
,
+
= fspine
2
∂t
δx
∂t
(8.44)
(8.45)
ext
ext is the external force applied
is the external force applied on the surface and fspine
where fsurface
on the spine.
Both the finite difference method and the finite element method can be used to compute
the numerical solution to the surface xsurface and spine xspine . The finite difference method
approximates the continuous function x as a set of discrete nodes in space. A disadvantage of
the finite difference approach is that the continuity of the solution between nodes is not made
explicitly. The finite-element method, on the other hand, provides continuous surface (or spine)
approximation by approximating the unknown function x in terms of combinations of the basis
functions. In the finite element method, we first tessellate the continuous material domain,
(u, v) for the surface and s for the spine in our case, into a mesh of m element subdomains
Dj , and then we approximate
x as a weighted sum of continuous basis functions Ni (so-called
shape functions): x ≈ xh = i xi Ni , where xi is a vector of nodal variables associated with
mesh node i. The shape functions Ni are fixed in advance and the nodal variables xi are the
unknowns. The motion equation can then be discretized as
M
∂x
∂ 2x
+ Kx = F ,
+C
2
∂t
∂t
(8.46)
where x = [x1T , . . . , xiT , . . . , xnT ], M is the mass matrix, C the damping matrix, K the stiff
matrix, and F the forcing matrix. M, C, and F can be obtained as follows:
µNjT Nj du dv ,
(8.47)
Mj =
Cj =
Fj =
© 2001 CRC Press LLC
Ej
Ej
Ej
γ NjT Nj du dv,
(8.48)
NjT fj du dv .
(8.49)
To compute K, use the following equation:
Kj =
NbT βNb + NsT αNs du dv ,
Ej
(8.50)
where
Nb =
Ns =
α=
β=
∂ 2N ∂ 2N ∂ 2N
,
,
∂u2 ∂u∂v ∂v 2
∂N ∂N T
,
∂u ∂v
w02 w22
w22 w20


w01 0 0
 0 w11 0 
0 0 w10
T
(8.51)
(8.52)
(8.53)
(8.54)
The deformable surface consists of a set of connected triangular elements chosen for their
ability to model a large range of topological shapes. Barycentric coordinates in two dimensions
are the natural choice for defining shape functions over a triangular domain. Barycentric
coordinates (L1 , L2 , L3 ) are defined by the following mapping with material coordinates
(u, v):
  
 
u
u1 u2 u3
L1
 v  =  v1 v 2 v3   L 2  ,
(8.55)
1
1 1 1
L3
where (u1 , v1 ), (u2 , v2 ), and (u3 , v3 ) are the coordinates of three vertex locations of the triangle.
We can use the 9-dof triangular element, which includes the position and its first parametric
partial derivatives at each triangle vertex, as shown in Figure 8.12a. The shape functions of
the first node in a 9-dof triangle are [91]:

N19
T



L1 + L21 L2 + L21 L3 − L1 L22 − L1 L23
  
2
2

 

=
 N2  =  c3 L1 L2 + 0.5L1 L2 L3 − c2 L1 L3 + 0.5L1 L2 L3  .
2
2
N3
−b3 L1 L2 + 0.5L1 L2 L3 + b2 L1 L3 + 0.5L1 L2 L3
N1
(8.56)
The triangle’s symmetry in Barycentric coordinates can be used to generate the shape function
for the second and third nodes in terms of the first. To generate N29 , use the above equations
but add a 1 to each index so that 1 → 2, 2 → 3, and 3 → 1. The N39 functions can be obtained
by adding another 1 to each index. Note that the shape functions for a 9-dof triangle do not
guarantee C 1 continuity between adjacent triangular elements. In [91], a 12-dof triangular
element can be made C 1 continuous by adding 1 dof on each edge of the triangle (see [9]
for details). An alternative to having a C 1 continuous triangular element is to use an 18-dof
element which includes the nodal location, with its first and second partial derivatives evaluated
at each node [43]. We use a 9-dof triangular element although the extension to 12- or 18-dof
triangular elements is straightforward.
The finite element of the spine has 4 dof between two nodes located at the ends of the
segment. The dof at each node correspond to its position and tangent. The spine segment can be
© 2001 CRC Press LLC
approximated as the weighted sum of a set of Hermite polynomials: x ≈ xh (s) =
where Ni , i = 0, . . . , 3 are given as follows:
3
i=0 xi Ni ,
N0 = 1 − 3(s/ h)2 + 2(s/ h)3 ,
N1 = h(s/ h − 2(s/ h)2 + (s/ h)3 ),
N2 = 3(s/ h)2 − 2(s/ h)3 ,
N3 = h(−(s/ h)2 + (s/ h)3 ) ,
(8.57)
where h is the parametric element length.
8.3.4
Numerical Implementation
The deformable surface–spine model can be stabilized during the fitting process if its motion
is critically damped to minimize vibrations. Critical damping can be achieved by appropriately balancing the mass and damping distributions. A simple way of eliminating vibration
while preserving useful dynamics is to set the mass density in equation (8.46) to zero, thus
reducing (8.46) to
C
∂x
+ Kx = F .
∂t
(8.58)
This first-order dynamic system governs the model which has no inertia and comes to rest
as soon as all the forces balance. We integrate equation (8.58) using an explicit first-order
Euler method. The method begins with a simple forward difference approximation. Consider
extrapolation from time level t to t + *t by forward differencing at t. The usual Taylor series
expansion at time t has the form
x(t + *t) = x(t) + *t
dx
(*t)2 d 2 x
(t) +
(A),
∂t
2! ∂t 2
A ∈ (0, t) ,
(8.59)
which yields the forward difference approximation
x(t + *t) − x(t)
dx
=
∂t
*t
(8.60)
and is only O(*t) accurate. Using this forward difference approximation and transposing
terms involving x(t), we have
Cx(t + *t) = (C − *tK)x(t) + *tF(t)
Thus, we obtain the updating formula for x from time t to t + *t as follows:
x(t + *t) = I − *tC−1 K x(t) + *tC−1 F(t)
(8.61)
(8.62)
It is well known that finite difference methods for initial-value systems yield expressions very
similar to the above results obtained by finite element schemes. A noteworthy distinction is that
the coefficient matrix C for finite differencing is diagonal in the usual difference approximation.
This leads, in the forward difference approximation, to more efficient algorithms for solving
the problem. In the finite element method, C is often sparse and ill conditioned, which causes
difficulty in the computation of C−1 . To obtain C−1 , then, computationally complex singular
decomposition methods have to be used. However, there is a physical solution in computational
mechanics, called the lumping procedure, which overcomes the difficulty with the sparseness
and ill-conditioning of matrix C. The idea can be interpreted physically as replacing the
© 2001 CRC Press LLC
continuous material with the distributed mass by the concentrated material with lumped mass
(beads) at the nodes. In practice, there are several ways to perform such a lumping procedure,
such as using modified shape functions or different numerical integral methods. Among those,
the easiest way is keeping only the diagonal coefficients of C and discarding all the off-diagonal
coefficients, which is the approach in solving the above-mentioned dynamic equation of the
deformable surface–spine model.
8.3.5
Applications
In this section, we present applications of the algorithms we presented for graphical modeling, reconstruction, and representation with both discontinuity-embedded and smooth objects.
The discontinuity-embedded deformable model defines a dynamic finite element representation with both continuous and discontinuous components as described in the previous section.
First, we apply our new deformable model to reconstruct several synthetic and range datasets
to illustrate its performance in recovering depth discontinuities in the final reconstructed surfaces (see Figure 8.13). In order to extract the contours of the object, we use Canny’s edge
operator to detect and locate depth discontinuities in the datasets (see Figure 8.14). It is not
a trivial task in general to detect surface discontinuities; however, we assume such location
information of depth discontinuities can be provided as a priori knowledge in our experiments
(see Figure 8.15). We then initialize the discontinuity-embedded deformable model by a finiteelement tessellation where the discontinuity path is identified within each element. The dataset
acts as the external force to dynamically deform the model in order to fit the surface to the
dataset. The final reconstructed surface is obtained when the dynamic motion equation reaches
it equilibrium. Figure 8.16 shows the reconstructed object surface.
FIGURE 8.13
Range image of a simply synthetic object, where the dataset contains both smooth and
discontinuous surfaces.
In Figures 8.17–8.21, we present the synthetic step data and the reconstruction results by
the elastically deformable model and our discontinuity-embedded deformable model. As we
can see, the elastically deformable model smooth over the depth discontinuity whereas the
discontinuity-embedded deformable model recovers the depth discontinuity explicitly. With
the discontinuity location information prescribed, the discontinuity-embedded deformable
model incorporates a discontinuity component into its representation and dynamically conforms to the data in both continuous and discontinuous parts. The example is a tool synthetic dataset illustrated in Figure 8.17. With the location of the depth discontinuity, the
© 2001 CRC Press LLC
FIGURE 8.14
Frame representation of the synthetic object, where the boundaries can be extracted by
various methods.
FIGURE 8.15
Gradient vector field of the synthetic object after an appropriate pre-processing step.
FIGURE 8.16
The reconstructed surface of the synthetic object, by incorporating the information regarding both continuous and discontinuous representations.
© 2001 CRC Press LLC
FIGURE 8.17
The contours of an object with deformable characteristics.
FIGURE 8.18
The reconstructed frame of the object by incorporating only the term representing the
deformation property with smoothness constraint.
FIGURE 8.19
The reconstructed surface of the object showing a clear mismatch from the original
contours.
© 2001 CRC Press LLC
FIGURE 8.20
The reconstructed frame of the object by incorporating both the terms representing the
deformation and discontinuity properties.
FIGURE 8.21
The reconstructed surface of the object showing a very satisfactory representation of the
original object.
discontinuity-embedded deformable model is capable of recovering the jump height on the
depth discontinuity (see Figure 8.20). As we see in the reconstructed surfaces by the elastically deformable model, the depth discontinuities are oversmoothed and hard to identify and
localize (see Figures 8.18 and 8.19). The loss of discontinuities will obviously affect the
outcome of those high-level processes such as object recognition. The depth discontinuities
are well recovered by our discontinuity-embedded deformable model, as seen in Figure 8.21.
With such a discontinuity-preserving surface reconstruction, high-level processes can easily
extract the object boundary information to achieve the ultimate goal–object recognition. Since
the discontinuity-embedded deformable model includes the conventional continuous component represented as in the elastically deformable model, all the advantages of the elastically
deformable model are kept in the discontinuity-embedded deformable model for representing
© 2001 CRC Press LLC
complex-structured smooth objects. Furthermore, the discontinuity-embedded deformable
model can achieve more accurate representation of surfaces with discontinuities than that of
the elastically deformable model.
We also applied our method to the reconstruction of the prostate model. A typical slice image
of the surgical prostate is shown in Figure 8.22, with the contours of the prostate capsule as
FIGURE 8.22
The contours of a three-dimensional prostate model consisting of multiple objects.
they are stacked in 3D. Next we can apply our elastic contour interpolation method to extracted
contours and then use our deformable surface–spine model to reconstruct 3D surgical prostate
models. We have developed reconstruction software with a graphical user interface (GUI)
that can interact with users to specify a number of parameters such as number of slices to
be inserted, the damping factor, and Gaussian smoothing variance in an elastic interpolation
algorithm. Figure 8.23 shows the interpolated contours of the prostate capsule with six slices
FIGURE 8.23
The 3D frame of the prostate model using an elastic interpolation method, where six
virtual slices are generated between the two adjacent original slices.
being inserted using elastic contour models. As we can see, the nonlinear elastic interpolation
method gives a very consistent result with original extracted contours. A triangular tiling
algorithm has also been implemented in our reconstruction software using minimizing area
© 2001 CRC Press LLC
as the cost function. The tiled triangular paths are then constructed, which result in a linear
surface model of the prostate capsule.
FIGURE 8.24
The reconstructed 3D model of the prostate.
FIGURE 8.25
The complete reconstructed 3D prostate model rendered transparently with all anatomical structures such as the prostate capsule, urethra, seminal vesicles ejaculatory ducts,
and carcinomas.
To improve accuracy in surface reconstruction, a sophisticated deformable surface–spine
model has been developed using the finite element method to obtain a continuous surface representation. The dynamics of the deformable model are governed by continuum mechanics,
known as the Lagrangian differential equation. In this experiment, external forces are determined by a sum of Gaussian weighted distances between goal contours and the surface of the
model. The coupled forces between the surface and spine are also computed according to
their relative positions to enforce the spine’s staying on the axial position of the surface. The
inflation or deflation forces are controlled in that we gradually reduce their strength. Initially
the surface of the model is expanded or contracted largely to the goal contours, and when
it is close to the goal contours we force the inflation or deflation forces to disappear. Since
our finite element method uses the position and tangents of each node as nodal parameters,
© 2001 CRC Press LLC
the reconstructed surface is of great smoothness, which outperforms the triangulation tiling
approach. Figure 8.24 shows our final reconstructed surface model of the prostate capsule that
possesses a very finely detailed surface description and a high fidelity to the original specimen.
Figures 8.25 and 8.26 show a completely reconstructed 3D prostate model rendered with all
anatomical structures such as the prostate capsule, urethra, seminal vesicles, ejaculatory ducts,
and carcinomas.
FIGURE 8.26
An transrectal ultrasound guided needle biopsy simulation interface, where the simulated
ultrasound probe and image beam is integrated with the reconstructed 3D virtual reality
scene.
FIGURE 8.27
Virtual flying-through the complete reconstructed 3D prostate model, where a mistargeted biopsy needle simulation is presented.
Interactive visualization of the 3D prostate model is achieved by using the state-of-the-art
graphics toolkit, object-oriented OpenInventor. With a sophisticated set of various kinds of
lights, 3D manipulators, and color and material editors, etc., we can examine the 3D prostate
model in any viewpoint and interactively walk through it to better understand the relationships
of the anatomical structures of the prostate and its tumors. The 3D stereo-glasses and a 3D
mouse and trackball allow a full view of the 3D surgical prostate model, allowing the viewer
© 2001 CRC Press LLC
(e.g., a surgeon) to examine the prostate. Furthermore, we have developed a system for imageguided needle biopsy simulation based on the reconstructed 3D surgical prostate model (see
Figure 8.27). Different ultrasound-like imaging probes are simulated to provide axially and/or
FIGURE 8.28
Display of an ultrasound image section.
FIGURE 8.29
Graphical representation of the same scene as shown in Figure 8.28.
longitudinally oriented sectional images for efficiently planning needle pathways. Figures 8.28
and 8.29 show a 2D sectional image of contours and a 3D view of the inside prostate by removing
the other half of the prostate. Needles with or without triggers are constructed and simulated
to perform the actual biopsy on 3D computerized prostate models according to the planned
needle pathways (see Figures 8.30 and 8.31). With an accurate 3D prostate model, realistic
imaging probes, and needles provided by our virtual simulation system, a surgeon can sit before
the computer to plan better needle paths and further to practice the actual biopsy procedure
before he/she actually performs on a patient. More important, by analyzing outcomes of this
simulation, we can validate the effectiveness of various biopsy techniques in prostate cancer
detection and tumor volume estimation [87].
© 2001 CRC Press LLC
FIGURE 8.30
An ultrasound image showing the prostate tumor, pointed out by the two arrows.
FIGURE 8.31
Demonstration of a planned needle biopsy procedure, including the target, needle tip,
and biopsy pathway.
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© 2001 CRC Press LLC
Chapter 9
Combined Motion Estimation and Transform
Coding in Compressed Domain
Ut-Va Koc and K.J. Ray Liu
9.1
Introduction
The motion-compensated discrete cosine transform (DCT) video compression scheme (MCDCT) is the basis of a number of international video coding standards, which are tabulated in Table 9.1, ranging from the low-bit-rate, and high-compression-rate videophone application to the
high-end, high-bit-rate, and high-quality HDTV application requiring a modest compression
rate. The MC-DCT scheme belongs to the class of hybrid spatial/temporal waveform-based
video compression approaches [1, 2, 3]. As illustrated in Figure 9.1, the MC-DCT scheme
employs motion estimation and compensation to reduce or remove temporal redundancy and
then uses DCT to exploit spatial correlation among the pixels of the motion-compensated predicted frame errors (residuals). Efficient coding is accomplished by adding the quantization
and variable-length coding steps after the DCT block. The coding model block rearranges the
2D DCT coefficients into a 1D order, usually in a zigzag manner. Basically all the standards
in Table 9.1 follow this procedure with modifications of each step to reach different targeted
bit rates and application goals. As these standards are becoming more and more prevalent in
various forms of video products, efficient and cost-effective implementations of the standards
become more important.
Table 9.1 DCT-Based Motion-Compensated Video Coding Standards
Standard
Application
H.261
MPEG-1
MPEG-2
HDTV
DVB
H.263
MPEG-4
Teleconferencing over ISDN
Video on CD-ROM
Generic high-bit-rate applications
U.S. terrestrial broadcast
European digital video broadcast
Low-bit-rate communications over PSTN
Object-based applications
Targeted
Bit Rate
p × 64 kbps
< 1.5 Mbps
> 1.5 Mbps
18 Mbps
6–38 Mbps
∼ 18 kbps
< 50 Mbps
Remark
Also called H.262
Based on MPEG-2
Based on MPEG-2
The implementation of a standard-compliant coder usually requires the conventional MCDCT video coder structure, as shown in Figure 9.2a, where the DCT unit and the block-based
motion estimation unit are two essential elements to achieve spatial and temporal compression,
© 2001 CRC Press LLC
FIGURE 9.1
Motion-compensated DCT (MC-DCT) scheme.
respectively. The feedback loop for temporal prediction consists of a DCT, an inverse DCT
FIGURE 9.2
Different MC-DCT video coder structures: (a) motion estimation/compensation in the
spatial domain; (b) motion estimation/compensation completely in the transform (DCT)
domain.
(IDCT) and a spatial domain motion estimator (SD-ME), which is usually the full-search blockmatching approach (BKM). This is undesirable. Besides adding complexity to the overall
architecture, this feedback loop limits the throughput of the coder and becomes the bottleneck
of a real-time high-end video codec. A compromise is to remove the loop and perform openloop motion estimation based on original images instead of reconstructed images in sacrifice
of the performance of the coder [2, 4]. The savings comes from the removal of the components
IDCT and Q−1 , but the quality of the reconstructed images at the decode may gradually degrade
because the decoder has no access to the original images, only the reconstructed images.
In this chapter, we propose a nonconventional coder structure, the fully DCT-based motioncompensated video coder structure, suitable for all the hybrid rectangularly shaped DCT
motion-compensated video coding standards. In Section 9.2, we discuss the advantages of
this proposed structure over the traditional architecture. The realization of this fully DCTbased architecture requires algorithms to estimate and compensate motions completely in the
© 2001 CRC Press LLC
DCT domain. In this way, we can combine motion estimation and transform coding completely
in the compressed (transform) domain without the need to convert the DCT coefficients back
to the spatial domain for motion estimation and compensation. In Section 9.4, we develop the
DCT-based motion estimation algorithms based on the DCT pseudo-phase techniques covered
in Section 9.3. Then we extend the DCT pseudo-phase techniques to the subpixel level in
Section 9.5. From these subpixel techniques, we can estimate subpixel motion in the DCT
domain without the need of image interpolation (Section 9.6). To complete the feedback
loop of the structure, we explore and devise the DCT-based motion compensation algorithms
(Section 9.7). Our conclusions are provided in Section 9.8.
9.2
Fully DCT-Based Motion-Compensated Video Coder Structure
In the conventional hybrid coder structure, as shown in Figure 9.2a, the presence of the
IDCT block inside the feedback loop of the conventional video coder design comes from the
fact that currently available motion estimation algorithms can estimate motion only in the
spatial domain rather than directly in the DCT domain. Therefore, developing a transform
domain motion estimation algorithm will eliminate this IDCT. Furthermore, the DCT block in
the feedback loop is used to compute the DCT coefficients of motion-compensated residuals.
However, for motion compensation in the DCT domain, this DCT block can be moved out of
the feedback loop. From these two observations, an alternative solution without degradation of
the performance is to develop motion estimation and compensation algorithms that can work in
the DCT domain. In this way, the DCT can be moved out of the loop as depicted in Figure 9.2b
and, thus, the operating speed of this DCT can be reduced to the data rate of the incoming
stream. Moreover, the IDCT is removed from the feedback loop, which now has only two
simple components Q and Q−1 (the quantizers) in addition to the transform domain motion
estimator (TD-ME). This not only reduces the complexity of the coder but also resolves the
bottleneck problem without any trade-off of performance. In this chapter, we demonstrate that
all the essential components (DCT-based motion estimation and compensation) of this fully
DCT-based coder provide comparable performance with less complexity than the pixel-based
full-search approaches. Furthermore, different components can be jointly optimized if they
operate in the same transform domain. It should be stressed that by using DCT-based estimation
and compensation methods, standard-compliant bitstreams can be formed in accordance with
the specification of any standard such as MPEG without the need to change the structure of any
standard-compliant decoder. To realize this fully DCT-based video coder architecture to boost
the system throughput and reduce the total number of components, we develop DCT-based
algorithms that perform motion estimation and compensation directly on the DCT coefficients
of video frames [1, 5, 6].
9.3
DCT Pseudo-Phase Techniques
As is well known, the Fourier transform (FT) of a signal x(t) is related to the FT of its
shifted (or delayed if t represents time) version, x(t − τ ), by the equation
F{x(t − τ )} = e−j ωτ F{x(t)} ,
© 2001 CRC Press LLC
(9.1)
where F{·} denotes the Fourier transform. The phase of Fourier transform of the shifted signal
contains the information about the amount of the shift, τ , which can easily be extracted. The
phase correlation method was developed to estimate motion from the Fourier phase [3, 7].
However, the DCT or its counterpart, the discrete sine transform (DST), does not have any
phase components as usually is found in the discrete Fourier transform (DFT); however, DCT
(or DST) coefficients of a shifted signal do carry this shift information. To facilitate explanation
of the DCT pseudo-phase techniques, let us first consider the case of one-dimensional discrete
signals. Suppose that the signal {x1 (n); n ∈ {0, . . . , N − 1}} is right shifted by an amount
m (in our convention, a right shift means that m > 0) to generate another signal {x2 (n); n ∈
{0, . . . , N − 1}}. The values of x1 (n) are all zero outside the support region S(x1 ). Therefore,
x2 (n) =
x1 (n − m), for n − m ∈ S(x1 ),
0,
elsewhere .
The above equation implies that both signals have resemblance to each other except that the
signal is shifted. It can be shown that, for k = 1, . . . , N − 1,
kπ
kπ
1
1
S
=
m+
− Z1 (k) sin
m+
,
N
2
N
2
1
1
kπ
kπ
m+
+ Z1C (k) sin
m+
.
X2S (k) = Z1S (k) cos
N
2
N
2
X2C (k)
Z1C (k) cos
(9.2)
(9.3)
Here X2S and X2C are DST (DST-II) and DCT (DCT-II) of the second kind of x2 (n), respectively,
whereas Z1S and Z1C are DST (DST-I) and DCT (DCT-I) of the first kind of x1 (n), respectively,
defined as follows [8]:
X2C (k) =
N−1
2
kπ
x2 (n) cos
C(k)
(n + 0.5) ; k ∈ {0, . . . , N − 1} ,
N
N
(9.4)
N−1
2
kπ
C(k)
x2 (n) sin
(n + 0.5) ; k ∈ {1, . . . , N} ,
N
N
(9.5)
N−1
2
kπ
C(k)
(n) ; k ∈ {0, . . . , N} ,
x1 (n) cos
N
N
(9.6)
N−1
2
kπ
x1 (n) sin
C(k)
(n) ; k ∈ {1, . . . , N − 1} ,
N
N
(9.7)
n=0
X2S (k) =
n=0
Z1C (k) =
n=0
Z1S (k) =
n=0
where
C(k) =
√1 ,
2
1,
for k = 0 or N,
otherwise .
s (k) = sin[ kπ (m + 1 )] and g c (k) =
The displacement, m, is embedded solely in the terms gm
m
N
2
kπ
1
cos[ N (m + 2 )], which are called pseudo-phases, analogous to phases in the Fourier transform
of shifted signals. To find m, we first solve (9.2) and (9.3) for the pseudo-phases and then use
© 2001 CRC Press LLC
the sinusoidal orthogonal principles as follows:
N
2 2
1
kπ
1
kπ
m+
sin
n+
C (k) sin
N
N
2
N
2
k=1
2
N
= δ(m − n) − δ(m + n + 1) ,
kπ
1
kπ
1
C 2 (k) cos
m+
cos
n+
N
2
N
2
(9.8)
= δ(m − n) + δ(m + n + 1) .
(9.9)
N−1
k=0
Here δ(n) is the discrete impulse function, defined as
1, for n = 0,
δ(n) =
0, otherwise .
(9.10)
1
kπ
1
Indeed, if we replace sin[ kπ
N (m + 2 )] and cos[ N (m + 2 )] by the computed sine and cosine
s
c
pseudo-phase components, ĝm (k) and ĝm (k), respectively, in (9.8) and (9.9), both equations
s (k) and ĝ c (k):
simply become IDST-II and IDCT-II operations on ĝm
m
N
2 2
kπ
1
s
=
C (k)ĝm (k) sin
n+
,
N
N
2
(9.11)
N−1
c
2 2
kπ
1
c
C (k)ĝm (k) cos
n+
.
IDCT-II ĝm =
N
N
2
(9.12)
IDST-II
s
ĝm
k=1
k=0
The notation ĝ is used to distinguish the computed pseudo-phase from the one in a noiseless
1
kπ
1
situation (i.e., sin[ kπ
N (m + 2 )] or cos[ N (m + 2 )]). A closer look at the right-hand side of (9.8)
tells us that δ(m − n) and δ(m + n + 1) have opposite signs. This property will help us
detect the shift direction. If we perform an IDST-II operation on the pseudo-phases found,
then the observable window of the index space in the inverse DST domain will be limited to
{0, . . . , N − 1}. As illustrated in Figure 9.3, for a right shift, one spike (generated by the
positive δ function) is pointing upward at the location n = m in the gray region (i.e., the
observable index space), whereas the other δ is pointing downward at n = −(m + 1) outside
the gray region. In contrast, for a left shift, the negative spike at n = −(m + 1) > 0 falls in
the gray region but the positive δ function at n = m stays out of the observable index space.
It can easily be seen that a positive peak value in the gray region implies a right shift and a
negative one means a left shift. This enables us to determine from the sign of the peak value
the direction of the shift between signals.
The concept of pseudo-phases plus the application of sinusoidal orthogonal principles leads
to the DCT pseudo-phase techniques, a new approach to estimate a shift or translational motion
between signals in the DCT domain, as depicted in Figure 9.4a:
1. Compute the DCT-I and DST-I coefficients of x1 (n) and the DCT-II and DST-II coefficients of x2 (n).
s (k) for k = 1, . . . , N by solving this equation:
2. Compute the pseudo-phase ĝm
 C
 Z1 (k)·X2S (k)−Z1S (k)·X2C (k) , for k = N,
s
[Z1C (k)]2 +[Z1S (k)]2
ĝm (k) =
 √1 ,
for k = N .
2
© 2001 CRC Press LLC
(9.13)
n
(a)
n
(b)
FIGURE 9.3
How the direction of shift is determined based on the sign of the peak value after application of the sinusoidal orthogonal principle for the DST-II kernel to pseudo-phases.
(a) How to detect right shift. (b) How to detect left shift.
FIGURE 9.4
Illustration of one-dimensional DCT pseudo-phase techniques. (a) DCT Pseudo-Phase
Techniques; (b) Right shift; (c) Left shift.
s (k); k = 1, . . . , N}, into an IDST-II decoder to
3. Feed the computed pseudo-phase, {ĝm
produce an output {d(n); n = 0, . . . , N − 1}, and search for the peak value. Then the
estimated displacement m̂ can be found by
if d(ip ) > 0 ,
i ,
(9.14)
m̂ = p
−(ip + 1), if d(ip ) < 0 ,
where ip = arg maxn |d(n)| is the index at which the peak value is located.
© 2001 CRC Press LLC
In step 1, the DCT and DST can be generated simultaneously with only 3N multipliers [9]–[11],
and the computation of DCT-I can be easily obtained from DCT-II with minimal overhead, as
will be shown later. In step 2, if noise is absent and there is only purely translational motion,
ĝm (k) will be equal to sin kπ
N (m + 0.5). The output d(n) will then be an impulse function
in the observation window. This procedure is illustrated by two examples in Figure 9.4b
and c with a randomly generated signal as input at a signal-to-noise ratio (SNR) of 10 dB.
These two examples demonstrate that the DCT pseudo-phase techniques are robust even in an
environment of strong noise.
9.4
DCT-Based Motion Estimation
The DCT pseudo-phase technique of extracting shift values from the pseudo-phases of the
DCT of one-dimensional signals can be extended to the two-dimensional case. Let us confine
the problem of motion estimation to this 2D translational motion model in which an object
moves translationally by mu in the x direction and mv in the y direction as viewed on the camera
plane and within the scope of a camera in a noiseless environment, as shown in Figure 9.5.
Then by means of the DCT pseudo-phase technique, we can extract the displacement vector
from the two consecutive frames of the images of that moving object by making use of the
sinusoidal orthogonal principles (9.8) and (9.9). The resulting novel algorithm for this twodimensional translational motion model is called the DXT-ME algorithm, which can estimate
translational motion in the DCT domain.
9.4.1
The DXT-ME Algorithm
Based on the assumption of 2D translational displacements, we can extend the DCT pseudophase technique to the DXT-ME algorithm depicted in Figure 9.6. The previous frame xt−1
and the current frame xt are fed into the 2D-DCT-II and 2D-DCT-I coders, respectively. A
2D-DCT-II coder computes four coefficients, DCCTII, DCSTII, DSCTII, and DSSTII, each of
which is defined as a two-dimensional separable function formed by 1D-DCT/DST-II kernels:
Xtcc (k, l) =
N−1
4
kπ
lπ
C(k)C(l)
x
(m,
n)
cos
(m
+
0.5)
cos
(n
+
0.5)
, (9.15)
t
N
N
N2
m,n=0
for k, l ∈ {0, . . . , N − 1} ,
N−1
4
kπ
lπ
cs
xt (m, n) cos
Xt (k, l) = 2 C(k)C(l)
(m + 0.5) sin
(n + 0.5) , (9.16)
N
N
N
m,n=0
for k ∈ {0, . . . , N − 1}, l ∈ {1, . . . , N} ,
N−1
4
kπ
lπ
Xtsc (k, l) = 2 C(k)C(l)
xt (m, n) sin
(m + 0.5) cos
(n + 0.5) , (9.17)
N
N
N
m,n=0
for k ∈ {1, . . . , N}, l ∈ {0, . . . , N − 1} ,
N−1
lπ
4
kπ
ss
Xt (k, l) = 2 C(k)C(l)
(m + 0.5) sin
(n + 0.5) , (9.18)
xt (m, n) sin
N
N
N
m,n=0
for k, l ∈ {1, . . . , N} ,
© 2001 CRC Press LLC
FIGURE 9.5
An object moves translationally by mu in the x direction and mv in the y direction, as
viewed on the camera plane.
FIGURE 9.6
Block diagram of DXT-ME. (a) Flowchart; (b) structure.
or symbolically,
Xtcc = DCCTII (xt ) , Xtcs = DCSTII (xt ) ,
Xtsc = DSCTII (xt ) , Xtss = DSSTII (xt ) .
In the same fashion, the two-dimensional DCT coefficients of the first kind (2D-DCT-I) are
calculated based on 1D-DCT/DST-I kernels:
© 2001 CRC Press LLC
cc
(k, l)
Zt−1
N−1
4
kπ
lπ
= 2 C(k)C(l)
xt−1 (m, n) cos
(m) cos
(n) ,
N
N
N
(9.19)
m,n=0
for k, l ∈ {0, . . . , N} ,
cs
Zt−1
(k, l) =
N−1
lπ
4
kπ
(m)
sin
(n)
,
C(k)C(l)
x
(m,
n)
cos
t−1
N
N
N2
(9.20)
m,n=0
for k ∈ {0, . . . , N}, l ∈ {1, . . . , N − 1} ,
N−1
4
kπ
lπ
sc
Zt−1 (k, l) = 2 C(k)C(l)
xt−1 (m, n) sin
(m) cos
(n) ,
N
N
N
(9.21)
m,n=0
for k ∈ {1, . . . , N − 1}, l ∈ {0, . . . , N} ,
N−1
4
kπ
lπ
ss
(k, l) = 2 C(k)C(l)
xt−1 (m, n) sin
Zt−1
(m) sin
(n) ,
N
N
N
(9.22)
m,n=0
for k, l ∈ {1, . . . , N − 1} ,
or symbolically,
cc
cs
= DCCTI (xt−1 ) , Zt−1
= DCSTI (xt−1 ) ,
Zt−1
sc
ss
Zt−1 = DSCTI (xt−1 ) , Zt−1 = DSSTI (xt−1 ) .
Similar to the one-dimensional case, assuming that only translational motion is allowed, one
can derive a set of equations to relate the DCT coefficients of xt−1 (m, n) with those of xt (m, n)
in the same way as in (9.2) and (9.3).
Zt−1 (k, l) · θ(k, l) = xt (k, l), for k, l ∈ N ,
where N = {1, . . . , N − 1},

 cc
cs (k, l) −Z sc (k, l) Z ss (k, l)
Zt−1 (k, l) −Zt−1
t−1
t−1
 Z cs (k, l) Z cc (k, l) −Z ss (k, l) −Z sc (k, l) 


t−1
t−1
t−1
Zt−1 (k, l) =  t−1
sc (k, l) −Z ss (k, l) Z cc (k, l) −Z cs (k, l)  ,

 Zt−1
t−1
t−1
t−1
ss (k, l) Z sc (k, l) Z cs (k, l) Z cc (k, l)
Zt−1
t−1
t−1
t−1
 CC
 

lπ
gmu mv (k, l)
+
0.5)
cos
cos kπ
(m
u
N
N (mv + 0.5)
 CS
 

lπ
 gmu mv (k, l)   cos kπ

N (mu + 0.5) sin N (mv + 0.5) 
=
θ(k, l) = 
 g SC (k, l)   sin kπ (m + 0.5) cos lπ (m + 0.5)  ,
u
v
 mu mv
 

N
N
kπ
lπ
SS (k, l)
sin
+
0.5)
sin
+
0.5)
gm
(m
(m
u
v
N
N
u mv
cc
T
cs
sc
ss
xt (k, l) = Xt (k, l) Xt (k, l) Xt (k, l) Xt (k, l)
.
(9.23)
(9.24)
(9.25)
(9.26)
Here Zt−1 (k, l) ∈ R 4×4 is the system matrix of the DXT-ME algorithm at (k, l). It can be
easily shown that Zt−1 (k, l) ∈ R 4×4 is a unitary matrix [1]. At the boundaries of each block in
the transform domain, the DCT coefficients of xt−1 (m, n) and xt (m, n) have a much simpler
one-dimensional relationship [5].
In a two-dimensional space, an object may move in four possible directions: northeast (NE:
mu > 0, mv > 0), northwest (NW: mu < 0, mv > 0), southeast (SE: mu > 0, mv < 0), and
southwest (SW: mu < 0, mv < 0). As explained in Section 9.3, the orthogonal equation for
s (k) to determine the sign of m
the DST-II kernel in (9.8) can be applied to the pseudo-phase ĝm
© 2001 CRC Press LLC
(i.e., the direction of the shift). In order to detect the signs of both mu and mv (or equivalently
the direction of motion), it becomes obvious from the observation in the one-dimensional case
SC (·, ·) and ĝ CS (·, ·) so that the signs of
that it is necessary to compute the pseudo-phases ĝm
mu mv
u mv
SC
CS (·, ·), respectively. Taking the block
mu and mv can be determined from ĝmu mv (·, ·) and ĝm
u mv
boundary equations into consideration, we define two pseudo-phase functions as follows:
 CS
for k, l ∈ N ,
ĝmu mv (k, l),



cc (k,l)X cs (k,l)−Z cs (k,l)X cc (k,l)

Z

t
t
1
t−1
t−1

, for k = 0, l ∈ N ,

cc (k,l))2 +(Z cs (k,l))2
 √2
(Zt−1
t−1
sc (k,l)X ss (k,l)
fmu mv (k, l) =
(9.27)
Z cc (k,l)Xcs (k,l)+Zt−1
t

√1 t−1 cc t
, for l = N, k ∈ N ,

2 +(Z sc (k,l))2
(Z
(k,l))

2
t−1
t−1



cs

 1 Xcct (k,l) ,
for k = 0, l = N
2 Zt−1 (k,l)
 SC
for k, l ∈ N ,
ĝmu mv (k, l),



cc (k,l)X sc (k,l)−Z sc (k,l)X cc (k,l)

Z

t
t
1
t−1
t−1

, for l = 0, k ∈ N ,

cc (k,l))2 +(Z sc (k,l))2
 √2
(Zt−1
t−1
cs (k,l)X ss (k,l)
gmu mv (k, l) =
(9.28)
Z cc (k,l)Xsc (k,l)+Zt−1
t

√1 t−1 cc t
, for k = N, l ∈ N ,

2 +(Z cs (k,l))2
(Z
(k,l))

2
t−1
t−1



sc

 1 Xcct (k,l) ,
for k = N, l = 0
2 Zt−1 (k,l)
In the computation of fmu mv (k, l) and gmu mv (k, l), if the absolute computed value is greater
than 1, then this value is ill conditioned and should be discarded. This ill-conditioned situation
occurs when the denominator in (9.27) and (9.28) is close to zero in comparison to the finite
machine precision or set to zero after the quantization step in the feedback loop of the encoder,
as shown in Figure 9.2. Due to the fact that neighboring image pixels are highly correlated, the
high-frequency DCT coefficients of an image tend to be very small and can be regarded as zero
after the quantization step, but the low-frequency DCT components usually have large values.
Therefore, the ill-conditioned situation happens more likely when k and l are both large. It is
desirable to set the value of fmu mv (k, l) (or gmu mv (k, l)) as close as possible to the ideal value
of fmu mv (k, l) (or gmu mv (k, l)) with the infinite machine precision and no quantization. Since
1
lπ
1
ideally fmu mv (k, l) = cos kπ
N (mu + 2 ) sin N (mv + 2 ),
1
1
1
lπ
kπ
fmu mv (k, l) =
sin
mv +
+
mu +
2
N
2
N
2
lπ
1
1
kπ
+ sin
mv +
−
mu +
.
N
2
N
2
For small values of mu and mv (slow motion) and large k and l (high-frequency DCT components), it is likely that
kπ
1
1
lπ
mv +
−
mu +
≈0,
N
2
N
2
and the first term in fmu mv (k, l) is bounded by 1. Therefore, it is likely that |fmu mv (k, l)| ≤ 0.5.
Without any other knowledge, it is reasonable to guess that fmu mv (k, l) is closer to zero than
to ±1. A similar argument follows for the case of gmu mv (k, l). Thus in our implementation,
we set the corresponding variable fmu mv (k, l) or gmu mv (k, l) to be zero when the magnitudes
of the computed values exceed 1. This setting for ill-conditioned computed fmu mv (k, l) and
gmu mv (k, l) values is found to improve the condition of fmu mv (k, l) and gmu mv (k, l) and also
the overall performance of the DXT-ME algorithm.
These two pseudo-phase functions pass through 2D-IDCT-II coders (IDCSTII and IDSCTII)
to generate two functions, DCS(·, ·) and DSC(·, ·), in view of the orthogonal property of DCT-II
© 2001 CRC Press LLC
and DST-II in (9.8) and (9.9):
DCS(m, n) = IDCSTII fmu mv
N−1 N
4 1
lπ
1
kπ
= 2
m+
sin
n+
C(k)C(l)fmu mv (k, l) cos
N
2
N
2
N
k=0 l=1
= [δ (m − mu ) + δ (m + mu + 1)] · [δ (n − mv ) − δ (n + mv + 1)] , (9.29)
DSC(m, n) = IDSCTII gmu mv
N N−1
4 kπ
1
lπ
1
= 2
C(k)C(l)gmu mv (k, l) sin
m+
cos
n+
N
2
N
2
N
k=1 l=0
= [δ (m − mu ) − δ (m + mu + 1)] · [δ (n − mv ) + δ (n + mv + 1)] . (9.30)
By the same argument as in the one-dimensional case, the 2D-IDCT-II coders limit the
observable index space {(i, j ) : i, j = 0, . . . , N −1} of DCS and DSC to the first quadrant of the
entire index space, shown as gray regions in Figure 9.7, which depicts (9.29) and (9.30). Similar
to the one-dimensional case, if mu is positive, the observable peak value of DSC(m, n) will be
positive regardless of the sign of mv since DSC(m, n) = δ(m−mu )·[δ(n−mv )+δ(n+mv +1)]
in the observable index space. Likewise, if mu is negative, the observable peak value of
DSC(m, n) will be negative because DSC(m, n) = δ(m+mu +1)·[δ(n−mv )+δ(n+mv +1)]
in the gray region. As a result, the sign of the observable peak value of DSC determines the
sign of mu . The same reasoning may apply to DCS in the determination of the sign of mv .
The estimated displacement, d̂ = (m̂u , m̂v ), can thus be found by locating the peaks of DCS
and DSC over {0, . . . , N − 1}2 or over an index range of interest, usually & = {0, . . . , N/2}2
for slow motion. How the peak signs determine the direction of movement is summarized in
Table 9.2. Once the direction is found, d̂ can be estimated accordingly:
Table 9.2 Determination of Direction of Movement (mu , mv ) from
the Signs of DSC and DCS
Sign of
DSC Peak
sign of
DCS Peak
Peak Index
Motion Direction
+
+
−
−
+
−
+
−
(mu , mv )
(mu , −(mv + 1))
(−(mu + 1), mv )
(−(mu + 1), −(mv + 1))
Northeast
Southeast
Northwest
Southwest
m̂u =
m̂v =
if DSC (iDSC , jDSC ) > 0 ,
iDSC = iDCS ,
− (iDSC + 1) = − (iDCS + 1) , if DSC (iDSC , jDSC ) < 0 ,
(9.31)
if DCS (iDCS , jDCS ) > 0 ,
jDCS = jDSC ,
− (jDCS + 1) = − (jDSC + 1) , if DCS (iDCS , jDCS ) < 0 ,
(9.32)
where
(iDCS , jDCS ) = arg max |DCS(m, n)| ,
(9.33)
(iDSC , jDSC ) = arg max |DSC(m, n)| .
(9.34)
m,n∈&
m,n∈&
Normally, these two peak indices are consistent, but in noisy circumstances they may not
agree. In this case, an arbitration rule must be made to pick the best index (iD , jD ) in terms
© 2001 CRC Press LLC
© 2001 CRC Press LLC
FIGURE 9.7
How the direction of motion is determined based on the sign of the peak value. (a) From DCS, (b) from DSC.
of minimum nonpeak-to-peak ratio (NPR):
(iDSC , jDSC ) if NPR(DSC) < NPR(DCS),
(iD , jD ) =
(iDCS , jDCS ) if NPR(DSC) > NPR(DCS) .
(9.35)
This index (iD , jD ) will then be used to determine d̂ by (9.31) and (9.32). Here NPR is defined
as the ratio of the average of all absolute nonpeak values to the absolute peak value. Thus,
0 ≤ NPR ≤ 1, and for a pure impulse function, NPR = 0. Such an approach to choose the
best index among the two indices is found empirically to improve the noise immunity of this
estimation algorithm.
In situations where slow motion is preferred, it is better to search the peak value in a zigzag
way, as is widely done in DCT-based hybrid video coding [12, 13]. Starting from the index
(0, 0), zigzagly scan all the DCS (or DSC) values and mark the point as the new peak index if
the value at that point (i, j ) is larger than the current peak value by more than a preset threshold
θ:
(iDCS , jDCS ) = (i, j ) if DCS(i, j ) > DCS (iDCS , jDCS ) + θ,
(iDSC , jDSC ) = (i, j ) if DSC(i, j ) > DSC (iDSC , jDSC ) + θ .
(9.36)
(9.37)
In this way, large spurious spikes at the higher index points will not affect the performance and
thus improve its noise immunity further. If there is no presence of slow motion in a fast-moving
picture, then simply no slow motion is preferred and the estimator will be able to find a peak
at the high-frequency region (i.e., large motion vector).
Figure 9.8 demonstrates the DXT-ME algorithm. Images of a rectangularly shaped moving object with arbitrary texture are generated (Figure 9.8a) and corrupted by additive white
Gaussian noise at SNR = 10 dB (Figure 9.8b). The resulting pseudo-phase functions f and
g, as well as DCS and DSC, are depicted in Figure 9.8c and d, correspondingly. Large peaks
can be seen clearly in Figure 9.8d on rough surfaces caused by noise in spite of noisy input
images. The positions of these peaks give us an accurate motion estimate (5, −3).
9.4.2
Computational Issues and Complexity
The block diagram in Figure 9.6a shows that a separate 2D-DCT-I is needed in addition to
the standard DCT (2D-DCT-II). This is undesirable from the complexity viewpoint. However,
this problem can be circumvented by considering the point-to-point relationship between the
2D-DCT-I and 2D-DCT-II coefficients in the frequency domain for k, l ∈ N :
 
 cc
kπ cos lπ
Zt−1 (k, l)
+ cos 2N
2N
 
 cs
 Z (k, l)   − cos kπ sin lπ
 
 t−1
2N
2N
=
 sc
 Z (k, l)   − sin kπ cos lπ
2N
2N
 
 t−1
kπ sin lπ
ss (k, l)
Zt−1
+ sin 2N
2N

 cc
Xt−1 (k, l)

 cs
 X (k, l) 

 t−1
×  sc

 X (k, l) 

 t−1
ss (k, l)
Xt−1
kπ sin lπ + sin kπ cos lπ + sin kπ sin lπ
+ cos 2N
2N
2N
2N
2N
2N
kπ cos lπ − sin kπ sin lπ
kπ cos lπ
+ cos 2N
+
sin
2N
2N
2N
2N
2N
kπ sin lπ
kπ cos lπ + cos kπ sin lπ
− sin 2N
+
cos
2N
2N
2N
2N
2N
kπ cos lπ − cos kπ sin lπ + cos kπ cos lπ
− sin 2N
2N
2N
2N
2N
2N







(9.38)
cc , X cs , X sc , and X ss are the 2D-DCT-II coefficients of the previous frame. A
where Xt−1
t−1
t−1
t−1
similar relation exists for the coefficients at the block boundaries. This observation results
in the simple structure in Figure 9.6b, where block T is a coefficient transformation unit
realizing (9.38).
© 2001 CRC Press LLC
FIGURE 9.8
DXT-ME performed on the images of an object moving in the direction (5, −3) with
additive white Gaussian noise at SNR = 10 dB. (a) Original inputs x1 and x2 ; (b) noise
added; (c) f and g; (d) DSC and DCS.
© 2001 CRC Press LLC
Table 9.3 Computational Complexity of Each Stage in DXT-ME
Stage
1
2
3
4
Component
Computational Complexity
2D-DCT-II
Coefficient transformation unit (T)
Pseudo-phase computation
2D-IDCT-II
Peak searching
Estimation
Odct = O(N )
O(N 2 )
O(N 2 )
Odct = O(N )
O(N 2 )
O(1)
If the DCT has computational complexity Odct , the overall complexity of DXT-ME is
O(N 2 ) + Odct with the complexity of each component summarized in Table 9.3. The computational complexity of the pseudo-phase computation component is only O(N 2 ) for an N × N
block and so is the unit to determine the displacement. For the computation of the pseudophase functions f (·, ·) in (9.27) and g(·, ·) in (9.28), the DSCT, DCST, and DSST coefficients
(regarded as DST coefficients) must be calculated in addition to the DCCT coefficients (i.e.,
the usual 2D DCT). However, all these coefficients can be generated with little overhead in the
course of computing 2D DCT coefficients. As a matter of fact, a parallel and fully pipelined
2D DCT lattice structure has been developed [9]–[11] to generate 2D DCT coefficients at a cost
of O(N ) operations. This DCT coder computes DCT and DST coefficients dually due to its
internal lattice architecture. These internally generated DST coefficients can be output to the
DXT-ME module for pseudo-phase computation. This same lattice structure can also be modified as a 2D IDCT, which also has O(N ) complexity. To sum up, the computational complexity
of this DXT-ME is only O(N 2 ), much lower than the O(N 4 ) complexity of BKM-ME.
Calculation of the actual number of computations, other than asymptotic complexity, requires the knowledge of specific implementations. In DCT-based motion-compensated video
coding, DCT, IDCT, and peak searching are required; therefore, we will count only the number
of operations required in the pseudo-phase computation. At each pixel position, we need to
solve a 4 × 4 linear equation by means of the Gauss elimination method with 4 divisions,
40 multiplications, and 30 additions/subtractions. Therefore, the total number of operations
is 18,944 for a 16 × 16 block and 75,776 for a corresponding overlapped block (32 × 32),
while the BKM-ME approach requires 130,816 additions/subtractions for block size 16 × 16
and search area 32 × 32. Still, the number of operations required by the DXT-ME algorithm
is smaller than BKM-ME. Further reduction of computations can be achieved by exploiting
various properties in the algorithm.
A closer look at (9.27), (9.28), and (9.38), reveals that the operations of pseudo-phase computation and coefficient transformation are performed independently at each point (k, l) in the
transform domain and therefore are inherently highly parallel operations. Since most of the operations in the DXT-ME algorithm involve mainly pseudo-phase computations and coefficient
transformations in addition to DCT and IDCT operations, which have been studied extensively, the DXT-ME algorithm can easily be implemented on highly parallel array processors
or dedicated circuits. This is very different from BKM-ME, which requires shifting of pixels
and summation of differences of pixel values and hence discourages parallel implementation.
9.4.3
Preprocessing
For complicated video sequences in which objects may move across the border of blocks in
a nonuniform background, preprocessing can be employed to enhance the features of moving
objects and avoid violation of the assumption made for DXT-ME before feeding the images
© 2001 CRC Press LLC
into the DXT-ME algorithm. Intuitively speaking, the DXT-ME algorithm tries to match
the features of any object on two consecutive frames so that any translation motion can be
estimated regardless of the shape and texture of the object as long as these two frames contain
the significant energy level of the object features. Due to the matching property of the DXTME algorithm, effective preprocessing will improve the performance of motion estimation
if preprocessing can enhance the object features in the original sequence. In order to keep
the computational complexity of the overall motion estimator low, the chosen preprocessing
function must be simple but effective in the sense that unwanted features will not affect the
accuracy of estimation. Our study found that both edge extraction and frame differentiation
are simple and effective schemes for extraction of motion information.
It is found that estimating the motion of an object from its edges is equivalent to estimating
from its image projection [14]. Furthermore, since the DXT-ME algorithm assumes that an object moves within the block boundary in a completely dark environment, its edge information
reduces the adverse effect of the object moving across the block boundary on the estimation
accuracy. The other advantage of edge extraction is that any change in the illumination condition does not alter the edge information and in turn makes no false motion estimates by the
DXT-ME algorithm. Since we only intend to extract the main features of moving objects while
keeping the overall complexity low, we employ a very simple edge detection by convolving
horizontal and vertical Sobel operators of size 3 × 3 with the image to obtain horizontal and
vertical gradients, respectively, and then combine both gradients by taking the square root of
the sum of the squares of both gradients [15]. Edge detection provides us the features of moving objects but also the features of the background (stationary objects), which is undesirable.
However, if the features of the background have smaller energy than those of moving objects
within every block containing moving objects, then the background features will not affect the
performance of DXT-ME. The computational complexity of this preprocessing step is only
O(N 2 ) and thus the overall computational complexity is still O(N 2 ).
Frame differentiation generates an image of the difference of two consecutive frames. This
frame-differentiated image contains no background objects but the difference of moving objects
between two frames. The DXT-ME estimator operates directly on this frame-differentiated
sequence to predict motion in the original sequence. The estimate will be good if the moving
objects are moving constantly in one direction in three consecutive frames. For 30 frames per
second, the standard NTSC frame rate, objects can usually be viewed as moving at a constant
speed in three consecutive frames. Obviously, this step also has only O(N 2 ) computational
complexity.
9.4.4
Adaptive Overlapping Approach
For fair comparison with BKM-ME, which has a larger search area than the block size, we
adopt the adaptive overlapping approach to enlarge adaptively the block area. The enlargement
of the block size diminishes the boundary effect that occurs when the displacement is very large
compared to the block size. As a result, the moving objects may move partially or completely
out of the block, making the contents in two temporally consecutive blocks very different.
However, this problem also exists for other motion estimation algorithms. That is why we
need to assume that objects in the scene are moving slowly. For rapid motion, it is difficult to
track motion.
Earlier in this section we mentioned that we search for peaks of DSC and DCS over a fixed
index range of interest & = {0, . . . , N/2}2 . However, if we follow the partitioning approach
used in BKM-ME, then we may dynamically adjust &. At first, partition the whole current
frame into bs × bs nonoverlapping reference blocks, shown as the shaded area in Figure 9.9a.
Each reference block is associated with a larger search area (of size sa) in the previous frame
© 2001 CRC Press LLC
FIGURE 9.9
Adaptive overlapping approach.
(the dotted region in the same figure) in the same way as for BKM-ME. From the position of
a reference block and its associated search area, a search range D = {(u, v) : −u1 ≤ u ≤
u2 , −v1 ≤ v ≤ v2 } can then be determined as in Figure 9.9b. In contrast to BKM-ME,
DXT-ME requires that the reference block size and the search area size be equal. Thus, instead
of using the reference block, we use the block of the same size and position in the current frame
as the search area of the previous frame. The peak values of DSC and DCS are searched in a
zigzag way over this index range, & = {0, . . . , max(u2 , u1 − 1)} × {0, . . . , max(v2 , v1 − 1)}.
In addition to the requirement that the new peak value be larger than the current peak value
by a preset threshold, it is necessary to examine if the motion estimate determined by the new
peak index lies in the search region D. Since search areas overlap one another, the DXT-ME
architecture utilizing this approach is called overlapping DXT-ME. Even though the block size
required by the overlapping DXT-ME algorithm is larger than the block size for one DCT
block, it is still possible to estimate motion completely in the DCT domain without going back
to the spatial domain by concatenating neighboring DCT blocks directly in the DCT domain
[16].
9.4.5
Simulation Results
A number of video sequences with different characteristics are used in our simulations to
compare the performance of the DXT-ME algorithm [1, 5] with the full-search block-matching
method (BKM-ME or BKM for the sake of brevity) as well as three commonly used fast-search
block-matching approaches such as the logarithmic search method (LOG), the three-step search
method (TSS), and the subsampled search approach (SUB) [17]. The performance of different
schemes is evaluated and compared
in terms of MSE (mean squared error per pel) and BPS
[x̂(m,n)−x(m,n)]2
(bits per sample) where MSE = m,n
and BPS is the ratio of the total number of
N2
bits required for each motion-compensated residual frame in JPEG format (BPS) converted by
the image format conversion program ALCHEMY with quality = 32 to the number of pixels.
As is widely used in the literature of video coding, all the block-matching methods adopt the
conventional mean absolute difference (MAD) optimization criterion:
m,n |x2 (m, n) − x1 (m − u, n − v)|
d̂ = (û, v̂) = arg min
,
N2
(u,v)∈S
where S denotes the set of allowable displacements depending on which block-matching
approach is in use.
© 2001 CRC Press LLC
The first sequence is the Flower Garden sequence, where the camera is moving before
a big tree and a flower garden in front of a house, as shown in Figure 9.10a. Each frame
has 352 × 224 pixels. Simple preprocessing is applied to this sequence: edge extraction or
frame differentiation as depicted in Figure 9.10b and c, respectively. Since macroblocks, each
consisting of 16 × 16 luminance blocks and two 8 × 8 chrominance blocks, are considered to
be the basic unit for motion estimation/compensation in MPEG standards [13], the following
simulation setting is adopted for simulations on the Flower Garden sequence and all subsequent
sequences: 16 × 16 blocks on 32 × 32 search areas. Furthermore, the overlapping DXT-ME
algorithm is used for fair comparison with block-matching approaches, which require a larger
search area.
FIGURE 9.10
Frame 57 in the Flower Garden sequence.
As can be seen in Figure 9.10b, the edge-extracted frames contain significant features of
moving objects in the original frames so that DXT-ME can estimate the movement of the
objects based on the information provided by the edge-extracted frames. Because the camera
is moving at a constant speed in one direction, the moving objects occupy almost the whole
scene. Therefore, the background features do not interfere with the operation of DXT-ME much
but still affect the overall performance of DXT-ME as compared to the frame-differentiated
preprocessing approach. The frame-differentiated images of the Flower Garden sequence, one
of which is shown in Figure 9.10c, have residual energy strong enough for DXT-ME to estimate
the motion directly on this frame-differentiated sequence due to the constant movement of the
camera.
The performances for different motion estimation schemes are plotted in Figure 9.11 and
summarized in Table 9.4 where the MSE and BPS values of different motion estimation approaches are averaged over the whole sequence from frame 3 to frame 99 for easy comparison.
It should be noted that the MSE difference in Table 9.4 is the difference of the MSE value
of the corresponding motion estimation scheme from the MSE value of the full-search blockmatching approach (BKM), and the MSE ratio is the ratio of the MSE difference to the MSE
of BKM. As indicated in the performance summary table (Table 9.5), the frame-differentiated
DXT-ME algorithm is 28.9% worse in terms of MSE than the full-search block-matching approach, whereas the edge-extracted DXT-ME algorithm is 36.0% worse. Surprisingly, even
though the fast-search block-matching algorithms (only 12.6% worse than BKM), TSS and
LOG, have smaller MSE values than the DXT-ME algorithm, TSS and LOG have larger BPS
values than the DXT-ME algorithm, as can clearly be seen in Table 9.4 and Figure 9.11. In
other words, the motion-compensated residual frames generated by TSS and LOG require more
bits than the DXT-ME algorithm to transmit/store after compression. This indicates that the
DXT-ME algorithm is better than the logarithmic and three-step fast-search block-matching
approaches for this Flower Garden sequence.
Another simulation is done on the Infrared Car sequence, which has the frame size 96 × 112
and one major moving object — the car moving along a curved road toward the camera fixed on
the ground. In the performance summary table, Table 9.5, the frame-differentiated DXT-ME
© 2001 CRC Press LLC
Table 9.4 Performance Summary of the Overlapping DXT-ME Algorithm with
Either Frame Differentiation or Edge Extraction as Preprocessing Against Full
Search and Fast Search Block-Matching Approaches (BKM, TSS, LOG, SUB) over
the Flower Garden Sequence
Approach
BKM
Frame-differentiated DXT-ME
Edge-extracted DXT-ME
TSS
LOG
SUB
MSE
MSE
Difference
MSE
Ratio
BPF
BPS
BPS
Ratio
127.021
163.712
172.686
143.046
143.048
127.913
0.000
36.691
45.665
16.025
16.026
0.892
0%
28.9%
36.0%
12.6%
12.6%
0.7%
63726
67557
68091
68740
68739
63767
0.808
0.857
0.864
0.872
0.872
0.809
0%
6.0%
6.8%
7.9%
7.9%
1%
Note: MSE difference is the difference from the MSE value of full-search block-matching
method (BKM), and MSE ratio is the ratio of MSE difference to the MSE of BKM.
Table 9.5 Performance Summary of the Overlapping DXT-ME Algorithm with
Either Frame Differentiation or Edge Extraction as Preprocessing Against Full
Search and Fast-Search Block-Matching Approaches (BKM, TSS, LOG, SUB)
over the Infrared Car Sequence
Approach
BKM
Frame-differentiated DXT-ME
Edge-extracted DXT-ME
TSS
LOG
SUB
MSE
MSE
Difference
MSE
Ratio
BPF
BPS
BPS
Ratio
67.902
68.355
72.518
68.108
68.108
68.493
0.000
0.453
4.615
0.206
0.206
0.591
0%
0.7%
6.8%
0.3%
0.3%
0.9%
10156
10150
10177
10159
10159
10159
0.945
0.944
0.946
0.945
0.945
0.945
0%
−0.1%
0.2%
0.0%
0.0%
0.0%
result is better than the SUB result in terms of MSE values and better than the full-search BKM
result in terms of BPS values. (See Figure 9.12.)
9.5
Subpixel DCT Pseudo-Phase Techniques
To further improve the compression rate, motion estimation with subpixel accuracy is essential because movements in a video sequence are not necessarily multiples of the sampling grid
distance in the rectangular sampling grid of a camera. It is shown that significant improvement
of coding gain can be obtained with motion estimation of half-pixel or finer accuracy [18]. Further investigation reveals that the temporal prediction error variance is generally decreased by
subpixel motion compensation, but beyond a certain “critical accuracy” the possibility of further improving prediction by more accurate motion compensation is small [19]. As suggested
in [18, 20], motion compensation with quarter-pel accuracy is sufficiently accurate for broadcast TV signals, but for videophone signals, half-pel accuracy is good enough. As a result,
motion compensation with half-pel accuracy is recommended in MPEG standards [13, 21].
© 2001 CRC Press LLC
FIGURE 9.11
Comparison of overlapping DXT-ME with block-matching approaches on the Flower
Garden sequence.
Implementations of half-pel motion estimation now exist [22]–[24]. However, many of these
implementations are based on the block-matching approach [20, 25, 26], which requires the
interpolation of images through bilinear or other interpolation methods [27]. However, interpolation not only increases the complexity and data flow of a coder but also may adversely
affect the accuracy of motion estimates from the interpolated images [20]. It is more desirable that subpixel accuracy of motion estimates be obtained without interpolating the images
at a low computational cost in the DCT domain so that seamless integration of the motion
compensation unit with the spatial compression unit is possible.
In this section, we extend the DCT pseudo-phase techniques discussed in Section 9.3 to the
subpixel level and show that if the spatial sampling of images satisfies the Nyquist criterion, the
subpixel motion information is preserved in the pseudo-phases of DCT coefficients of moving
images. Furthermore, it can be shown that with appropriate modification, the sinusoidal orthogonal principles can still be applicable except that an impulse function is replaced by a sinc
function whose peak position reveals subpixel displacement. Therefore, exact subpixel motion
displacement can be obtained without the use of interpolation. From these observations, we
can develop a set of subpixel DCT-based motion estimation algorithms that are fully compatible with the integer-pel motion estimator, for low-complexity and high-throughput video
applications.
© 2001 CRC Press LLC
FIGURE 9.12
Infrared Car sequence.
Without loss of generality, let us consider the one-dimensional model in which a continuous
signal xc (t) and its shifted version xc (t − d) are sampled at a sampling frequency 1/T to
generate two sample sequences {x1 (n) = xc (nT )} and {x2 (n) = xc (nT − d)}, respectively.
Let us define the DCT and DST coefficients as
N−1
2C(k) kπ
1
C
Xi (k) DCT {xi } =
xi (n) cos
n+
, (9.39)
N
N
2
n=0
N−1
2C(k)
kπ
DST {xi } =
xi (n) sin
N
N
n=0
√1 , for k = 0 or N ,
2
C(k) =
1, otherwise ,
XiS (k)
where
1
n+
2
,
for i = 1 or 2. By using the sinusoidal relationship:
kπ
1
1 j kπ
n+ 21
−j kπ n+ 21
cos
,
+e N
n+
=
e N
N
2
2
kπ 1 kπ
1
1
j
−j kπ n+ 21
n+ 2
sin
,
n+
=
e N
−e N
N
2
2j
we can show that the DCT/DST and DFT coefficients are related as follows:
kπ
kπ
C(k) Z
XiC (k) =
X̃i (−k)ej 2N + X̃iZ (k)e−j 2N , for k = 0, . . . , N − 1 ,
N
kπ
kπ
C(k)
XiS (k) =
X̃iZ (−k)ej 2N − X̃iZ (k)e−j 2N , for k = 1, . . . , N ,
jN
where {X̃iZ (k)} is the DFT of the zero-padded sequence {xiZ (n)} defined as
xi (n), for n = 0, . . . , N − 1 ,
xiZ (n) =
0,
for n = N, . . . , 2N − 1 ,
(9.40)
(9.41)
(9.42)
(9.43)
(9.44)
(9.45)
so that
N−1
2kπn
X̃iZ (k) DFT xiZ =
xi (n)e−j 2N , for k = 0, . . . , 2N − 1 .
n=0
© 2001 CRC Press LLC
(9.46)
From the sampling theorem, we know that the discrete time Fourier transform (DTFT) of
sequences x1 (n) and x2 (n) is related to the Fourier transform of xc (t), Xc (-), in the following
way:
ω − 2π l
1 Xc
,
T
T
l
l
1 ω − 2π l
−j ω−2π
d
T
X2 (ω) DTFT {x2 } =
Xc
.
e
T
T
X1 (ω) DTFT {x1 } =
(9.47)
(9.48)
l
Furthermore, if Xc (-) is bandlimited in the baseband (− Tπ , Tπ ), then for - =
1
Xc (-) ,
T
1
X2 (-T ) = Xc (-) e−j -d .
T
X1 (-T ) =
ω
T
∈ (− Tπ , Tπ ),
(9.49)
(9.50)
Thus, the DFT of x1 (n) and x2 (n) are
X̃1 (k) DFT {x1 } =
= X1
2π k
N
X̃2 (k) DFT {x2 } =
= X2
2π k
N
N−1
x1 (n)e−j
2π kn
N
n=0
1
= Xc
T
N−1
2π k
NT
,
x2 (n)e−j
2π kn
N
n=0
1
= Xc
T
2π k
NT
e−j
(9.51)
2π kd
NT
,
(9.52)
whereas the DFT of x1Z (n) and x2Z (n) become
πk
=
= X1
N
πk
X̃2Z (k) = X2
=
N
X̃1Z (k)
πk
1
Xc
,
T
NT
π kd
πk
1
Xc
e−j N T .
T
NT
(9.53)
(9.54)
Therefore,
X2
πk
N
= X1
πk
N
π kd
e−j N T .
(9.55)
Substituting (9.55) back into (9.43) and (9.44), we get
kπd
kπ
kπd
kπ
C(k) Z
X̃1 (−k)ej N T ej 2N + X̃1Z (k)e−j N T e−j 2N ,
N
for k = 0, . . . , N − 1 ,
kπd
kπ
kπd
kπ
C(k) Z
X2S (k) =
X̃1 (−k)ej N T ej 2N − X̃1Z (k)e−j N T e−j 2N ,
jN
for k = 1, . . . , N .
X2C (k) =
© 2001 CRC Press LLC
(9.56)
(9.57)
Using the sinusoidal relationship in (9.42) to change natural exponents back to cosine/sine, we
finally obtain the relationship between x1 (n) and x2 (n) in the DCT/DST domain:
X2C (k)
N−1
d
1
2C(k) kπ
n+ +
, for k = 0, . . . , N − 1 ,
=
x1 (n) cos
N
N
T
2
(9.58)
N−1
2C(k) kπ
d
1
=
x1 (n) sin
n+ +
, for k = 1, . . . , N .
N
N
T
2
(9.59)
n=0
X2S (k)
n=0
We conclude the result in the following theorem:
THEOREM 9.1
If a continuous signal xc (t) is Tπ bandlimited and the sampled sequences of xc (t) and xc (t −d)
are {xc (nT )} and {xc (nT − d)}, respectively, then their DCT and DST are related by
DCT {xc (nT − d)} = DCT d {xc (nT )} ,
(9.60)
DST {xc (nT − d)} = DST d {xc (nT )} ,
(9.61)
N−1
2C(k) kπ
1
DCTα {x} x(n) cos
n+α+
,
N
N
2
(9.62)
N−1
2C(k) kπ
1
x(n) sin
n+β +
,
DSTβ {x} N
N
2
(9.63)
T
T
where
n=0
n=0
are the DCT and DST with α and β shifts in their kernels, respectively. Here d is the shift
amount and T is the sampling interval, but d/T is not necessarily an integer.
9.5.1
Subpel Sinusoidal Orthogonality Principles
The sinusoidal orthogonal principles in (9.8) and (9.9) are no longer valid at the subpixel
level. However, we can extend these sinusoidal orthogonal equations suitable for subpixel
motion estimation.
Let’s replace, in (9.8) and (9.9), the integer variables m and n by the real variables u and v
and define
L̄c (u, v) N−1
C 2 (k) cos
kπ
N
C 2 (k) sin
kπ
N
k=0
L̄s (u, v) N−1
k=0
u+
1
kπ
1
cos
v+
,
2
N
2
(9.64)
u+
1
kπ
1
sin
v+
.
2
N
2
(9.65)
Since
L̄c (u, v) N−1
k=0
kπ
C (k) cos
N
2
1
kπ
1
u+
cos
v+
2
N
2
1 1
= − + [ξ(u − v) + ξ(u + v + 1)] ,
2 2
© 2001 CRC Press LLC
and
kπ
1
kπ
1
L̄s (u, v) C (k) sin
u+
sin
v+
N
2
N
2
k=1
1
1
1
1
sin π v +
+ [ξ(u − v) − ξ(u + v + 1)] ,
= sin π u +
2
2
2
2
N
2
we show that
1 1
L̄c (u, v) = − + [ξ(u − v) + ξ(u + v + 1)] ,
2 2 1
1
1
L̄s (u, v) = sin π u +
sin π v +
2
2
2
1
+ [ξ(u − v) − ξ(u + v + 1)] ,
2
(9.66)
(9.67)
where
ξ(x) N−1
k=0
kπ
cos
x
N
πx cos 2N
1
.
=
1 − cos π x + sin π x ·
πx
2
sin 2N
(πx)
If (2N
) is so small that the second and higher order terms of
πx
πx
1, sin 2N
≈ 2N
. Thus,
ξ(x) ≈
(πx)
(2N )
(9.68)
πx
can be ignored, then cos 2N
≈
1
[1 − cos π x] + N sinc(x) ,
2
(9.69)
where sinc(x) sin(π x)/(π x). For large N , ξ(x) is approximately a sinc function whose
largest peak can be identified easily at x = 0, as depicted in Figure 9.13a, where ξ(x) closely
resembles N · sinc(x), especially when x is small. The slope of ξ(x) is also plotted in Figure 9.13b, which shows the sharpness of ξ(x).
FIGURE 9.13
kπ
Plot of ξ(x) = N−1
k=0 cos( N x) and its slope for N = 16. Observe the similarity between
the curves of N *sinc(x) and the last term of ξ . (a) ξ(x); (b) slope of ξ(x).
A closer look at (9.66) and (9.67) reveals that either L̄c (u, v) or L̄s (u, v) consists of ξ
functions and one extra term, which is not desirable. In order to obtain a pure form of sinc
© 2001 CRC Press LLC
functions similar to (9.8) and (9.9), we define two modified functions Lc (u, v) and Ls (u, v)
as follows:
N−1
kπ
1
kπ
1
cos
u+
cos
v+
,
(9.70)
Lc (u, v) N
2
N
2
k=0
Ls (u, v) N−1
sin
k=1
kπ
N
u+
1
kπ
1
sin
v+
.
2
N
2
(9.71)
Then we can show that
1
[ξ(u − v) + ξ(u + v + 1)] ,
(9.72)
2
1
Ls (u, v) = [ξ(u − v) − ξ(u + v + 1)] .
(9.73)
2
Equations (9.70)–(9.73) are the equivalent form of the sinusoidal orthogonal principles (9.8)
and (9.9) at the subpixel level. The sinc functions at the right-hand side of the equations are
the direct result of the rectangular window inherent in the DCT [28]. Figure 9.14a and b
illustrate Ls (x, −3.75) and Lc (x, −3.75), respectively, where two ξ functions are interacting
with each other but their peak positions clearly indicate the displacement. However, when the
displacement v is small (in the neighborhood of −0.5), ξ(u − v) and ξ(u + v + 1) move close
together and addition/subtraction of ξ(u − v) and ξ(u + v + 1) changes the shape of Ls and Lc .
As a result, neither Ls nor Lc looks like two ξ functions and the peak positions of Ls and Lc
are different from those of ξ(u − v) and ξ(u + v + 1), as demonstrated in Figure 9.14c and d,
respectively, where the peak positions of Ls (x, −0.75) and Lc (x, −0.75) are −1.25 and −0.5,
differing from the true displacement −0.75. In the extreme case, ξ(u − v) and ξ(u + v + 1)
cancel out each other when the displacement is −0.5 such that Ls (x, −0.5) ≡ 0, as shown in
Figure 9.14e.
Fortunately, we can eliminate the adverse interaction of the two ξ functions by simply adding
Lc to Ls since Lc (x, v) + Ls (x, v) = ξ(x − v), as depicted in Figure 9.14f, where the sum
Lc (x, −0.75) + Ls (x, −0.75) behaves like a sinc function and its peak position coincides with
the displacement. Furthermore, due to the sharpness of this ξ function, we can accurately
pinpoint the peak position under a noisy situation and in turn determine the motion estimate.
This property enables us to devise flexible and scalable subpixel motion estimation algorithms
in the subsequent sections.
Lc (u, v) =
9.6
DCT-Based Subpixel Motion Estimation
Consider a moving object casting a continuous intensity profile It (u, v) on a camera plane of
the continuous coordinate (u,v) where the subscript t denotes the frame number. This intensity
profile is then digitized on the fixed sampling grid of the camera with a sampling distance d
to generate the current frame of pixels xt (m, n) shown in Figure 9.15a where m and n are
integers. Further assume that the displacement of the object between the frames t − 1 and t
is (du , dv ) such that It (u, v) = It−1 (u − du , v − dv ) where du = (mu + νu )d = λu d and
dv = (mv + νv )d = λv d. Here mu and mv are the integer components of the displacement,
and νu and νv ∈ [− 21 , 21 ]. Therefore,
xt (m, n) = It (md, nd) = It−1 (md − du , nd − dv ) ,
xt−1 (m, n) = It−1 (md, nd) ,
© 2001 CRC Press LLC
FIGURE 9.14
Illustration of sinusoidal orthogonal principles at the subpixel level for different displacements. (a) Ls (x, −3.75); (b) Lc (x, −3.75); (c) Ls (x, −0.75); (d) Lc (x, −0.75);
(e) Ls (x, −0.5); (f) Lc (x, −0.75) + Ls (x, −0.75).
as in Figure 9.15b. Unlike the case of integer-pel movement, the displacement is not necessarily
multiples of the sampling distance d. In other words, νu and νv do not necessarily equal zero.
For integer-pel displacements (i.e., λu = mu and λv = mv ), the pseudo-phases are computed
by solving the pseudo-phase motion equation at (k, l):
Zt−1 (k, l) · θmu ,mv (k, l) = xt (k, l), for k, l ∈ N
(9.74)
where N = {1, . . . , N − 1}, θmu ,mv is the pseudo-phase vector, and the 4 × 4 system matrix
Zt−1 and the vector xt are composed from the 2D-DCT-II of xt−1 (m, n) and the 2D-DCT-I of
xt (m, n), respectively:

 cc
cs (k, l) −Z sc (k, l) +Z ss (k, l)
Zt−1 (k, l) −Zt−1
t−1
t−1
 cs
cc (k, l) −Z ss (k, l) −Z sc (k, l) 

 Zt−1 (k, l) +Zt−1
t−1
t−1
,

Zt−1 (k, l) =  sc

ss
cc
cs
 Zt−1 (k, l) −Zt−1 (k, l) +Zt−1 (k, l) −Zt−1 (k, l) 
ss (k, l) +Z sc (k, l) +Z cs (k, l) +Z cc (k, l)
Zt−1
t−1
t−1
t−1
 CC

 cc

gmu mv (k, l)
Xt (k, l)
 CS

 X cs (k, l) 
 gmu mv (k, l) 
 t
 
 .
xt (k, l) =  sc
 , θmu ,mv (k, l) =  SC

 Xt (k, l) 
 gmu mv (k, l) 
SS (k, l)
Xtss (k, l)
gm
u mv
Here the 2D-DCT-I of xt−1 (m, n) and the 2D-DCT-II of xt (m, n) are defined in (9.15)–(9.18)
xx ; xx = cc, cs, sc, ss} can be obtained by a simple
and (9.19)–(9.23), respectively, where {Zt−1
xx
rotation (9.38) from {Xt−1 ; xx = cc, cs, sc, ss}, which are computed and stored in memory
in the previous encoding cycle.
However, for noninteger-pel movement, we need to use (9.60) and (9.61) in Theorem 9.1
to derive the system equation at the subpixel level. If the Fourier transform of the continuous
© 2001 CRC Press LLC
FIGURE 9.15
(a) The black dots and the gray squares symbolize the sampling grids for frames It−1 (u, v)
and It (u, v), respectively, at a sampling distance d. These two frames are aligned on the
common object displaced by (du , dv ) in the continuous coordinate (u, v). (b) Two digitized
images of consecutive frames, xt−1 (m, n) and xt (m, n), are aligned on the common object
moving (λu , λv ) = (du /d, dv /d) pixels southeast.
intensity profile It (u, v) is πd bandlimited and It (u, v) = It−1 (u − du , v − dv ), then according
to Theorem 9.1, we can obtain the following 2D relations:
Xtcc (k, l) =
N−1
lπ
4
kπ
1
1
m
+
λ
cos
n
+
λ
C(k)C(l)
x
(m,
n)
cos
+
+
u
v
t−1
N
2
N
2
N2
m,n=0
for k, l ∈ {0, . . . , N − 1} ,
Xtcs (k, l) =
4
C(k)C(l)
N2
N−1
m,n=0
xt−1 (m, n) cos
kπ
N
m + λu +
1
2
sin
lπ
N
n + λv +
1
2
(9.75)
for k ∈ {0, . . . , N − 1}, l ∈ {1, . . . , N} ,
(9.76)
N−1
lπ
4
kπ
1
1
m + λu +
cos
n + λv +
xt−1 (m, n) sin
Xtsc (k, l) = 2 C(k)C(l)
N
2
N
2
N
m,n=0
for k ∈ {1, . . . , N}, l ∈ {0, . . . , N − 1} ,
(9.77)
N−1
lπ
4
kπ
1
1
m + λu +
sin
n + λv +
xt−1 (m, n) sin
Xtss (k, l) = 2 C(k)C(l)
N
2
N
2
N
m,n=0
for k, l ∈ {1, . . . , N} .
(9.78)
Thus, we can obtain the pseudo-phase motion equation at the subpixel level:
Zt−1 (k, l) · θλu ,λv (k, l) = xt (k, l), for k, l ∈ N ,
(9.79)
(k, l), gλCS
(k, l), gλSC
(k, l), gλSSu ,λv (k, l)]T . A similar relationwhere θλu ,λv (k, l) = [gλCC
u ,λv
u ,λv
u ,λv
ship between the DCT coefficients of xt (m, n) and xt−1 (m, n) at the block boundary can be
obtained in the same way.
In (9.79), the pseudo-phase vector θλu ,λv (k, l) contains the information of the subpixel
movement (λu , λv ). In an ideal situation where one rigid object is moving translationally
© 2001 CRC Press LLC
within the block boundary without observable background and noise, we can find θλu ,λv (k, l)
explicitly in terms of λu and λv as such:

gλCC
(k, l)
u ,λv


1
lπ
1
cos kπ
N (λu + 2 ) cos N (λv + 2 )

 CS
 
1
lπ
1 
 gλu ,λv (k, l)   cos kπ

N (λu + 2 ) sin N (λv + 2 ) 



θλu ,λv (k, l) =  SC
=
.

kπ
1
lπ
1 
 gλu ,λv (k, l)   sin N (λu + 2 ) cos N (λv + 2 ) 
1
lπ
1
sin kπ
gλSSu ,λv (k, l)
N (λu + 2 ) sin N (λv + 2 )
9.6.1
(9.80)
DCT-Based Half-Pel Motion Estimation Algorithm (HDXT-ME)
From (9.79), we know that the subpixel motion information is hidden, although not obvious,
in the pseudo-phases. To obtain subpixel motion estimates, we can directly compute the
pseudo-phases in (9.79) and then locate the peaks of the sinc functions after applying the
subpixel sinusoidal orthogonal principles (9.70)–(9.73) to the pseudo-phases. Alternatively,
we can have better flexibility and scalability by first using the DXT-ME algorithm to get an
integer-pel motion estimate and then utilizing the pseudo-phase functions f (k, l) and g(k, l)
computed in the DXT-ME algorithm to increase estimation accuracy to half-pel, due to the fact
that (9.79) has exactly the same form as (9.74). Specifically, based on the subpixel sinusoidal
orthogonal principles (9.70)–(9.73), the subpixel motion information can be extracted in the
form of impulse functions with peak positions closely related to the displacement.
For the sake of flexibility and modularity in design and further reduction in complexity,
we adopt the second approach to devise a motion estimation scheme with arbitrary fractional
pel accuracy by applying the subpixel sinusoidal orthogonal principles to the pseudo-phase
functions passed from the DXT-ME algorithm. The limitation of estimation accuracy will only
be determined by the interaction effects of the ξ functions and the slope of the ξ function at and
around zero and how well the subpixel motion information is preserved in the pseudo-phases
after sampling.
We define DCS(u, v) and DSC(u, v) as follows:
DCS(u, v) N−1
N−1
k=0 l=1
DSC(u, v) N−1
N−1
k=1 l=0
f (k, l)
kπ
1
lπ
1
cos
u+
sin
v+
,
C(k)C(l)
N
2
N
2
(9.81)
g(k, l)
kπ
1
lπ
1
sin
u+
cos
v+
.
C(k)C(l)
N
2
N
2
(9.82)
Thus, from the subpixel sinusoidal orthogonal principles (9.70)–(9.73) and the definitions of
f (k, l) and g(k, l), we can show that
1
[ξ(u − λu ) + ξ(u + λu + 1)] · [ξ(v − λv ) − ξ(v + λv + 1)] , (9.83)
4
1
DSC(u, v) = [ξ(u − λu ) − ξ(u + λu + 1)] · [ξ(v − λv ) + ξ(v + λv + 1)] . (9.84)
4
DCS(u, v) =
The rules to determine subpixel motion direction are summarized in Table 9.6 and are similar
to the rules for determining integer-pel motion direction.
Figure 9.16 illustrates how to estimate subpixel displacements in the DCT domain. Figure 9.16c and d depict the input images x1 (m, n) of size 16 × 16 (i.e., N = 16) and x2 (m, n)
displaced from x1 (m, n) by (2.5, −2.5) at SNR = 50 dB. These two images are sampled on
a rectangular grid at a sampling distance d = 0.625 from the continuous intensity profile
xc (u, v) = exp(−(u2 + v 2 )) for u, v ∈ [−5, 5] in Figure 9.16a, whose Fourier transform is
© 2001 CRC Press LLC
Table 9.6 Determination of Direction of Movement (λu , λv ) from
the Signs of DSC and DCS
Sign of
DSC Peak
Sign of
DCS Peak
Peak Index
Motion Direction
+
+
−
−
+
−
+
−
(λu , λv )
(λu , −(λv + 1))
(−(λu + 1), λv )
(−(λu + 1), −(λv + 1))
Northeast
Southeast
Northwest
Southwest
bandlimited as in Figure 9.16b to satisfy the condition in Theorem 9.1. Figure 9.16e and f
are the 3D plots of the pseudo-phases f (k, l) and g(k, l) provided by the DXT-ME algorithm,
which also computes DSC(m, n) and DCS(m, n) as shown in Figure 9.16g and h, with peaks
positioned at (3, 1) and (2, 2) corresponding to the integer-pel estimated displacement vectors
(3, −2) and (2, −3), respectively, because only the first quadrant is viewed. As a matter of
fact, DSC(m, n) and DSC(m, n) have large magnitudes at {(m, n); m = 2, 3, n = 1, 2}.
To obtain an estimate at half-pel accuracy, we calculate DSC(u, v) and DCS(u, v) in (9.81)
and (9.82), respectively, for u, v = 0 : 0.5 : N − 1 as depicted in Figure 9.16i and j, where the
peaks can clearly be identified at (2.5, 1.5) corresponding to the motion estimate (2.5, −2.5)
exactly equal to the true displacement vector even though the two input images do not look alike.
Note that the notation a : r : b is an abbreviation of the range {a +i ·r for i = 0, . . . , b−a
r } =
{a, a + r, a + 2r, . . . , b − r, b}. For comparison, DSC(u, v) and DCS(u, v) are also plotted
in Figure 9.16k and l, respectively, for u, v = 0 : 0.25 : N − 1 = 0, 0.25, 0.5, . . . , N −
1.25, N − 1 where smooth ripples are obvious due to the ξ functions inherent in the DCS and
DSC of (9.83) and (9.84) and have peaks also at (2.5, 1.5).
Therefore, the DCT-based half-pel motion estimation algorithm (HDXT-ME) comprises
three steps:
1. The DXT-ME algorithm estimates the integer components of the displacement
as (m̂u , m̂v ).
2. The pseudo-phase functions from the DXT-ME algorithm, f (k, l) and g(k, l), are used
to compute DCS(u, v) and DSC(u, v) for u ∈ {m̂u − 0.5, m̂u , m̂u + 0.5} and v ∈
{m̂v − 0.5, m̂v , m̂v + 0.5} from (9.81) and (9.82), respectively.
3. Search the peak positions of DCS(u, v) and DSC(u, v) for the range of indices, & =
{(u, v) : u ∈ {m̂u − 0.5, m̂u , m̂u + 0.5}; v ∈ {m̂v − 0.5, m̂v , m̂v + 0.5}}, to find
(9.85)
uDCS , vDCS = arg max DCS(u, v) ,
u,v∈&
uDSC , vDSC = arg max DSC(u, v) .
u,v∈&
(9.86)
These peak positions determine the estimated displacement vector (λ̂u , λ̂v ). However, if
the absolute value of DSC(u, v) is less than a preset threshold 5D > 0, then λ̂u = −0.5;
likewise, if |DCS(u, v)| < 5D , λ̂v = −0.5. Therefore,
uDSC = uDCS , if |DSC(uDSC , vDSC )| > 5D ,
(9.87)
λ̂u =
−0.5,
if |DSC(uDSC , vDSC )| < 5D ,
vDCS = vDSC , if |DCS(uDCS , vDCS )| > 5D ,
λ̂v =
(9.88)
−0.5,
if |DCS(uDCS , vDCS )| < 5D .
© 2001 CRC Press LLC
FIGURE 9.16
Illustration of DCT-based half-pel motion estimation algorithm (HDXT-ME). (a) Continuous intensity profile xc (u, v); (b) FT of xc (u, v), Xc (-u , -v ); (c) 16 × 16 x1 (m, n);
(d) 16 × 16 x2 (m, n); (e) pseudo-phase f (k, l); (f) pseudo-phase g(k, l); (g) DSC(m, n);
(h) DCS(m, n); (i) DSC(u, v) for u, v = 0 : 0.5 : 15; (j) DCS(u, v) for u, v = 0 : 0.5 : 15;
(k) DSC(u, v) for u, v = 0 : 0.25 : 15; (l) DCS(u, v) for u, v = 0 : 0.25 : 15.
© 2001 CRC Press LLC
In step 2, only those half-pel estimates around the integer-pel estimate (m̂u , m̂v ) are considered due to the fact that the DXT-ME algorithm finds the nearest integer-pel motion estimate
(m̂u , m̂v ) from the subpixel displacement. This will significantly reduce the number of computations without evaluating all possible half-pel displacements.
In step 3, the use of 5D deals with the case of zero pseudo-phases when the displacement is
−0.5. Specifically, if λu = −0.5, then gλSC
(k, l) = 0, ∀k, l which leads to g(k, l) = 0 and
u ,λv
DSC(u, v) = 0. However, in a noisy situation, it is very likely that g(k, l) is not exactly zero
and, thus, neither is DSC(u, v). Therefore, 5D should be set very small but large enough to
accommodate the noisy case. In our experiment, 5D is empirically chosen to be 0.08. Similar
consideration is made on DCS(u, v) for λv = −0.5. It is also possible that the peak positions
of DCS(u, v) and DSC(u, v) differ in the noisy circumstances. In this case, the arbitration
rule used in the DXT-ME algorithm may be applied.
To demonstrate the accuracy of this HDXT-ME algorithm, we use a 16 × 16 dot image x1
in Figure 9.17a as input and displace x1 to generate the second input image x2 according to
the true motion field {(λu , λv ) : λu , λv = −5 : 0.5 : 4} (shown in Figure 9.17b) through
the bilinear interpolating function specified in the MPEG standard [13], which interpolates
the value x(m + u, n + v) from four neighboring pixel values for m, n being integers and
u, v ∈ [0, 1) in the following way:
x(m + u, n + v) = (1 − u) · (1 − v) · x(m, n) + (1 − u) · v · x(m, n + 1)
+ u · (1 − v) · x(m + 1, n) + u · v · x(m + 1, n + 1) .
(9.89)
Figure 9.17c shows the estimated motion field by the HDXT-ME algorithm, which is exactly
the same as the true motion field.
Figure 9.18a–c further illustrate estimation accuracy for half-pel motion estimation schemes
using peak information from Ls (u, v), Lc (u, v), and Lc (u, v) + Ls (u, v), respectively. In
Figure 9.18a, the “+” line indicates peak positions of Ls (u, v) found in the index range {0 : 0.5 :
15} for a block size N = 16 with respect to different true displacement values {−7 : 0.5 : 7}.
The “o” line specifies the final estimates after determination of motion directions from the
peak signs of Ls (u, v) according to the rules in Table 9.6. These estimates are shown to align
with the reference line u = v, implying their correctness. For the true displacement = −0.5,
Ls (−0.5, v) ≡ 0 for all v and 5D is used to decide whether the estimate should be set to
−0.5. In Figure 9.18b, Lc (u, v) is used instead of Ls (u, v) but Lc (u, v) is always positive,
inferring that no peak sign can be exploited to determine motion direction. In Figure 9.18c,
Lc (u, v) + Ls (u, v) provides accurate estimates without adjustment for all true displacement
values, but the index range must include negative indices (i.e., [−15 : 0.5 : 15]).
In the HDXT-ME algorithm, step 2 involves only nine DCS(u, v) and DSC(u, v) values
at and around (m̂u , m̂v ). Since DCS(u, v) and DSC(u, v) are variants of inverse 2D-DCT-II,
the parallel and fully pipelined 2D DCT lattice structure proposed in [9]–[11] can be used to
compute DCS(u, v) and DSC(u, v) at a cost of O(N ) operations in N steps. Furthermore,
the searching in step 3 requires O(N 2 ) operations for one step. Thus, the computational
complexity of the HDXT-ME algorithm is O(N 2 ) in total.
9.6.2
DCT-Based Quarter-Pel Motion Estimation Algorithm (QDXT-ME and
Q4DXT-ME)
The interaction of two ξ functions in Lc (u, v) and Ls (u, v) from (9.66) and (9.67) disassociates the peak locations with the displacement (λu , λv ) for λu , λv ∈ [−1.5, 0.5]. In spite of
this, in the HDXT-ME algorithm, we can still accurately estimate half-pel displacements by
locating the peaks of Ls (λ, v) for true displacements λ = −N + 1 : 0.5 : N − 1 and indices
v = 0 : 0.5 : N − 1 if 5D is introduced to deal with the case for λ = −0.5. However, at
© 2001 CRC Press LLC
FIGURE 9.17
Estimated motion fields (c) and (e) of HDXT-ME and QDXT-ME by moving a dot image
(a) according to the true motion fields (b) and (d).
© 2001 CRC Press LLC
FIGURE 9.18
Relation between true displacements and peak positions for half-pel and quarter-pel
estimation. The signs of peak values in Ls (u, v) indicate the motion directions and
are used to adjust the peak positions for motion estimates. (a) Ls (u, v) for half-pel
estimation; (b) Lc (u, v) for half-pel estimation; (c) Lc (u, v) + Ls (u, v) for half-pel estimation; (d) Ls (u, v) for quarter-pel estimation; (e) Lc (u, v) for quarter-pel estimation;
(f) Lc (u, v) + Ls (u, v) for quarter-pel estimation.
the quarter-pel level, it does cause estimation errors around λ = −0.5, as indicated in Figure 9.18d, where the peaks of Ls (λ, v) stay at v = 0 for true displacements λ varying over
[−1, 0]. The sum of Lc (λ, v) and Ls (λ, v) is a pure ξ function and thus the adverse interaction
is eliminated. As a result, the peak position of this sum can be used to predict precisely the
displacement at either the half-pel level or quarter-pel level, as demonstrated in Figure 9.18c
and f, respectively. However, for two-dimensional images, DCS or DSC has four ξ functions
as in (9.83) or (9.84). The DXT-ME algorithm provides two pseudo-phase functions f (k, l)
and g(k, l), but only DCS and DSC are available for subpixel estimation. In this case, the sum
of DCS and DSC can only annihilate two ξ functions, leaving two ξ functions as given by:
DCS(u, v) + DSC(u, v) =
1
[ξ (u − λu ) ξ (v − λv ) − ξ (u + λu + 1) ξ (v + λv + 1)] .
2
(9.90)
Even though this sum is not a single ξ function, the estimation error of using this sum is limited
to 1/4 pixel for the worst case when true displacements are either −0.75 or −0.25.
The above discussion leads to the DCT-based quarter-pel motion estimation algorithm
(QDXT-ME) as follows:
1. The DXT-ME algorithm computes the integer-pel estimate (m̂u , m̂v ).
2. DCS(u, v) and DSC(u, v) are calculated from f (k, l) and g(k, l) in (9.81) and (9.82),
respectively, for the range of indices, & = {(u, v) : u = m̂u − 0.75 : 0.25 : m̂u +
0.75; v = m̂v − 0.75 : 0.25 : m̂v + 0.75}.
© 2001 CRC Press LLC
3. Search the peak position of D2 (u, v) DCS(u, v) + DSC(u, v) over &; that is,
(uD2 , vD2 ) = arg max |D2 (u, v)| .
u,v∈&
The estimated displacement vector is obtained as follows:
(u , v ), if |D (u , v )| > 5 ,
D2 D2
2 D2 D2
D
λ̂u , λ̂v =
(−0.5, −0.5), if |D2 (uD2 , vD2 )| < 5D .
(9.91)
(9.92)
Step 3 is based on the fact that |D2 (λu , λv )| = 0 if and only if (λu , λv ) = −0.5. This
QDXT-ME algorithm follows the same procedure as HDXT-ME except for the search region
and using the sum of DCS and DSC. Therefore, QDXT-ME has the same computational
complexity, O(N 2 ), as HDXT-ME.
If we modify the DXT-ME algorithm to provide the other two pseudo-phase functions g CC
and g SS in addition to f and g, we can compute DCC and DSS in the following way:
DCC(u, v) N−1
N−1
g CC (k, l) cos
k=0 l=0
DSS(u, v) N−1
N−1
k=1 l=1
g SS (k, l) sin
kπ
N
kπ
N
u+
u+
1
lπ
1
cos
v+
,
2
N
2
1
lπ
1
sin
v+
.
2
N
2
(9.93)
(9.94)
Then we can show that
D4 (u, v) DCC(u, v) + DCS(u, v) + DSC(u, v) + DSS(u, v)
= ξ(u − λu )ξ(v − λv ) .
(9.95)
(9.96)
This sum contains only one ξ without any negative interaction effect whose peak is sharp at
(λu , λv ). This leads to another quarter-pel motion estimation algorithm (Q4DXT-ME), which
can estimate accurately for all displacements at the quarter-pel or even finer level.
1. Find the integer-pel estimate (m̂u , m̂v ) by the DXT-ME algorithm.
2. Obtain four pseudo-phases g CC , g CS , g SC , and g SS from the modified DXT-ME algorithm. Compute DCS(u, v), DSC(u, v), DCC(u, v), and DSS(u, v) for the range of
indices, & = {(u, v) : u = m̂u − 0.75 : 0.25 : m̂u + 0.75; v = m̂v − 0.75 : 0.25 :
m̂v + 0.75}.
3. Search the peak position of D4 (u, v) over &:
(uD4 , vD4 ) = arg max |D4 (u, v)| .
u,v∈&
The estimated displacement vector is then the peak position:
λ̂u , λ̂v = (uD4 , vD4 ) .
9.6.3
Simulation Results
The Miss America image sequence, with slow head and shoulder movement accompanying
occasional eye and mouth opening, is used for simulation [6]. The performance of the DCTbased algorithms is compared with BKM-ME and its subpixel counterparts in terms of MSE per
© 2001 CRC Press LLC
pixel and BPS. For all the MSE values computed in the experiment, the bilinear interpolation
in (9.89) is used for comparison to reconstruct images displaced by a fractional pixel because
the bilinear interpolation is used in MPEG standards for motion compensation [13, 21].
As usual, the integer-pel BKM-ME algorithm minimizes the MAD function of a block over
a larger search area. Its subpixel versions have two levels implemented: HBKM-ME and
QBKM-ME. Both algorithms optimize the MAD value around the subpixel displacements
around the integer-pel estimate. In addition, we also compare with three kinds of fast-search
block-matching algorithms with integer-pel, half-pel, and quarter-pel accuracy: the threestep search algorithm (TSS, HTSS, QTSS), the logarithmic search algorithm (LOG, HLOG,
QLOG), and the subsampled search algorithm (SUB, HSUB, QSUB) [17].
The simulation results are summarized by averaging over the sequence in terms of the MSE
and BPS values in Table 9.7. The coding gain from subpixel motion estimation is obvious
when we compare how much improvement we can have from integer-pel accuracy to half-pel
and even quarter-pel accuracy:
• HBKM-ME has 47.03% less of MSE value or 12.24% less of BPS value than BKM-ME,
whereas QBKM-ME has 60.76% less of MSE or 17.78% less of BPS than BKM-ME.
• Edge-extracted HDXT-ME has 45.36% less of MSE value or 12.95% less of BPS value
than edge-extracted DXT-ME, whereas edge-extracted QDXT-ME has 59.79% less of
MSE or 18.18% less of BPS.
Table 9.7 Performance Summary of DCT-Based Algorithms and
Block-Matching Algorithms (BKM, TSS, LOG, SUB) at Different Accuracy
Levels on Miss America Sequence
Approach
Integer-Pel Accuracy
BKM
Frame-differentiated DXT
Edge-extracted DXT
TSS
LOG
SUB
Half-Pel Accuracy
HBKM
Frame-differentiated HDXT
Edge-extracted HDXT
HTSS
HLOG
HSUB
Quarter-Pel Accuracy
QBKM
Frame-differentiated QDXT
Edge-extracted QDXT
Frame-differentiated Q4DXT
Edge-extracted Q4DXT
QTSS
QLOG
QSUB
© 2001 CRC Press LLC
MSE
MSE
Difference
MSE
Ratio
BPF
BPS
BPS
Ratio
7.187
7.851
9.363
7.862
7.862
7.202
0.000
0.664
2.176
0.675
0.675
0.015
0.0%
9.2%
30.3%
9.4%
9.4%
0.2%
8686
8855
9200
8910
8910
8684
0.343
0.349
0.363
0.352
0.352
0.343
0.0%
1.9%
5.9%
2.6%
2.6%
0.0%
3.807
5.598
5.116
3.877
3.877
3.810
0.000
1.791
1.308
0.070
0.070
0.002
0.0%
47.0%
34.4%
1.8%
1.8%
0.1%
7628
8216
8000
7676
7676
7628
0.301
0.324
0.316
0.303
0.303
0.301
0.0%
7.7%
4.9%
0.6%
0.6%
0.0%
2.820
4.728
3.899
4.874
3.765
2.843
2.843
2.825
0.000
1.908
1.079
2.054
0.945
0.023
0.023
0.005
0.0%
67.7%
38.3%
72.8%
33.5%
0.8%
0.8%
0.2%
7146
7758
7578
7785
7532
7162
7162
7144
0.282
0.306
0.299
0.307
0.297
0.283
0.283
0.282
0.0%
8.6%
6.0%
8.9%
5.4%
0.2%
0.2%
0.0%
9.7
DCT-Based Motion Compensation
Manipulation of compressed video data in the DCT domain has been recognized as an
important component in many advanced video applications [29]–[33]. In a video bridge,
where multiple sources of compressed video are combined and retransmitted in a network,
techniques of manipulation and composition of compressed video streams entirely in the DCT
domain eliminate the need to build a decoding/encoding pair. Furthermore, manipulation in
the DCT domain provides flexibility to match heterogeneous quality of service requirements
with different network or user resources, such as prioritization of signal components from loworder DCT coefficients to fit low-end communication resources. Finally, many manipulation
functions can be performed in the DCT domain more efficiently than in the spatial domain
[30] due to a much lower data rate and removal of the decoding/encoding pair. However, all
earlier work has been focused mainly on manipulation at the decoder side.
To serve the purpose of building a fully DCT-based motion-compensated video coder, our
aim is to develop the techniques of motion compensation in the DCT domain without converting back to the spatial domain before motion compensation. In [30], the method of pixelwise
(integer-pel) translation in the DCT domain is proposed for extracting a DCT block out of four
neighboring DCT blocks at an arbitrary position. Although addressing a different scenerio,
this method can be applied after modification to integer-pel motion compensation in the DCT
domain. For subpel motion compensation, we derive an equivalent form of bilinear interpolation in the DCT domain and then show that it is possible to perform other interpolation
functions for achieving more accurate and visually better approximation in the DCT domain
without increasing the complexity.
9.7.1
Integer-Pel DCT-Based Motion Compensation
As illustrated in Figure 9.19a, after motion estimation, the current block C of size N × N
in the current frame It can be best predicted from the block displaced from the current block
position by the estimated motion vector (du , dv ) in the spatial domain. This motion estimate
determines which four contiguous predefined DCT blocks are chosen for the prediction of
the current block out of eight surrounding DCT blocks and the block at the current block
position. To extract the displaced DCT block in the DCT domain, a direct method is used to
obtain separately from these four contiguous blocks four subblocks that can be combined to
form the final displaced DCT block (as shown in Figure 9.19b, with the upper-left, lower-left,
upper-right, and lower-right blocks from the previous frame It−1 labeled as B1 , B2 , B3 , and
B4 , respectively, in the spatial domain [30]). Subblocks Si are extracted in the spatial domain
from these four blocks by pre-multiplication and post-multiplication of the windowing/shifting
matrices, Hi and Vi :
Sk = Hk Bk Vk , for k = 1, . . . , 4 ,
where Hk and Vk are the N × N windowing/shifting matrices defined as
0 Ih1
0 0
0 0
0 0
H1 =
, V1 =
, H2 =
, V2 =
,
Iv1 0
Ih2 0
Iv2 0
00
0 Ih3
0 Iv3
0 0
0 Iv4
H3 =
, V3 =
, H4 =
.
, V4 =
00
00
00
Ih4 0
(9.97)
(9.98)
(9.99)
Here In is the n × n identity matrix (i.e., In = diag{1, . . . , 1}) and n is determined by the
height/width of the corresponding subblock. These pre- and post-multiplication matrix opera-
© 2001 CRC Press LLC
tions can be visualized in Figure 9.19c, where the overlapped gray areas represent the extracted
subblock. Then these four subblocks are summed to form the desired translated block B̂ref .
FIGURE 9.19
(a) Prediction of current block in current frame from four contiguous DCT blocks selected
among nine neighboring blocks in previous frame based on the estimated displacement
vector for current block. (b) Schematic diagram of how a pixelwise translated DCT
block is extracted from four contiguous DCT blocks. (c) Decomposition of integer-pel
DCT-based translation as four matrix multiplication operations. (d) Decomposition of
half-pel DCT-based translation as four matrix multiplication operations.
If we define the DCT operation on an N × N matrix B as
DCT{B} = DBDT ,
where the (k, m) element of D is the DCT-II kernel:
2
kπ
1
D(k, m) = C(k) cos
m+
, for k, m = 0, . . . , N − 1 .
N
N
2
© 2001 CRC Press LLC
Therefore, DT D = N2 IN . The formation of the DCT of B̂ref in the DCT domain can be
described in this equation:
DCT{B̂ref } =
N
2
2 4
DCT{Hk }DCT{Bk }DCT{Vk } .
(9.100)
k=1
This corresponds to pre- and post-multiplication of the DCT transformed Hk and Vk with the
DCT of Bk since DCT is a unitary orthogonal transformation and is guaranteed to be distributive
to matrix multiplications. The DCT of the motion-compensated residual (displaced frame
difference, DFD) for the current block C is, therefore,
DCT{DF D} = DCT{B̂ref } − DCT{C} .
(9.101)
DCT{Hk } and DCT{Vk } can be precomputed and stored in the memory. Furthermore, many
high-frequency coefficients of DCT{Bk } or displacement estimates are zero (i.e., sparse and
block-aligned reference blocks), making the actual number of computations in (9.100) small.
In [30], simulation results show that the DCT domain approach is faster than the spatial domain
approach by about 10 to 30%. Further simplication is also possible (as seen from Figure 9.19b):
HU = H1 = H3 , HL = H2 = H4 , VL = V1 = V2 , VR = V3 = V4 .
(9.102)
Therefore, only four windowing/shifting matrices need to be accessed from the memory instead
of eight.
At first sight, the DCT-based approach requires more computations than the pixel-based
approach. The pixel-based approach includes the IDCT and DCT steps. Thus, it requires
n2 additions (or subtractions) and 4 n × n matrix multiplications (for IDCT and DCT) to
calculate the DCT coefficients of the motion-compensated residue. In contrast, the DCTbased approach needs 4n2 additions plus 8 n × n matrix multiplications. However, savings can
be achieved by employing the sparseness of the DCT coefficients, which is the basis of DCT
compression. More savings come at the subpixel level. The DCT-based approach requires no
extra operations, whereas the pixel-based approach needs interpolation (i.e., to handle 4 (2×2)
times bigger images). In [34], further savings in the computation of the windowing/shifting
matrices is made by using fast DCT. A 47% reduction in computational complexity has been
reported with fast DCT over the brute-force method without the assumption of sparseness, and
a 68% reduction with only the top-left 4 × 4 subblocks being nonzero can be achieved with
the use of fast DCT.
9.7.2
Subpixel DCT-Based Motion Compensation
For the case of subpixel motion, interpolation is used to predict interpixel values. According
to the MPEG standards, bilinear interpolation is recommended for its simplicity in implementation and effectiveness in prediction [13, 21], although it is well known that a range of other
interpolation functions, such as cubic, spline, Gaussian, and Lagrange interpolations, can provide better approximation accuracy and more pleasant visual quality [15, 27, 35, 36]. The
complexity argument is true if the interpolation operation is performed in the spatial domain,
but in the DCT domain, it is possible to employ better interpolation functions than the bilinear
interpolation without any additional computational load increase.
Interpolation Filter
For simplicity of derivations, we start with the one-dimensional half-pel bilinear interpolation and then proceed to the two-dimensional case of quarter-pel accuracy with other
© 2001 CRC Press LLC
FIGURE 9.20
Illustration of extraction of the subpel displaced block, x2 (n), from two adjacent 1D
blocks, x1a (n) and x1b (n), with bilinear interpolation.
interpolation functions. Consider two one-dimensional adjacent blocks, x1a (n) and x1b (n)
for n = 0, . . . , N − 1, as shown in Figure 9.20. We want to extract a block {x2 (n)}N−1
n=0 of
displaced u pixels to the right of x1a (0) where u is supposed to be an odd multiple of 0.5 (i.e.,
half-pel motion). Therefore, we can show that


 1 [x1a (N + n − i) + x1a (N + n − i + 1)], 0 ≤ n ≤ i − 2 ,



2


1
x2 (n) =
(9.103)
[x1a (N − 1) + x1b (0)],
n=i−1,

2




1

 [x1b (n − i) + x1b (n − i + 1)],
N −1≥n≥i ,
2
where i = u. In the matrix form,
x2 = GBL (i)x1a + GBR (i)x1b ,
(9.104)
where x2 , x1a , and x1b are the column vectors of x2 (n), x1a (n), and x1b (n), respectively, and
GBL (i) and GBR (i) are defined as follows:
1
0 Ii
0 Ii−1
GBL (i) =
+
,
00
00
2
1
0
0
0
0
GBR (i) =
+
.
(9.105)
IN−i+1 0
IN−i 0
2
In the DCT domain,
DCT{x2 } = DCT{GBL (i)}DCT{x1a } + DCT{GBR (i)}DCT{x1b } .
(9.106)
Here GBL (i) and GBR (i) can be regarded as bilinear interpolation filter matrices, which act as
a linear filter or transform. Therefore, GBL (i) and GBR (i) can be replaced by any FIR filter
or interpolation function of finite duration (preferably with the length much smaller than the
block size N ).
Bilinear Interpolated Subpixel Motion Compensation
For the 2D case, if (u, v) is the displacement of the reconstructed block B̂ref measured from
the upper left corner of the block B1 , then for hU = u and vL = v,
DCT{B̂ref } =
4
k=1
© 2001 CRC Press LLC
DCT{Hk }DCT{Bk }DCT{Vk } ,
(9.107)
where
H1 = H3 = HU = GBL (hU ), H2 = H4 = HL = GBR (hU ),
T
T
V1 = V2 = VL = GBL
(vL ), V3 = V4 = VR = GBR
(vL ) .


0 · · · 0 0.5 0.5 0 · · ·
 .. . . ..

[GBL (hU ) GBR (hU )] =  . . . 0 . . . . . . 0  .
(9.108)
(9.109)
(9.110)
0 · · · 0 · · · 0 0.5 0.5
Once again, GBL (·) and GBR (·) can be precomputed and stored in the memory as in the case
of integer-pel motion compensation and thus the extra computational load for doing bilinear
interpolation is eliminated.
Cubic Interpolated Subpixel Motion Compensation
Three different interpolation functions, namely cubic, cubic spline, and bilinear interpolations, are plotted in Figure 9.21a. As can be seen, the bilinear interpolation has the shortest
filter length and the cubic spline has the longest ripple but the cubic spline also has the smallest
FIGURE 9.21
(a) Plots of different interpolation functions. (b), (c), and (d) depict how to form a
pre- or post-multiplication matrix for half-pel or even quarter-pel DCT-based motion
compensation.
© 2001 CRC Press LLC
approximation error of the three [36]. To compromise between filter length and approximation
accuracy, we choose the cubic interpolation in the simulation. By choosing the resolution of
the filter as half a pixel length, the bilinear interpolation fhb (n) = [0.5, 1, 0.5] and the cubic interpolation fhc (n) = [−0.0625, 0, 0.5625, 1.0000, 0.5625, 0, −0.0625]. From Figure 9.21b,
it is clear that the contributions at the half-pel position from all the pixel values are summed
up and give rise to the bilinear filter matrices GBL (·) and GBR (·). In a similar way, as in
Figure 9.21c, the cubic filter matrices GCL (·) and GCR (·) can be defined as
0 −0.0625Ii+1
0 0.5625Ii
GCL (i) =
+
0
0
0
0
0 0.5625Ii−1
0 −0.0625Ii−2
+
+
,
0
0
0
0
0
0
0
0
GCR (i) =
+
0.5625IN−i 0
−0.0625IN−i−1 0
0
0
0
0
+
+
.
0.5625IN−i+1 0
−0.0625IN−i+2 0
Here GCL (·) and GCR (·) can be precomputed and stored. Therefore, its computational complexity remains the same as both integer-pel and half-pel bilinear interpolated DCT-based
motion compensation methods. The reconstructed DCT block and the corresponding motioncompensated residual can be obtained in a similar fashion:
DCT{B̂ref } =
4
DCT{Hk }DCT{Bk }DCT{Vk },
(9.111)
k=1
DCT{DF D} = DCT{B̂ref } − DCT{C} ,
(9.112)
where
H1 = H3 = HU = GCL (hU ), H2 = H4 = HL = GCR (hU ),
T
T
V1 = V2 = VL = GCL
(vL ), V3 = V4 = VR = GCR
(vL ) .
(9.113)
(9.114)
This idea can be extended to other interpolation functions such as sharped Gaussian [35]
and quarter-pel accuracy.
9.7.3
Simulation
Simulation is performed on the Infrared Car and Miss America sequences to demonstrate
the effectiveness of our bilinear and cubic motion compensation methods.
The first set of simulations subsamples each picture It (i, j ) from the sequences (i.e., y(i, j )
= It (2 ∗ i, 2 ∗ j )) and then this shrunken picture y(i, j ) is displaced by a half-pel motion vector
(arbitrarily chosen as (2.5, 1.5)) with both bilinear and
cubic interpolated motion compensation
[x̂(i,j )−x(i,j )]2
by treating the original
methods. The MSEs are computed as MSE = i,j N 2
unsampled pixels It (2 ∗ i + 1, 2 ∗ j + 1) as the reference picture x(i, j ) = It (2 ∗ i + 1, 2 ∗ j + 1)
where x̂(i, j ) is the predicted pixel value from y(i, j ). As shown in Figure 9.22, the zeroorder interpolation is also simulated for comparison. The zero-order interpolation, also called
sample-and-hold interpolation, simply takes the original pixel value as the predicted half-pel
pixel value [15]. As can be seen in Figure 9.22, both the bilinear and cubic methods have
much lower MSE values than the zero-order method, and the cubic method performs much
better than the bilinear counterpart without increased computational load.
© 2001 CRC Press LLC
FIGURE 9.22
Pictures from the Infrared Car and Miss America sequences are subsampled and displaced by a half-pel motion vector with different motion compensation methods. The
MSE-per-pixel values are obtained by comparing the original unsampled pixel values
with the predicted pixel values of the motion-compensated residuals. Zero-order interpolation means replication of sampled pixels as the predicted pixel values.
Figure 9.23 shows the results of another set of simulations in which the subpixel DCT-based
motion compensation algorithms generate motion-compensated residuals of the Infrared Car
and Miss America sequences based on the displacement estimates of the full-search blockmatching algorithm, where the residuals are used to compute the MSE and BPS values for
comparison. It can be seen that the cubic interpolation approach achieves lower MSE and BPS
values than the bilinear interpolation.
9.8
Conclusion
In this chapter, we propose a fully DCT-based motion-compensated video coder structure
that not only is compliant with the hybrid motion-compensated DCT video coding standards but
also has a higher system throughput and a lower overall complexity than the conventional video
coder structure due to removal of the DCT and IDCT from the feedback loop. Therefore, it is
© 2001 CRC Press LLC
FIGURE 9.23
Pictures from the Infrared Car and Miss America sequences are subsampled and displaced by a half-pel motion vector with different motion compensation methods. The
MSE-per-pixel values are obtained by comparing the original unsampled pixel values
with the predicted pixel values of the motion-compensated residuals. Zero-order interpolation means replication of sampled pixels as the predicted pixel values.
more suitable for high-quality and high-bit-rate video applications such as HDTV or low-cost
video coder implementation. To realize such a fully DCT-based coder, we develop DCT-based
motion estimation and compensation algorithms.
The DCT pseudo-phase techniques that we develop provide us the means to estimate shifts
in the signals in the DCT domain. The resulting DXT-ME algorithm has low computational
complexity, O(N 2 ), as compared to O(N 4 ) for BKM-ME. Its performance over several image
sequences is comparable with that achieved by BKM-ME and some fast-search approaches
such as TSS, LOG, and SUB. Furthermore, its DCT-based nature enables us to incorporate
its implementation with the DCT codec design to gain further savings in complexity and take
advantage of advances in research on the DCT codec design. Finally, the DXT-ME algorithm
has inherently highly parallel operations in computing the pseudo-phases and thus it is very
suitable for VLSI implementation [37].
To deal with subpixel motion, we extend the DCT techniques to the subpixel level and derive
the subpel sinusoidal orthogonal principles. We demonstrate that subpixel motion information
is preserved in the DCT coefficients under the Nyquist condition. This fact enables us to
develop the DCT-based half-pel and quarter-pel motion estimation algorithms to estimate
© 2001 CRC Press LLC
subpixel motion in the DCT domain without any interpixel interpolation at a desired level
of accuracy. This results in significant savings in computational complexity for interpolation
and far less data flow compared to the conventional block-matching methods on interpolated
images. Furthermore, it avoids the deterioration of estimation precision caused by interpolation
and provides flexibility in the design in such a way that the same hardware can support different
levels of required accuracy with a complexity O(N 2 ), far less than O(N 4 ) for BKM-ME and
its subpixel versions.
Finally, we discuss the integer-pel DCT-based motion compensation method and develop
the subpel DCT-based motion compensation schemes using the bilinear and cubic interpolation
functions. We show that without increasing the number of computations, the cubic interpolated
half-pel and quarter-pel schemes exhibit higher coding gain in terms of smaller MSE and BPS
values than the bilinear interpolated counterparts.
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Processing and Communications, New Brunswick, September 1994, pp. 63–68.
[5] U.V. Koc and K.J.R. Liu, “DCT-based motion estimation,” IEEE Trans. Image Processing, vol. 7, no. 7, pp. 948–965, July 1998.
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[7] M. Ziegler, “Hierarchical motion estimation using the phase correlation method in
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[8] P. Yip and K.R. Rao, “On the shift property of DCT’s and DST’s,” IEEE Trans. Acoustics,
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[11] K.J.R. Liu, C.T. Chiu, R.K. Kologotla, and J.F. JaJa, “Optimal unified architectures for
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© 2001 CRC Press LLC
[12] CCITT Recommendation H.261, Video Codec for Audiovisual Services at p × 64 kbit/s,
CCITT, August 1990.
[13] CCITT Recommendation MPEG-1, Coding of Moving Pictures and Associated Audio for
Digital Storage Media at up to about 1.5 Mbit/s, ISO/IEC 11172, Geneva, Switzerland,
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[14] A. Zakhor and F. Lari, “Edge-based 3-D camera motion estimation with application to
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[15] A.K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs,
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[16] W.D. Kou and T. Fjallbrant, “A direct computation of DCT coefficients for a signal block
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IEEE Trans. Circuits and Systems for Video Technology, vol. 3, no. 2, pp. 148–157,
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[18] S.-L. Iu, “Comparison of motion compensation using different degrees of sub-pixel
accuracy for interfield/interframe hybrid coding of HDTV image sequences,” in 1992
IEEE Int. Conf. Acoustics, Speech, Signal Processing, San Francisco, CA, 1992, vol. 3,
pp. 465–468.
[19] B. Girod, “Motion compensation: Visual aspects, accuracy, and fundamental limits,” in
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chapter 5, Kluwer Academic, 1993.
[20] B. Girod, “Motion-compensating prediction with fractional-pel accuracy,” IEEE Trans.
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[21] CCITT Recommendation MPEG-2, Generic Coding of Moving Pictures and Associated
Audio, ISO/IEC 13818, Geneva, Switzerland, 1994, H.262.
[22] S.-I. Uramoto, A. Takabatake, and M. Yoshimoto, “A half-pel precision motion estimation processor for NTSC-resolution video,” IEICE Trans. Electronics, vol. 77, no. 12,
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[23] T. Akiyama, H. Aono, K. Aoki, K.W. Ler, B. Wilson, T. Araki, T. Morishige, H. Takeno,
A. Sato, S. Nakatani, and T. Senoh, “MPEG2 video codec using image compression
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[24] D. Brinthaupt, L. Letham, V. Maheshwari, J. Othmer, R. Spiwak, B. Edwards, C. Terman, and N. Weste, “A video decoder for H.261 video teleconferencing and MPEG
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[25] G. Madec, “Half pixel accuracy in block matching,” in Picture Coding Symp., Cambridge,
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[26] G. de Haan and W.A.C. Biezen, “Sub-pixel motion estimation with 3-D recursive search
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© 2001 CRC Press LLC
[29] S.F. Chang and D.G. Messerschmitt, “A new approach to decoding and compositing
motion-compensated DCT-based images,” in Proc. IEEE Int. Conf. Acoustics, Speech,
Signal Processing, 1993, vol. 5, pp. 421–424.
[30] S.-F. Chang and D.G. Messerschmitt, “Manipulation and compositing of MC-DCT compressed video,” IEEE Journal on Selected Areas in Communications, vol. 13, no. 1, p. 1,
January 1995.
[31] Y.Y. Lee and J.W. Woods, “Video post-production with compressed images,” SMPTE
Journal, vol. 103, pp. 76–84, February 1994.
[32] B.C. Smith and L. Rowe, “Algorithms for manipulating compressed images,” IEEE
Comput. Graph. Appl., pp. 34–42, September 1993.
[33] J.B. Lee and B.G. Lee, “Transform domain filtering based on pipelining structure,” IEEE
Trans. Signal Processing, vol. 40, pp. 2061–2064, August 1992.
[34] N. Merhav and V. Bhaskaran, “A fast algorithm for DCT-domain inverse motion compensation,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, 1996, vol. 4,
pp. 2309–2312.
[35] W.F. Schreiber, Fundamentals of Electronic Imaging Systems — Some Aspects of Image
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© 2001 CRC Press LLC
Chapter 10
Object-Based Analysis–Synthesis Coding Based
on Moving 3D Objects
Jörn Ostermann
10.1
Introduction
For the coding of moving images with low data rates between 64 kbit/s and 2 Mbit/s, a
block-based hybrid coder has been standardized by the ITU-T [8] where each image of a
sequence is subdivided into independently moving blocks of size 16×16 picture elements
(pels). Each block is coded by 2D motion-compensated prediction and transform coding [56].
This corresponds to a source model of “2D square blocks moving translationally in the image
plane,” which fails at boundaries of naturally moving objects and causes coding artifacts known
as blocking and mosquito effects at low data rates.
In order to avoid these coding distortions, several different approaches to video coding have
been proposed in the literature. They can be categorized into four methods: region-based
coding, object-based coding, knowledge-based coding, and semantic coding.
Region-based coding segments an image into regions of homogeneous texture or color [36].
Usually, these regions are not related to physical objects. Regions are allowed to move and
change their shape and texture over time. Some recent proposals merge regions with dissimilar
texture but with similar motion into one entity in order to increase coding efficiency [12].
The concept of object-based analysis–synthesis coding (OBASC) aiming at a data rate of
64 kbit/s and below was proposed in [44]. A coder based on this concept divides an image
sequence into moving objects. An object is defined by its uniform motion and described by motion, shape, and color parameters, where color parameters denote luminance and chrominance
reflectance of the object surface. Those parts of an image that can be described with sufficient
accuracy by moving objects require the transmission of motion and shape parameters only,
since the texture of the previously coded objects can be used. The remaining image areas are
called areas of model failure. They require the transmission of shape and texture parameters
in order to generate a subjectively correct decoded image. This detection of model failures
can be adapted to accommodate properties of the human visual system. OBASC in its basic
form does not require any a priori knowledge of the moving objects. A first implementation of
an OBASC presented in 1991 was based on the source model of “moving flexible 2D objects”
(F2D) and was used for coding image sequences between 64 and 16 kbit/s [3, 18, 26]. Implementations of an OBASC based on the source models of “moving rigid 3D objects” (R3D)
and “moving flexible 3D objects” (F3D) using 3D motion and 3D shape are presented in [48]
and [49], respectively. In [13], an implicit 3D shape representation is proposed.
© 2001 CRC Press LLC
Knowledge-based coders [31] use a source model which is adapted to a special object. In
contrast to OBASC, this allows the encoding of only a special object like a face. However, due
to this adaptation of the source model to the scene contents, a more efficient encoding becomes
possible. A recognition algorithm is required to detect the object in the video sequence. For
encoding of faces, a predefined 3D face model gets adapted to the face in the sequence. Then,
the motion parameters of the face are estimated and coded. Perhaps the most challenging task
for a knowledge-based encoder is the reliable detection of the face position [20, 34, 60].
Semantic coders [15] are modeled after knowledge-based coders. Until now, semantic
coding was mainly investigated for the encoding of faces using high-level parameters such
as the facial action coding system [2, 11, 14, 21, 22, 68]. Alternatively, facial animation
parameters as defined by MPEG-4 can be used [30, 53]. By using high-level parameters we
limit the degrees of freedom of the object and achieve a higher data reduction.
Knowledge-based as well as semantic coders use three-dimensional source models. Since
they can encode only a particular object, they require a different algorithm for encoding the
image areas outside this object. An OBASC based on a 3D source model seems to be a natural
choice.
The purpose of this chapter is twofold. First, the concept of object-based analysis–synthesis
coding is reviewed. Second, different source models used for OBASC and their main properties
are compared. In order to use 3D source models, a reliable motion estimation algorithm is
required. Here, we develop a robust gradient-based estimator that is able to track objects. The
coding efficiencies obtained with the source models F2D [27], R3D, and F3D are compared in
terms of data rate required for the same picture quality. The coding schemes will be evaluated
using videophone test sequences. As a well-known reference for picture quality, the blockbased hybrid coder H.261 [8, 9] is used.
In Section 10.2, the principles of object-based analysis–synthesis coding are reviewed. In
Section 10.3, different 2D and 3D source models for OBASC and their implementations in an
OBASC are presented. Since the data rate of an OBASC depends mainly on how well the image
analysis can track the moving objects, we present in Section 10.4 details of image analysis. A
robust 3D motion estimator is developed. Model failure detection is discussed in more detail.
In Section 10.5, we present an overview of parameter coding. The coding efficiency of the
different source models is compared in Section 10.6. A final discussion concludes this book
chapter.
10.2
Object-Based Analysis–Synthesis Coding
The goal of OBASC is the efficient encoding of image sequences. Each image of the
sequence is called a real image. OBASC [44] subdivides each real image into moving objects
called real objects. A real object is topologically connected and characterized by its uniform
motion. A real object is modeled by a model object as defined by the source model of the
encoder. Hence, one real object is described by one model object, whereas region-based
coding describes regions with homogeneous textures as separate entities. Each model object
m is described by three sets of parameters, A(m) , M (m) , and S (m) , defining its motion, shape,
and color, respectively. Motion parameters define the position and motion of the object; shape
parameters define its shape. Color parameters denote the luminance as well as the chrominance
reflectance on the surface of the object. In computer graphics, they are sometimes called texture.
The precise meaning of the three parameter sets depends on the source model employed (see
Section 10.3).
© 2001 CRC Press LLC
Figure 10.1 is used to explain the concept and structure of OBASC. An OBASC consists
of five parts: image analysis, parameter coding, parameter decoding, parameter memory, and
image synthesis. Instead of the frame memory used in block-based hybrid coding, OBASC
requires a memory for parameters in order to store the coded, transmitted, and decoded parameters A , M , and S for all objects. Whereas the double prime ( ) symbol marks the
transmitted parameters used to update the parameter memory, the prime ( ) symbol marks the
decoded parameters at the output of the parameter memory.
FIGURE 10.1
Block diagram of an object-based analysis–synthesis coder.
The parameter memories in the coder and decoder contain the same information. Evaluating these parameter sets, image synthesis produces a model image sk , which is displayed
at the decoder. In order to avoid annoying artifacts at object boundaries, a shape-dependent
antialiasing filter may be applied at object boundaries [58].
At time instant k + 1, the image analysis has to evaluate the current image sk+1 considering
the parameter sets A , M , and S estimated for image sk . The task of image analysis is to
track each object known from previous frames and detect new moving objects. Each object m
is described by three sets of parameters, Am,k+1 , Mm,k+1 , and Sm,k+1 . These parameter sets
are available at the output of the image analysis in PCM format. Considering the previously
estimated and coded parameter sets A , M , and S creates a feedback loop in the encoder. This
allows the image analysis to compensate for previous estimation errors as well as shape and
motion quantization errors introduced by the lossy encoding of the parameters by parameter
coding. Hence, an accumulation of estimation and quantization errors is avoided.
Figure 10.2 serves as an example to describe the parameter sets in the case of a source model
of rigid 2D objects with 2D motion. The color parameters of an object can be covered and
uncovered due to (1) camera motion, (2) a new object entering the scene, (3) motion of another
object, or (4) egomotion. In this chapter, we focus on (2) to (4). The extension of OBASC to
consider camera motion is straightforward on a conceptual level. Mech and Wollborn describe
an implementation in [40]. In the example of Figure 10.2, areas of object 1 get uncovered
due to the motion of object 2. Assuming that these parts of object 1 have not been visible
before, the color parameters of these uncovered areas (UAs) have to be transmitted. Similarly,
a rotating 3D object might uncover previously not visible areas for which the transmission
of color parameters is required. The color parameters of an uncovered area can uniquely be
associated with one object. Therefore, the color parameter Sm,k+1 of object m at time instant
UA
k + 1 consists of its color parameter Sm,k at time instant k and the color parameters Sm,k+1
of
© 2001 CRC Press LLC
FIGURE 10.2
Image analysis demonstrated using rigid translaterally moving objects in the image plane.
The dashed lines denote the image boundaries in order to show the positions of object 2. It
is not necessary to know the motion parameters at time instant tk . The texture parameters
of object 1 change due to the uncovered areas.
its uncovered area:
UA
Sm,k+1 = Sm,k ∪ Sm,k+1
.
(10.1)
If several objects are moving, an uncovered area can belong to a moving object. The shape of
the uncovered area can be derived from object motion and object shape. In Figure 10.2, the
shape of the uncovered area of object 1 is determined by the shape and motion of object 2 [64].
Most implementations of an image analysis for an OBASC assume moving objects in front
of a static background. In the current image, moving and static objects are detected first
by means of change detection [23, 45, 52, 64]. For moving objects, new motion and shape
parameters are estimated in order to reuse most of the already transmitted color parameters
(m)
Sk . As pointed out in [25], the estimation of motion and shape parameters are mutually
dependent problems. However, in the case of a static background, the correct estimation of
motion is the more challenging task of the two [64]. Objects for which motion and shape
parameters can be estimated successfully are referred to as MC objects (model compliance).
In the final step of image analysis, image areas that cannot be described by MC objects using
(m)
(m)
(m)
the transmitted color parameters Sk and the new motion and shape parameters Ak+1 Mk+1 ,
respectively, are detected. Areas of model failure (MF) [46] are derived from these areas.
They are defined by 2D shape and color parameters only and are referred to as MF objects.
The detection of MF objects takes into account that small position and shape errors of the MC
objects — referred to as geometrical distortions — do not disturb subjective image quality.
Therefore, MF objects are limited to those image areas with significant differences between the
motion- and shape-compensated prediction image and the current image sk+1 . They tend to
be small in size. This allows coding of color parameters of MF objects with high quality, thus
avoiding subjectively annoying quantization errors. Since the transmission of color parameters
© 2001 CRC Press LLC
is expensive in terms of data rate, the total area of MF objects should not be larger than 4% of
the image area, assuming 64 kbit/s, CIF (common intermediate format, 352×288 luminance
and 176×144 chrominance pels/frame), and 10 Hz.
Depending on the object class MC/MF, the parameter sets of each object are coded by parameter coding using predictive coding techniques (Figure 10.3). Motion and shape parameters
are encoded and transmitted for MC objects and shape and color parameters for MF objects.
For MC objects, motion parameters are quantized and encoded. The motion information is
used to predict the current shape of the MC object. After motion compensation of the shape,
only the shape prediction error has to be encoded. The shape of uncovered areas is derived
FIGURE 10.3
Block diagram of parameter coding.
from the shape and motion of the MC objects. In the case of uncovered areas being visible for
the first time, color parameters have to be transmitted. For MF objects, a temporal prediction
of the shape is not useful, since areas of model failure are not temporally correlated. Hence,
the shape parameters of MF objects are encoded in intra mode. For the color parameters, the
motion-compensated prediction error is computed using the motion parameters of the underlying MC object. Then the prediction error is quantized and encoded. Table 10.1 summarizes
which parameter sets have to be transmitted for each object class. As in any block-based coder,
color parameters have to be transmitted only in the case of a scene cut. Since the coding of
color parameters generally requires a bit rate of more than 1 bit/pel in active areas, the size
of the MF objects determines to a large extent the bit rate required for encoding an image
sequence. Hence, image analysis should be optimized such that the size of the MF objects
becomes small. Furthermore, parameter coding for MF and MC objects has to be optimized in
order to minimize the overall data rate R = RA + RM + RS for coding all parameter sets [50].
Parameter decoding decodes the two parameter sets transmitted for each object class. In the
memory for parameters, the position and shape of MC objects are updated. Furthermore, in
areas of model failure, color parameters of MC objects are substituted by the color parameters
of the transmitted MF objects. Therefore, only MC objects are available at the output of the
parameter memory.
In OBASC, the suitability of source models can be judged by comparing the data rates
required for coding the same image sequence with the same image quality. Image quality is
influenced mainly by the algorithm for detecting model failures and by the bit rate available
© 2001 CRC Press LLC
Table 10.1 Parameter Sets That Must Be
Coded for MC Objects, Uncovered Areas
(UAs), and MF Objects and in the Case of a
Scene Cut (SC)
Coder Mode
Parameter Set
Motion parameters A
Shape parameter M
Color parameter S
MC Object
Mode
MC UA
×
×
×
MF Object
Mode
MF SC
×
×
×
for coding the color parameters of model failures. Assuming an image format of CIF with a
reduced frame frequency of 10 Hz, an average area of MF objects of 4% of the image area
should be sufficient in order to encode a videophone sequence with good subjective quality at
a bit rate of 64 kbit/s.
10.3
Source Models for OBASC
In this section, the different source models applied to OBASC, their main properties, as
well as some implementation details are presented. In order to highlight commonalities and
differences between source models used for OBASC, it is useful to subdivide a source model
into its main components, namely the camera model, the illumination model, the scene model,
and the object model. The source model used here assumes a 3D real world that has to be
modeled by a model world. Whereas the real image is taken by a real camera looking into the
real world, a model image is synthesized using a model camera looking into the model world.
A world is described by a scene, its illumination, and its camera. A scene consists of objects,
their motion, and their relative position. Initially, the source models are distinguished from
each other by the object model. For simplicity, we name the source models according to the
name of their object model. Recent research also has focused on illumination models [62, 63].
The goal of the modeling is to generate a model world, Wk , with a model image identical
to the real image, sk , at a time instance k. This implies that the model objects may differ from
the real objects. However, similarity between the real object and the model object generally
helps in performing proper image analysis.
The following sections will describe the different parts of a source model. After the review
of the camera, illumination, and scene model, different object models as applied to OBASC are
explained. These object models are used to describe the real objects by means of MC objects.
For each object model, parameter coding and some implementation details are highlighted.
10.3.1
Camera Model
The real camera is modeled by a static pinhole camera [65]. Whereas a real image is
generated by reading the target of the real camera, a model image is read off the target of the
model camera. Assuming a world coordinate system (x, y, z) and an image coordinate system
(i)
(i)
(i)
(X, Y ), this camera projects the point P(i) = (Px , Py , Pz )T on the surface of an object in
© 2001 CRC Press LLC
(i)
(i)
the scene onto the point p(i) = (pX , pY )T of the image plane according to
(i)
pX
=F ·
(i)
Px
(i)
Pz
,
(i)
pY
(i)
=F ·
Py
(i)
Pz
(10.2)
where F is the focal length of the camera (Figure 10.4). This model assumes that the image
plane is parallel to the (x, y) plane of the world coordinate system. For many applications,
this camera model is of sufficient accuracy. However, in order to incorporate camera motion,
a CAHV camera model [69] allowing for arbitrary camera motion and zoom should be used.
FIGURE 10.4
Camera model.
10.3.2
Scene Model
The scene model describes the objects of a world using an object model and the relationship
between objects (Figures 10.2 and 10.5). It allows an explanation of the effects of covered and
uncovered areas. In the case of uncovered areas, the relative position of the objects to each
other and to the image plane allow the correct assignment of the area to one object.
10.3.3
Illumination Model
The illumination model describes the temporal changes in the video sequence caused by the
changing illumination of the real world. The interaction of incident light from a light source
with a point P of an object is described by the distribution Lr,λ of reflected radiance from an
object surface depending on the distribution Ei,λ of incident irradiance and the object surface
reflectance function R at this point according to
Lr,λ (L, V , N, P , λ) = R(L, V , N, P , λ) · Ei,λ (L, N, λ)
(10.3)
Here N is the surface normal vector, L the illumination direction, V the viewing direction (to
the focal point of the camera), and λ the wavelength of light (Figure 10.6). With simplifying
© 2001 CRC Press LLC
FIGURE 10.5
Scene model.
FIGURE 10.6
Surface patch dA with normal vector N illuminated from direction L by a point light
source with the infinitesimal small area dAs . The patch is viewed from direction V .
assumptions such as opaque object surfaces and temporally invariant illumination direction as
well as viewing direction, (10.3) simplifies to
Lr,λ (N, P , λ) = R(N, P , λ) · Ei,λ (N, λ)
(10.4)
Assuming the scene to be illuminated by a point light source and ambient diffuse light simplifies
the description of the incident irradiance to the shading model of Phong used in early computer
graphics [58]:
Ei (N ) = cambient + clambert · max(0, LN )
(10.5)
with cambient the ambient irradiance and clambert the point light source irradiance. Assuming
that N and therefore the object shape is known, this simple illumination model requires three
parameters to be estimated: the ratio between ambient and direct irradiance, cambient /clambert ,
and the two angles describing the direction of the direct point light source irradiation. This
model according to (10.5) has been implemented in an OBASC by Stauder [62]. In the image
© 2001 CRC Press LLC
plane, Stander assumes that the luminance l(p) of a point moving from pk to pk+1 changes
according to
l (pk+1 ) = l (pk ) ·
Ei (Nk+1 )
.
Ei (Nk )
(10.6)
Pearson and others proposed to model the irradiance by a discrete irradiance map [55]. This
map gives an irradiance value for each patch of the incident light on a Gaussian sphere. It is not
restricted to any illumination situation. Several light sources can be handled. For a reasonable
approximation of an illumination situation, the number of patches should be 9 × 9 or higher.
This method is especially useful if the irradiance values can be measured.
Another simple illumination model assumes that the image signal l(p) depends on the illumination E(p) and the bidirectional reflection function R(p). R(p) accounts for the wavelength
of the illumination, surface material, and the geometric arrangement of illumination, camera,
and surface. The illumination Ei (p) depends on ambient and direct light. Assuming diffuse
illumination, diffuse reflecting surfaces, parallel projection, and a constant kb , the image signal
is given by the reflection model
l(p) = kB · Ei (p) · R(p) .
(10.7)
In the image plane, the luminance l(p) of a point moving from pk to pk+1 changes according
to
l (pk+1 ) = kb · E (pk+1 ) · R (pk ) .
(10.8)
This reflection model indicates that illumination can be modeled by a multiplicative factor.
This has proven to be useful in block matching, 3D motion estimation, and change detection [7,
19, 52, 63].
The simplest, yet most widely used illumination model simply assumes for the luminance
of a moving point
l (pk+1 ) = l (pk ) .
(10.9)
Sometimes, this model is referred to as the constant intensity assumption. The implicit assumptions are diffuse illumination, diffuse reflecting surfaces, and no temporal variation in the
illumination. Here, we select the simple illumination model according to (10.9).
10.3.4
Object Model
The object model describes the assumptions of the source model about the real objects.
In order to do so, shape, motion, and surface models are required. While all object models
described here use the same surface model, they employ different motion and shape models
as discussed below.
As far as the surface model is concerned, it is assumed that object surfaces are opaque
and have a diffuse reflecting surface. The surface of an object m is described by the color
parameters Sm . These color parameters contain the luminance as well as the chrominance
reflectance.
Moving Rigid 2D Objects (R2D) with 3D Motion
Object Model
This object model assumes rigid 2D arbitrarily shaped objects. Hence, each object can
be perceived as a part of a plane. Its projection into the image plane is the 2D silhouette
© 2001 CRC Press LLC
of the object. Each object is allowed to move in 3D space. Allowing two parameters to
describe the orientation of the plane in space and six parameters to describe object motion,
the functional relationship between a point P on the object surface projected onto the image
plane as p = (X, Y )T and p = (X , Y )T before and after motion, respectively, is described
by eight parameters (a1 , . . . , a8 ) [23, 65]:
p = (X , Y )T =
a1 X + a2 Y + a3 a4 X + a5 Y + a6
,
a7 X + a 8 + 1
a7 X + a 8 + 1
T
.
(10.10)
Implementation
Image analysis for this source model estimates motion hierarchically for an object, which
initially is the entire frame, sk+1 . Then, the motion-compensated prediction ŝ of the object is
computed using an image synthesis algorithm, which fetches the luminance for a point p in
frame sk+1 from point p of frame sk according to the inverse of (10.10). Finally, the estimated
motion parameters are verified by comparing the original image and the predicted image,
and detecting those areas where the motion parameters do not allow for a sufficiently precise
approximation of the original image. These areas are the objects where motion parameters are
estimated in the next step of the hierarchy. This verification step allows the segmentation of
moving objects using the motion as the segmentation criterion. Hötter [23] implemented three
steps of the hierarchy. The image areas that could not be described by motion parameters at
the end of this object segmentation and motion compensation process are the MFR2D objects.
In order to achieve robust estimates, Hötter allowed the motion model to reduce the number
of parameters from eight to an affine transformation with six (a1 , . . . , a6 ) parameters or to a
displacement with two parameters (a1 , a2 ). This adaptation is especially important as objects
become smaller [23].
In order to increase the efficiency of parameter coding, especially the shape parameter
coding, the segmentation of the current frame into moving objects takes the segmentation
of the previous image as a starting point. In the image plane, this increases the temporal
consistency of the 2D shape of the objects. Since eight parameters are estimated for each
frame in which an object is visible, this model allows the object to change its orientation in
space arbitrarily from one frame interval to the next. No temporal coherence for the orientation
of the object is required or enforced.
Moving Flexible 2D Objects (F2D) with 2D Motion
Object Model
This source model assumes that the motion of a real object can be described by a homogeneous displacement vector field. This displacement vector field moves the projection of the
real object into the image plane to its new position. Assuming a point P on the object surface
moving from P to P , its projection into the image plane moves from p to p . p and p are
) = (DX (p ), DY (p ))T :
related by the displacement vector D(p
) .
p = p − D(p
(10.11)
The motion parameters of an MC object m are the displacement vectors of those points p that
belong to the projection of object m into the image plane. The shape of this object is defined
by the 2D silhouette that outlines the projection of object m in the image plane.
Implementation
For estimating the changed area due to object motion, Hötter applies a change detector to
the coded image sk and the real current image sk+1 . This change detector is initialized with the
© 2001 CRC Press LLC
silhouette of the objects as estimated for image sk . This allows object tracking and increases
the coding efficiency for shape parameters. For displacement estimation, a hierarchical blockmatching technique is used [4]. Experimental investigations show that an amplitude resolution
of a half pel and a spatial resolution of one displacement vector for every 16×16 pel result
in the lowest overall data rate for encoding. This implementation of image analysis is only
able to segment moving objects in front of a static background. Since an OBASC relies on the
precise segmentation of moving objects and their motion boundaries, this restriction on the
image analysis limits the coding efficiency for scenes with complex motion.
To compute the motion-compensated prediction image, the texture of the previously decoded
image can be used similar to the MPEG-1 and MPEG-2 standards. The displacement vector
field is bilinearly interpolated inside an object. The vectors are quantized to half-pel accuracy.
In order to accomplish prediction using these half-pel motion vectors, the image signal is
bilinearly interpolated. In [27], the disadvantage of this image synthesis by filter concatenation
was noted. Assuming that all temporal image changes are due to motion, the real image sk (p )
is identical to s0 (p). If displacement vectors with subpel amplitude resolution are applied, the
displaced position does not necessarily coincide with the sampling grid of s0 . In that case,
a spatial interpolation filter h is required to compute the missing sample. In Figure 10.7,
the luminance value of sk+1 (y1 ) requires two temporal filter operations. Assuming bilinear
FIGURE 10.7
Image synthesis by filter concatenation (one-dimensional case) [27].
interpolation and the displacement vector field as depicted in Figure 10.7, sk+1 (y1 ) is given by
sk+1 (y1 ) =
1
1
1
s0 (y1 ) + s0 (y2 ) + s0 (y3 ) .
4
2
4
(10.12)
The disadvantage of this method is that repeated interpolation results in severe low-pass filtering
of the image. Hötter suggests image synthesis by parameter concatenation using an object
memory for the texture parameters of each object (Figure 10.8). Thus, the interpolation filter h
has to be applied only once. In order to synthesize the image sk+1 (y1 ), the total displacement
© 2001 CRC Press LLC
FIGURE 10.8
Image synthesis by parameter concatenation (one-dimensional case) using an object
memory for color parameters [27].
between s0 and sk+1 is computed by concatenating the displacement vectors. In this case,
sk+1 (y1 ) is given by
sk+1 (y1 ) = s0 (y2 ) .
(10.13)
This is basically a method of texture mapping, as known from computer graphics, and OBASC
based on 3D source models (see Section 10.3.4) [67]. Indeed, Hötter’s implementation uses a
mesh of triangles in order to realize this object memory, or texture memory as it is known in
computer graphics (Figure 10.9). In [28], Hötter develops a stochastic model describing the
synthesis errors due to spatial interpolation and displacement estimation errors. The model was
verified by experiments. Use of this texture memory gives a gain of 1 dB in the signal-to-noise
ratio for every 14 frames encoded. This gain is especially relevant since the improvement is
only due to the less frequent use of the interpolation filter, thus resulting in significantly sharper
images.
© 2001 CRC Press LLC
FIGURE 10.9
Image synthesis for MC objects using a triangle-based mesh as texture memory.
MF objects are detected as described in Section 10.4.3. The shape parameter coding is
presented in Section 10.5.2. The 2D displacement vector fields are DPCM coded using spatial
prediction.
Moving Rigid 3D Objects (R3D)
Object Model
In the model used here, the 3D shape is represented by a mesh of triangles, which is put
(i)
up by vertices referred to as control points, PC . The appearance of the model object surface
is described by the color parameters S (m) . In order to limit the bit rate for coding of shape
parameters, the shape parameters M (m) of an object m represent a 2D binary mask, which
defines the silhouette of the model object in the model image. During initialization, the 3D
shape of an object is completely described by its 2D silhouette (i.e., there is an algorithm
that computes a generalized 3D cylinder from a 2D silhouette (Figure 10.10) using a distance
transform to determine the object depth (Figure 10.10b) [45]). The distance transform assigns
two depth values wF ±Z to each point h of the object silhouette. Each depth value depends
on the Euclidian distance between point h and the silhouette boundary. Depending on the
application, an appropriate mapping d → wF ±Z can be selected. Using a mapping derived
from an ellipse is suitable for the modeling of head and shoulder scenes. The object width b
and the object depth h are related according to (Figure 10.11):
√
h
d(b − d) for d < b2
wF ±Z (d) = F ± hb
(10.14)
otherwise .
2
In order to determine the object width, we use the four measurements b1 , b2 , b3 , and b4
according to Figure 10.12 in order to determine the maximum elongation of the object in the
direction of the x and y axes as well as the two diagonals. We determine the object width b as
© 2001 CRC Press LLC
FIGURE 10.10
Processing steps from object silhouette to model object: (a) object silhouette; (b) 3D
object shape with required silhouette rotated by 30◦ and illuminated; (c) contour lines
approximating the object shape; (d) polygons approximating the contour lines; (e) mesh
of triangles using polygon points as vertices; (f) model object with color parameters
projected onto it.
the minimum of the four values according to
b = min(b1 , b2 , b3 , b4 ) .
(10.15)
wF ±Z is constant for d > b/2. Hence, the surface of the model object is parallel to the image
plane where the object is wide. In order to automatically adapt the object depth to the width
of different objects, we set the ratio β = b/ h instead of h to a fixed value:
√
1
d(b − d) for d < b2
wF ±Z (d) = F ± βb
(10.16)
otherwise .
2β
In Figure 10.10, the ratio β between object width and object depth is set to 1.5. The maximum
distance between the estimated silhouette and the silhouette of the model object does not exceed
dmax ≤ 1.4 pel (Figure 10.10d). After initialization, the shape parameters M (m) are used as
update parameters to the model object shape.
An object may consist of one, two, or more rigid components [6]. The subdivision of an
object into components is estimated by image analysis. Each component has its own set of
motion parameters. Since each component is defined by its control points, the components are
linked by those triangles of the object having control points belonging to different components.
Due to these triangles, components are flexibly connected. Figure 10.13 shows a scene with
the objects Background and Claire. The model object Claire consists of the two components
Head and Shoulder.
© 2001 CRC Press LLC
FIGURE 10.11
A 3D shape symmetric to the image plane Z = F is created. (a) Distance transform
according to (10.14); d is the smallest distance to the border of the object silhouette, b is
set according to (10.15), and β = b/ h = 1.5. In the example, the object width is larger
than b according to 10.15. (b) Cut through a model object (top); view from the focal point
of the camera onto the contour lines (bottom). For computing object depth, we always
measure the distance d to the closest boundary point [51].
FIGURE 10.12
Determining the object width. The distances b1 , b2 , b3 , and b4 are measured by determining the parallel projection of the silhouette onto the x and y axes and onto the image
diagonals. We use the minimum as object width, here b1 .
(m)
(m)
(m)
(m)
(m)
(m)
3D motion is described by the parameters A(m) = (Tx , Ty , Tz , Rx , Ry , Rz )
defining translation and rotation. A point P (i) on the surface of object m with N control points
(i)
PC is moved to its new position P (i) according to
(m)
P (i) = RC · P (i) − C (m) + C (m) + T (m)
(10.17)
(m)
with the translation vector T (m) = (Tx
© 2001 CRC Press LLC
(m)
, Ty
(m) T
) , the object center C
, Tz
= (Cx , Cy , Cz ) =
FIGURE 10.13
Model scene and model object Claire subdivided into two flexibly connected components:
(a) scene consisting of two objects; (b) components of model object Claire.
FIGURE 10.14
Triangular mesh with color parameter on the skin of the model object.
FIGURE 10.15
Triangular mesh after flexible shape compensation by flexible shift vector F (n) .
© 2001 CRC Press LLC
N
(i)
(C)
(C)
(C)
T
i=1 PC , the rotation angles RC = (Rx , Ry , Rz ) , and the rotation matrix [RC ]
defining the rotation in the mathematically positive direction around the x, y, and z axes with
the rotation center C:
1
N


cos Ry cos Rz , sin Rx sin Ry cos Rz − cos Rx sin Rz , cos Rx sin Ry cos Rz + sin Rx sin Rz
[RC ] =  cos Ry sin Rz , sin Rx sin Ry sin Rz + cos Rx cos Rz , cos Rx sin Ry sin Rz − sin Rx cos Rz 
− sin Ry ,
sin Rx cos Ry ,
cos Rx cos Ry
(10.18)
Implementation
An overview of the image analysis and a detailed description of motion estimation is given
in Section 10.4. Parameter coding is presented in Section 10.5.
Moving Flexible 3D Objects (F3D)
Object Model
In addition to the properties of the source model R3D, the source model F3D allows for
local flexible shifts on the surface of the model object shape. This is modeled by a flexible
skin (Figure 10.14). This flexible skin can be moved tangentially to the surface of the object
(Figure 10.15). It allows modeling of local deformations. In the model world, the flexible
(n)
surface is modeled by a shift of control points PC in the tangential surface plane. The normal
(n)
vector to this tangential surface plane is computed by averaging the normal vectors nDj to the
(n)
J triangles to which the control point PC belongs:
(n)
nt
=
J
j =1
(n)
nDj
(10.19)
(n)
This tangential surface plane with the normal vector nt
(n)
(n)
is spanned by Rf u and Rf v . These
(n)
vectors are of unit length and are orthogonal to each other. For each control point PC , two
(n)
(n)
(n)
flexible shape parameters Sf = (Sf u , Sf v )T have to be estimated.
(n)
(n)
PC
= PC + F (n)
PC
= PC + Sf u Rf u + Sf v Rf v
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(10.20)
(n)
with F the flexible shift vector and PC = (Px , Py , Pz )T and PC = (PX , PY , PZ )T being
a control point before and after shift, respectively. The flexible shift vectors F (n) can be
(m)
interpreted as local motion parameters, in contrast to the global motion parameters, RC and
T (m) .
Implementation
The flexible shift vectors require additional data rates for encoding. Hence they are estimated
and transmitted only for those image areas that cannot be described with sufficient accuracy
using the source model R3D (see Section 10.4.1) [49]. These areas are the MFR3D objects.
The shift parameters are estimated for one MFR3D object at a time. For all control points of
an MC object that are projected onto one MFR3D object, the shift parameters are estimated
jointly because one control point affects the image synthesis of all the triangles to which it
belongs. For estimation, the image signal is approximated by a Taylor series expansion and
the parameters are estimated using a gradient method similar to the 3D motion estimation
© 2001 CRC Press LLC
algorithm described in Section 10.4.2. Since the robust 3D motion estimation as described
in Section 10.4.2 is not influenced by model failures, it is not necessary to estimate the 3D
motion parameters again or to estimate them jointly with the flexible shift parameters.
10.4
Image Analysis for 3D Object Models
The goal of image analysis is to gain a compact description of the current real image sk+1 ,
(m)
(m)
(m)
taking the transmitted parameter sets Ak , Mk , andSk
and subjective image quality
requirements into account. The image analysis consists of the following parts: image synthesis,
change detection, 3D motion estimation, detection of object silhouettes, shape adaptation, and
model failure detection. Whereas Section 10.4.1 gives a short overview of image analysis, the
sections that follow describe 3D motion estimation and detection of model failures in more
detail.
10.4.1
Overview
Figure 10.16 shows the structure of image analysis. The inputs to image analysis are the
(m)
(m)
current real image, sk+1 , and the model world, Wk , described by its parameters Ak , Mk ,
(m)
and Sk for each object m. First, a model image, sk , of the current model world is computed
by means of image synthesis.
In order to compute the change detection mask, Bk+1 , the change detection evaluates the
images sk and sk+1 on the hypothesis that moving real objects generate significant temporal
changes in the images [23, 64], that they have occluding contours [45], and that they are
opaque [45, 63]. This mask Bk+1 marks the projections of moving objects and the background
uncovered due to object motion as changed. Areas of moving shadows or illumination changes
are not marked as changed because illumination changes can be modeled by semitransparent
objects [62].
Since change detection accounts for the silhouettes of the model objects, the changed areas in
mask Bk+1 will be at least as large as these silhouettes. Figure 10.17 gives the change detection
mask Bk+1 for frames 2 and 18 of the test sequence Claire [10]. Due to little motion of Claire
at the beginning of the sequence, only parts of the projection of the real object are detected
during the first frames. The other parts of the person are still considered static background.
In order to compensate for real object motion and to separate the moving objects from
the uncovered background, 3D motion parameters are estimated for the model objects [1],
[32]–[35], [37, 38, 48]. The applied motion estimation algorithm requires motion, shape,
and color parameters of the model objects and the current real image sk+1 as input. Motion
parameters Ak+1 are estimated using a Taylor series expansion of the image signal, linearizing
the rotation matrix (10.18) assuming small rotation angles and maximum likelihood estimation
(see Section 10.4.2) [29].
The resulting motion parameters are used to detect the uncovered background, which is
included in mask Bk+1 . The basic idea for the detection of uncovered background is that the
projection of the moving object before and after motion has to lie completely in the changed
area [23]. Subtracting the uncovered background from mask Bk+1 gives the new silhouette
Ck+1 for all model objects (Figure 10.17).
The silhouette of each model object m is then compared and adapted to the real silhouette
(m)
Ck+1 [45]. Differences occur either when parts of the real object start moving for the first
time or when differences between the shape of the real and the model object become visible
© 2001 CRC Press LLC
FIGURE 10.16
Block diagram of image analysis: Ak , Mk , and Sk stored motion, shape, and color parameters; sk+1 real image to be analyzed; sk , s ∗ model images; Bk+1 change detection mask;
Ck+1 object silhouettes; Mk+1 shape parameters for MC and MF objects; Sk+1 color
parameters for MC and MF objects. Arrows indicate the information used in various
parts of image analysis. Semicircles indicate the output of processing steps.
during rotation. In order to compensate for the differences between the silhouettes of the model
objects and Ck+1 , the control points close to the silhouette boundary are shifted perpendicular
to the model object surface such that the model object gets the required silhouette. This gives
MC , where MC denotes model compliance.
the new shape parameters Mk+1
For the detection of model failures, a model image s ∗ is synthesized using the previous color
MC , respectively. The
parameters Sk and the current motion and shape parameters Ak+1 and Mk+1
∗
differences between the images s and sk+1 are evaluated for determining the areas of model
failure. The areas of model failure cannot be compensated for using the source model of
“moving rigid 3D objects.” Therefore, they are named rigid model failures (MFR3D ) and are
MF
represented by MFR3D objects. These MF objects are described by 2D shape parameters Mk+1
MF
and color parameters Sk+1 only.
In case the source model F3D is used, three more steps have to be added to the image analysis
(Figure 10.18). As explained in Section 10.3.4, flexible shift parameters are estimated only
for those parts of the model object that are projected onto areas of MFR3D . Following the
example of Figure 10.23, the estimation is limited to control points that are projected onto
the eye, mouth, and right ear area. After estimation of the shift parameters, a model image is
© 2001 CRC Press LLC
FIGURE 10.17
Object silhouettes of Claire: (a) frame 2; (b) frame 18. Gray and black areas are marked
changed by change detection. Detection of object silhouettes gives the object silhouette
(gray) and the uncovered background (black).
synthesized and the areas of MFF3D are estimated using the same algorithm for the detection
of model failures as for the R3D source model.
10.4.2
Motion Estimation for R3D
Image analysis of the OBASC must compensate for the motion of the objects in the scene
in order to provide a motion-compensated prediction of the current image sk+1 . The 3D real
objects are modeled by 3D model objects. Usually, the motion and the shape of the real objects
are unknown. In order to ensure a reliable and robust estimation, methods for robust estimation
are used. In Section 10.4.2, the basic motion estimation algorithm, which enables tracking
of real objects with model objects, is reviewed [48]. In Section 10.4.2, methods for robust
estimation are developed and compared. Since the shape of the real objects is not known and
these objects tend not to be completely rigid due to facial expressions and hair motion, we
developed a gradient-based motion estimator instead of a feature-based estimator.
Basic Motion Estimation
In order to derive the motion estimation algorithm, it is assumed that differences between
two consecutive images sk and sk+1 are due to object motion only. In order to estimate these
motion parameters, a gradient method is applied here.
During motion estimation, each object is represented by a set of observation points. Each
observation point O (j ) = (Q(j ) , g (j ) , I (j ) ) is located on the model object surface at position
(j )
(j ) (j )
Q(j ) and holds its luminance value I (j ) and its linear gradients gQ = (gx , gy )T . gQ
are the horizontal and vertical luminance gradients from the image which provided the color
parameters for the object. For simplicity, we use sk here. The gradients are computed by
convoluting the image signal with the Sobel operator


1 0 −1
1
(10.21)
E = · 2 0 −2
8
1 0 −1
giving the gradients
gx (x, y) = l(x, y)∗ E
gy (x, y) = l(x, y)∗ E T .
© 2001 CRC Press LLC
(10.22)
FIGURE 10.18
Block diagram of image analysis: Ak , Mk , and Sk stored motion, shape, and color paMC
rameters; sk+1 real image to be analyzed; s ∗ model image; Ak+1 = AMC
R3D + AF 3D global
MC
MF shape
(R, T ) and local (Sf ) motion parameters of MC objects; Mk+1 = MR3D + Mk+1
MF color parameters of MC and
parameters of MC and MF objects; Sk+1 = Sk + Sk+1
MF objects. Arrows indicate the information used in various parts of image analysis.
Semicircles indicate the output of processing steps.
The measure for selecting observation points is a high spatial gradient. This adds robustness
against noise to the estimation algorithm (see Section 10.4.2). Figure 10.19 shows the location
of all observation points belonging to the model object Claire. If parts of the object texture
are exchanged due to MF objects, the observation points for the corresponding surface of the
object are updated. The observation points are also used for the estimation of flexible shift
parameters.
FIGURE 10.19
The position of all observation points of model object Claire.
© 2001 CRC Press LLC
Since some triangles of a model object can be deformed due to flexible shifts or due to its
control points belonging to different components of the model object, we define the position
of an observation point Q(j ) relative to the position of the control points P (0) , P (1) , and P (2)
of its triangle using barycentric coordinates c0 , c1 , and c2 of the coordinate system defined by
P (0) , P (1) , and P (2) :
Q(j ) = c0 P (0) + c1 P (1) + c2 P (2) .
(10.23)
It is assumed that objects are rigid and have diffuse reflecting surfaces. Furthermore, diffuse illumination of the scene is assumed.1 Hence, color parameters are constant. With an
(j )
(j ) (j )
observation point Ok = (Qk , gQ , I (j ) ) at time instant k projected onto the image plane
(j )
(j )
(j )
at qk and the same observation point after motion Ok+1 = (Qk+1 , g (j ) , I (j ) ) projected onto
(j )
(j )
qk+1 , the luminance difference between image k and image k + 1 at position qk is
(j )
(j )
(j )
9I qk
= sk+1 qk
− s k qk
(j )
(j )
= sk+1 qk
− sk+1 qk+1 ,
(10.24)
assuming that the luminance difference is only due to object motion with sk+1 (pk+1 ) =
sk (pk ) = I . According to [4], we can approximate the image signal using a Taylor expansion
of second order without explicitly computing the second derivative. We compute the secondorder gradient ḡ by averaging the linear gradients of the observation point and the image
signal
ḡ =
1
gQ + gk+1 (qk ) .
2
(10.25)
Approximating the image signal with a Taylor series and stopping after the linear term gives
sk+1 (qk+1 ) ≈ sk+1 (qk ) + ḡ · (qk+1 − qk ) .
(10.26)
Now, we can express the luminance difference according to (10.24) as
(j ) (j ) T
(j )
(j )
9I (qk ) = −ḡ · (qk+1 − qk ) = gx , gy
.
· qk+1 − qk
(10.27)
Substituting image coordinates by model world coordinates with equation (10.2) yields
(j )
(j )
(j )
(j )
Qy,k
Qx,k
(j ) Qx,k+1
(j ) Qy,k+1
(j )
9I = F · gx
(10.28)
− (j ) + F · gy
− (j )
(j )
(j )
Qz,k+1
Qz,k
(j )
Qz,k+1
Qz,k
The position Qk of the observation point O (j ) is known. By relating Qk to Qk+1 by means
of the motion equation (10.17), a nonlinear equation with the known parameters 9I , g, and F
and the six unknown motion parameters results. This equation is linearized by linearizing the
rotation matrix RC (10.18), assuming small rotation angles


1 −Rz Ry
(10.29)
RC =  Rz 1 −Rx ,
1
−Ry Rx
1 See [7] and [62] on how to consider illumination effects.
© 2001 CRC Press LLC
giving
Qk+1 = RC · (Qk − C) + C + T
(10.30)
Substituting (10.30) into (10.28), the linearized equation for one observation point is
9I = F · gx /Qz · Tx
+ F · gy /Qz · Ty
− Qx gx + Qy gy F /Q2z + 9I /Qz · Tz
− Qx gx Qy − Cy + Qy gy Qy − Cy + Qz gy (Qz − Cz ) F /Q2z
+ 9I /Qz Qy − Cy · Rx
+ Qy gy (Qx − Cx ) + Qx gx (Qx − Cx ) + Qz gx (Qz − Cz ) F /Q2z
+ 9I /Qz (Qx − Cx ) · Ry
− gx Qy − Cy − gy (Qx − Cx ) F /Qz · Rz
(10.31)
with the unknown motion parameters T = (Tx , Ty , Tz )T and RC = (Rx , Ry , Rz )T and the
observation point Ok = (Qk , g, I ) at position Qk = (Qx , Qy , Qz )T . In order to get reliable
estimates for the six motion parameters, equation (10.31) has to be established for many
observation points, resulting in an overdetermined system of linear equations
A·x−b =r ,
(10.32)
with the residual r = (r1 , . . . , rJ )T , x = (TX , TY , TZ , RX , RY , RZ )T , b = (9I (q (1) ), . . . ,
9I (q (J ) )T , and A = (a1 , . . . , aJ )T , and aj according to (10.31). The equations are solved
by minimization of r:
→ min ,
|r|2 = r T · r −
x
(10.33)
which corresponds to a minimization of the prediction error of the observation points
9I (j )
2
→ min .
(10.34)
0(j )
The motion parameters are given by
−1
· AT · b
x̂ = AT · A
(10.35)
In order to avoid the inversion of large matrices, we do not compute A but immediately compute
the 6 × 6 matrix AT · A.
Due to the linearizations in (10.26) and (10.29), motion parameters have to be estimated
iteratively for each model object. After every iteration, the model object is moved according
to (10.17) using the estimated motion parameters x̂. Then, a new set of motion equations is
established, giving new motion parameter updates. Since the motion parameter updates approach zero during the iterations, the introduced linearizations do not harm motion estimation.
The iteration process terminates if the decrease of the residual error |r|2 becomes negligible.
© 2001 CRC Press LLC
Robust Motion Estimation
Equation (10.32) is solved such that the variance of the residual errors 9I is minimized.
However, this approach is sensitive to measurement errors [41]. Measurement errors occur
because (10.32) is based on several model assumptions and approximations that tend to be
valid for the majority of observation points but not all. Observation points that violate these
assumptions are named outliers [59]. When using (10.34) for solving (10.32), outliers have a
significant influence on the solution. Therefore, we have to take measures that limit the influence of these outliers on the estimation process [51]. Sometimes, the following assumptions
are not valid:
1. Rigid real object
2. Quadratic image signal model
3. Small deviations of model object shape from real object shape
Each of these cases is discussed below.
If parts of the real object are nonrigid (i.e., the object is flexible), we have image areas that
cannot be described by the current motion and shape parameters Ak and Mk , respectively, and
the already transmitted color parameters Sk . These image areas can be detected due to their
potentially high prediction error 9I . Observation points in these areas can be classified as
outliers. For iteration i of (10.34), we will consider only observation points for which the
following holds true:
(j )
9Ii
< σ9I · TST
(10.36)
with
σ9I
J 1 (j ) 2
=
.
9Ii
J
(10.37)
j =0
The threshold TST is used to remove the outliers from consideration.
According to (10.26), motion estimation is based on the gradient method, which allows for
(j )
(j )
estimating only small local displacements (qk+1 −qi ) in one iteration step [43]. Given an im(j )
age gradient gi and a maximum allowable displacement Vmax = |vmax | = |(vx,max , vy,max )T |,
(j )
(j )
we can compute a maximum allowable frame difference 9Ilimit (qi ) at an image location qi
(j )
(j ) (10.38)
= vmax · gi .
9Ilimit qi
(j )
(j )
Observation points with |9I (qi )| > |9Ilimit (qi )| are excluded from consideration for the
ith iteration step. We assume that they do not conform to the image signal model assumption.
Considering image noise, we can derive an additional criterion for selecting observation
points. Assuming white additive camera noise n, we measure the noise of the image difference
signal as
2
σ9I
= 2 · σn2 .
(10.39)
(j )
(j )
According to (10.27), we represent the local displacement (qk+1 − qi ) as a function of the
noiseless luminance signal. Therefore, the luminance difference 9I (q (j ) ) and the gradient
© 2001 CRC Press LLC
g (j ) should have large absolute values in order to limit the influence of camera noise. Hence,
we select as observation points only points with a gradient larger than a threshold TG :
(j ) (10.40)
g > TG .
Relatively large gradients allow also for a precise estimation of the motion parameters. Summarizing these observations, we conclude that we should select observation points with large
absolute image gradients according to (10.40). Equations (10.36) and (10.38) are the selection
criteria for the observation points we will use for any given iteration step.
Instead of using the binary selection criteria for observation points according to (10.36)
and (10.38), we can use continuous cost functions to control the influence of an observation
point on the parameter estimation. We use the residuum r according to (10.32) as measure for
the influence of an observation point [57, 70]. Assuming that the probability density function
f (r) of the residuals rj according to (10.32) is Gaussian, (10.34) is a maximum-likelihood
estimator or M estimator [29].
Now, we will investigate how different assumptions about f (r) influence the M estimator.
Let us assume that one f (r) is valid for all observation points. A critical point for selecting an
appropriate probability density function is the treatment of outliers. Ideally, we want outliers
to have no influence on the estimated motion parameters.
The M estimator minimizes the residuum rj according to (10.32) using a cost function <(rj ):
J
→ min .
< rj −
x
j =1
(10.41)
With
δ(<(rj ))
= rj =
δx
(10.42)
the solution of (10.41) becomes
J
= rj = 0 .
(10.43)
j =1
Equation (10.43) becomes an M estimator for the probability density function f (r) if we set
<(r) = − log f (r) .
(10.44)
This M estimator is able to compute the correct solution for six motion parameters with up
to 14% of the observation points being outliers [70]. Some authors report success with up to
50% outliers [38].
Let us assume that ε% of our measurement data represents outliers that do not depend
on the observable motion. Now, we can choose a separate probability density function for
the residuals of the outliers. Let us assume that the residuals of the inliers (non-outliers)
are Gaussian distributed and that the outliers have an arbitrary Laplace distribution. We can
approximate the probability density function of the residuals with [29, 70]
f (r) =
© 2001 CRC Press LLC

2
1−ε − r2
√
e
2π
2
√
1−ε −(a|r|− a2 )
e
2π
if |r| < a
otherwise .
(10.45)
The cost function is
<(r) =
r2
2
for |r| < a
a · |r| −
a2
2
otherwise
(10.46)
with the associated M estimator
=(rj ) = max −a, min(rj , a) ,
(10.47)
where a is the threshold for detecting outliers. In order to adapt the outlier detection to the
image difference signal 9I , we select a proportional to σ9I (10.37).
Often, the probability density function f (r) is unknown. Therefore, heuristic solutions for
=(r) such as the cost function (1 − r 2 /b2 )2 according to Turkey were found [38, 70]:

2

r2

if |rj | < b
rj · 1 − bj2
(10.48)
=(rj ) =

0
otherwise .
The cost (1 − r 2 /b2 )2 increases to 1 when |r| decreases. Observation points with |r| ≥ b are
excluded from the current iteration; b is the threshold for detecting outliers. In order to adapt
the outlier detection to the image difference signal 9I , we select b proportional to σ9I (10.37).
The shape difference between the model object and the real object can be modeled by means
of a spatial uncertainty of an observation point along the line of sight. This can be considered
using a Kalman filter during motion estimation [39].
Experimental Results
Ideally, each iteration i of our motion estimation would consist of four steps. First, we
solve (10.35) with all observation points. Then we select the observation points that fulfill
the criteria for robust motion estimation. In the third step, we estimate the motion parameters
using (10.35) again with these selected observation points. Finally, we use these parameters
for motion compensation according to (10.17). In order to avoid solving (10.35) twice, we
use the observation points that fulfilled the criteria for robust estimation in iteration i − 1.
Hence, each iteration i consists of three steps: (1) solve (10.35) using the observation points
selected in iteration i − 1, (2) motion compensate the model object with the estimated motion
parameters, and (3) select the new observation points to be used in iteration i + 1.
In order to evaluate the motion estimation algorithm and the robust estimation methods, we
test the algorithm on synthetic image pairs that were generated with known parameters [51].
First, we create a model object using the test sequence Claire (Figure 10.10f). We create the
first test image by projecting the model image into the image plane. Before synthesizing the
second test image, we move the model object and change its facial expression, simulating
motion and model failures, respectively. Finally, we add white Gaussian noise with variance
σn2 to the test images. In the following tests, we use the model object that corresponds to the
first test image. We estimate the motion parameters that resulted in the second test image.
2
As a first quality measure, we compute the average prediction error variance σdiff
inside
the model object silhouette between the motion-compensated model object and the second
test image. As a second quality measure, we compute the error of the model object position,
measured as the average position error dmot of all vertices between the estimated position P̂ (n)
and the correct position P (n) :
dmot =
N
1 (n)
P − P̂ (n) .
N
n=1
© 2001 CRC Press LLC
(10.49)
With (10.49), we can capture all motion estimation errors. The smaller dmot gets, the better the
estimator works in estimating the true motion between two images. Estimation of true motion
is required in order to be able to track an object in a scene. We do not compare directly the
estimated motion parameters because they consist of translation and rotation parameters that
are not independent of each other, thus making it difficult to compare two motion parameter
sets.
2
Figure 10.20 shows the prediction error σdiff
in relation to the image noise for three thresholds. Using the threshold TST = ∞ according to (10.36) allows us to consider every observation point. We use only observation points with a small luminance difference when setting
TST = 1. Alternatively, we can use (10.38). Hence, we select only observation points that
FIGURE 10.20
Prediction error variance as a function of image noise. The different curves were measured using different criteria for selecting observation points: Vmax according to (10.38),
TST according to (10.36).
indicate a small local motion. As can be seen, removing the outliers from the measurement
2
data reduces the prediction error. Due to the model failure, the prediction error variance σdiff
does not decrease to 0. As can be seen in Figure 10.20, the prediction error increases with the
image noise.
If we compare the average position error dmot , we see it decreases significantly when we use
the criteria according to (10.36) and (10.38) (Figure 10.21). dmot is an important criterion for
estimating the true motion of an object. According to the experiments shown in Figure 10.21,
using 9I (q (j ) ) with (10.36) and the threshold TST = 1 as the control criterion results in the
smallest position error dmot .
Using an M estimator further decreases the position error dmot . Using the probability density
function (10.45) with
a = σ9I · TST ,
(10.50)
2 of all observation points during iteration i, gives the
and the prediction error variance σ9I
best results for TST = 0.2. The most precise estimation was measured using the cost function
according to Turkey, (10.48), with
b = σ9I · TST ,
(10.51)
2 of all observation points during iteration i, and T
the prediction error variance σ9I
ST = 1.
The position error dmot is not influenced much by image noise. This is because the noise is
Gaussian and the model object covers a relatively large area of the image (30% for Claire).
© 2001 CRC Press LLC
FIGURE 10.21
Average deviation of control point position dmot according to (10.49) as a function of image
noise σn . The curves were created using different criteria: Vmax according to (10.19), TST
according to (10.17), a according to (10.45) and (10.50), b according to (10.48) and (10.51),
and σ9I according to (10.37).
Segmentation into Components
The initial segmentation of moving objects into components was developed by Busch [6].
After motion compensating the entire rigid model object, the segmentation algorithm clusters
neighboring triangles with similar 2D motion parameters. If these clusters allow for an improved motion compensation, a model object is subdivided into flexibly connected components.
This decision is based on the evaluation of two consecutive images only.
In [39], Martínez proposes a different approach to segmenting an object into components.
The 3D motion is measured for each triangle. In order to achieve a reliable estimate for
the triangles, a Kalman filter modeling model object shape errors and the camera noise is
adopted. Triangles with similar 3D motion are stored in a cluster memory. As soon as a
cluster in the memory is stable for several images, this cluster is used to define a component
of the object. This algorithm yields segmentation of persons into head, shoulders, and arms.
Further improvements in motion estimation are achieved by enforcing spatial constraints due
to spherical joints between components.
After segmenting an object into rigid components, 3D motion is estimated iteratively for
each individual component as well as for the entire object.
10.4.3
MF Objects
MF objects are not related to real objects. They are just used to cover the deficiencies of the
object and illumination models as well as estimation errors. Therefore, MF objects are always
detected at the end of image analysis. This procedure can be seen as the final verification step
of image analysis. According to the scene model, MF objects exist only in the model image
plane.
Detection of MF Objects
∗ ,
MF objects are estimated by comparing the current real image with the model image sk+1
which is synthesized using previously transmitted color parameters Sk and the current motion
MC
and shape parameters AMC
k+1 and Mk+1 , respectively (Figures 10.16 and 10.18). As a result of
© 2001 CRC Press LLC
this comparison, we will segment those image areas that cannot be described with sufficient
subjective quality using MC objects as defined by the source model. Each MF object is
described by its 2D silhouette and the color parameters inside the silhouette.
The detection of MF objects implies a receiver model. The following list gives some
qualitative properties of the receiver model. It is assumed that the subjective image quality is
not disturbed by:
1. Camera noise
2. Small position errors of the moving objects
3. Small shape errors of the moving objects
4. Small areas with erroneous color parameters inside a moving object
The errors listed as items (2) to (4) are referred to as geometrical distortions. Properties of
the human visual system such as the modulation transfer function and spatiotemporal masking
are not considered.
The following algorithm implicitly incorporates the above-mentioned assumptions. The
∗
difference image between the prediction image sk+1
and the current image sk+1 is evaluated
by binarizing it using an adaptive threshold Te such that the error variance of the areas that
are not declared as synthesis errors is below a given allowed noise level Ne . Ne = 6/255 is a
commonly used threshold. The resulting mask is called the synthesis error mask. Figure 10.22a
and b show a scaled difference image and the resulting synthesis error mask, respectively.
∗
The synthesis error mask marks those pels of image sk+1
which differ significantly from the
corresponding pels of sk+1 . Since the areas of synthesis errors are frequently larger than 4% of
the image area, it is not possible to transmit color parameters for these areas with a sufficiently
high image quality (i.e., visible quantization errors would occur). However, from a subjective
point of view it is not necessary to transmit color parameters for all areas of synthesis errors.
∗
Due to the object-based image description, the prediction image sk+1
is subjectively pleasant.
There are no block artifacts, and object boundaries are synthesized properly.
There are two major reasons for synthesis errors. First of all, synthesis errors are due
to position and shape differences between a moving real object and its corresponding model
object. These errors are caused by motion and shape estimation errors. They displace contours
in the image signal and will produce line structures in the synthesis error mask. Due to
the feedback of the estimated and coded motion and shape parameters into image analysis
(Figure 10.1), these estimation errors tend to be small and unbiased and they do not accumulate.
Therefore, it is reasonable to assume that these errors do not disturb subjective image quality.
They are classified as geometrical distortions. As a simple detector of geometrical distortions,
a median filter of size 5 × 5 pel is applied to the mask of synthesis errors (Figure 10.22c).
Second, events in the real world that cannot be modeled by the source model will contribute to
synthesis errors. Using the source model R3D, it is not possible to model changing human facial
expressions or specular highlights. Facial expressions in particular are subjectively important.
In order to be of subjective importance, it is assumed that an erroneous image region has to
be larger than 0.5% of the image area (Figure 10.22c). Model failures are those image areas
∗
where the model image sk+1
is subjectively wrong (Figure 10.22d). Each area of model failure
is modeled by an MFR3D object, defined by color and 2D shape parameters (Figure 10.23a
and b). Applying the F3D object model to the MFR3D objects (Figure 10.18) can compensate
for some of the synthesis errors such that the MFF3D objects tend to be smaller (Figure 10.23).
© 2001 CRC Press LLC
FIGURE 10.22
Detection of model failures: (a) scaled difference image between real image sk+1 and
∗ ; (b) synthesis error mask; (c) gemodel image after motion and shape compensation sk+1
ometric distortions and perceptually irrelevant regions; (d) mask MFR3D with model
failures of the source model rigid 3D object.
10.5
Optimization of Parameter Coding for R3D and F3D
The task of parameter coding is the efficient coding of the parameter sets motion, shape, and
color provided by image analysis. Parameter coding uses a coder mode control to select the
appropriate parameter sets to be transmitted for each object class. The priority of the parameter
sets is arranged by a priority control.
10.5.1
Motion Parameter Coding
The unit of the estimated object translation T = (tx , ty , tz )T is pel. The unit of the estimated
(C)
(C)
(C)
object rotation RC = (Rx , Ry , Rz )T is degree. These motion parameters are PCM coded
by quantizing each component with 8 bit within an interval of ±10 pel and degrees, respectively.
This ensures a subjectively lossless coding of motion parameters.
© 2001 CRC Press LLC
FIGURE 10.23
Detection of model failures: MFR3D objects with (a) shape and (b) color parameters. After
the source model F3D is applied to the MFR3D objects, the MFF3D objects are detected
with the (c) shape and (d) texture parameters. (a) is an enlargement of Figure 10.22d.
10.5.2
2D Shape Parameter Coding
Since the model object shape is computed and updated from its silhouette, shape parameters
are essentially 2D. The principles for coding the shape parameters of MF and MC objects
are identical. Shape parameters are coded using a polygon/spline approximation developed
by Hötter [24]. A measure dmax describes the maximum distance between the original and
approximated shape. First, an initial polygon approximation of the shape is generated using
∗ is not satisfied, the approximation
four points (Figure 10.24a). Where the quality measure dmax
FIGURE 10.24
Polygon approximation: (a) initial polygon; (b) insertion of a new polygon point.
is iteratively refined through insertion of additional polygon points until the measure fulfills
∗
dmax ≤ dmax
(Figure 10.24b). In case the source model F2D is used, we check for each
line of the polygon, whether an approximation of the corresponding contour piece by a spline
© 2001 CRC Press LLC
∗ . If so, the spline approximation is used, giving a natural
approximation also satisfies dmax
shape approximation for curved shapes (Figure 10.25). In order to avoid visible distortions at
FIGURE 10.25
∗
Combination of polygon and spline approximation for a quality measure dmax
= 15.
∗
= 1.4 pel. Experimental results showed
object boundaries, MC objects are coded with dmax
∗
that MF objects should be coded with dmax about 2.1 pel in order to minimize the overall bit
rate required for coding MF objects [46].
The coordinates of the polygon points are coded relative to their perspective predecessor.
In the case of the source model F2D, the curve type line/spline is coded for each line of the
polygon.
The data rate for coding shape parameters of MC objects is cut to half by using the motioncompensated coded silhouette of the last image as a prediction of the current silhouette. Starting
with this approximation, only shape update parameters have to be transmitted.
10.5.3
Coding of Component Separation
The split of an object into components is defined on a triangle basis. Whenever a new
component is defined by the encoder, its shape is encoded losslessly with a flag for each visible
triangle. The encoder and decoder then define the shape of the component by connecting the
visible triangles with the ones that they occlude at the back of the object.
10.5.4
Flexible Shape Parameter Coding
A list of all currently visible control points having flexible shape parameters Sfn = 0 is
transmitted using a run-length code. In a second step, the components of the corresponding
vectors Sfn are linearly quantized using 16 representation levels within an interval of ±5 pel.
The quantized vector components are entropy coded.
10.5.5
Color Parameters
Conventional DCT is not suitable for the coding of color parameters of arbitrarily shaped
regions. New algorithms have been developed for this application [17, 61]. Here the special
type of DCT for arbitrarily shaped regions developed by Gilge [17] is improved by applying a
segmentation of the color parameters into homogeneous regions prior to transform coding [47].
The segmentation is based on the minimum spanning tree [42] using the signal variance as criterion. The boundaries of the regions are coded using a chain code [16]. The DCT coefficients
are quantized with a linear quantizer of signal-dependent step size. The advantage of this
scheme using segmentation prior to transform coding is that errors due to coarse quantization
© 2001 CRC Press LLC
are mainly concentrated at the boundaries of the segmented regions, where they are less visible
due to masking of the human visual system in areas of high local activity.
10.5.6
Control of Parameter Coding
Due to limited data rate, a transmission of all parameter sets cannot be guaranteed. Coder
control is used to overcome this difficulty. It consists of coder mode control and priority control.
Coder mode control selects the relevant parameter sets and coder adjustments for each object,
and priority control arranges these parameter sets for transmission (see Table 10.1).
Depending on the model object class MF or MC, the coder mode control selects two param(m)
eter sets for transmission. For MC objects, only motion Ak+1 and shape update parameters
(m)
Mk+1 are coded. Coding of color parameters is not necessary because the existing color pa(m)
rameters Sk of the model objects are sufficient to synthesize the image properly. 2D shape
parameters, defining the location of the model failures in the image plane, and color parameters
are coded for MF objects.
Priority control guarantees that the motion parameters of all MC objects are transmitted
first. In a second step, the shape parameters of the MC objects are transmitted. Finally, the
shape and color parameters of the MF objects are transmitted until the available data rate is
exhausted.
10.6
Experimental Results
The object-based analysis–synthesis coder based on the source models R3D and F3D is
applied to the test sequences Claire [10] and Miss America [5], with a spatial resolution
corresponding to CIF and a frame rate of 10 Hz. The results are compared to those of an
H.261 coder [8, 9] and OBASC based on the source model F2D as presented by Hötter [26]–
[28]. As far as detection of model failures and coding of shape parameters are concerned, the
same algorithms and coder adjustments are applied. Parameter coding aims at a data rate of
approximately 64 kbit/s. However, the bit rate of the coder is not controlled and no buffer is
implemented. In the experiments, the allowed noise level Ne for detection of model failures is
set to 6/255. Color parameters of model failures are coded according to Section 10.5.5 with a
peak signal-to-noise ratio (PSNR) of 36 dB. In all experiments the coders are initialized with
the first original image of the sequence (i.e., the frame memory is initialized with the first
original image for the block-based coder H.261). For the two object-based analysis–synthesis
coders, the model object Background in the memory for parameters is initialized with the first
original image.
For head and shoulder scenes the 3D model object is usually divided into two to three
components. Applying the estimated motion parameter sets to the model object gives a natural
impression of object motion. This indicates that the estimated motion parameters are close to
the real motion parameters and that the distance transform applied to the object silhouette for
generating the 3D model object shape is suitable for the analysis of head and shoulder scenes.
The area of rigid model failures MFR3D is on average less than 4% of the image area
(Figure 10.26). Generalizing this, for head and shoulder scenes the rigid model failures can
perhaps be expected to be less than 15% of the moving area. The exact figures of model failure
area as related to moving area are 12% for Claire and 7% for Miss America. The test sequence
Claire seems to be more demanding, due to the fast rotation of the subject’s head, whereas
Miss America’s motion is almost 2D.
© 2001 CRC Press LLC
FIGURE 10.26
Area of rigid model failures MFR3D in pel for the test sequence Claire. The total area is
101,376 pel. The average area of model failures is 3.5% of the image area.
Table 10.2 compares the average bit rate for the different parameter sets motion, shape,
and color and the source models F2D, R3D, and F3D. Coding of the head and shoulder test
sequences Claire and Miss America and the MPEG-4 test sequence Akiyo will not exceed
the data rates given in Table 10.2. The source models F2D and R3D need approximately the
same data rate. Due to the displacement vector field, OBASC based on the source model F2D
requires a relatively high amount of motion information. Shape parameters include the shape
of MC and MF objects. Shape parameters of MC objects require similar data rates for both
source models. However, the source model F2D causes only a few large MF objects, whereas
the source model R3D causes smaller but more MF objects. This larger number of MFR3D
objects is due to the applied source model assuming rigid shapes. Shape differences between
real and model objects as well as small flexible motion on the surface of real objects cannot
be compensated for. These effects cause small local position errors of the model objects. If
texture with high local activity is displaced for more than 0.5 pel, model failures are detected
due to the simple filter for the detection of geometric distortions. Since these small position
errors can be compensated for when using the source model F2D, the overall data rate for
shape parameters is 750 bit higher for the source model R3D. Since this is due to the more
local motion model of F2D, we have to also look at comparing the sum of motion and shape
data rates. Here, the source model R3D requires 7.5% fewer bits than the source model F2D.
Table 10.2 Average Bit Rate of Parameter Sets for Different Source Models
Source
Model
F2D
R3D
F3D
Motion:
(bit/frame)
Shape of MC
Objects: RM,MC
(bit/frame)
Shape of MF
Objects: RM,MF
(bit/frame)
Area of MF Objects and
Uncovered Background
(% of image area)
1100
200
200
900
500
950
900
1150
1000
4%
4%
3%
RA
Note: The coders use the same algorithm for detection of model failures.
When comparing F3D with R3D, we notice that the average area of MF objects decreases
from 4 to 3%. For CIF sequences, this results in bit savings of at least 1000 bit/frame for
the texture parameters of MF objects. At the same time, the data rate for MF object shape
decreases from 1150 to 1000 bit/frame. This is mainly due to the smaller number of MF objects
in the case of F3D. Since the use of F3D requires the transmission of the flexible shift vectors
as an additional dataset, the rate for MC object shape parameters increases by 450 bit/frame.
© 2001 CRC Press LLC
This indicates that by spending an additional 450 bit/frame on the shape parameters of an MC
object, we save 150 bit/frame on the MF object shape and reduce the area of MF objects by 1%.
Therefore, the bit savings is significantly higher than the costs for this additional parameter
set of F3D.
Figure 10.27 shows part of the 33rd decoded frame of the test sequence Claire using the
source models F2D, R3D, F3D, and H.261. Subjectively, there is no difference between the
source models F2D, R3D, and F3D. However, the source model F3D requires only 56 kbit/s
instead of 64 kbit/s. When compared to decoded images of an H.261 coder [9], picture
quality is improved twofold (Figure 10.28). At the boundaries of moving objects, no block or
mosquito artifacts are visible, due to the introduction of shape parameters. Image quality in the
face is improved, because coding of color parameters is limited to model failures, which are
mainly located in the face. Since the average area of model failures (i.e., the area where color
parameters have to be coded) covers 4% of the image area, color parameters can be coded at
a data rate higher than 1.0 bit/pel. This compares to 0.1 to 0.4 bit/pel available for coding of
color parameters with an H.261/RM8 encoder.
FIGURE 10.27
Part of the 33rd decoded frame of test sequence Claire at a data rate of 64 kbit/s: (a) blockbased hybrid coder H.261 (RM8), (b) F2D, (c) R3D, (d) F3D at 56 kbit/s.
10.7
Conclusions
In this chapter the concept and implementation of an object-based analysis–synthesis coder
based on the source models of moving rigid 3D objects (R3D) and moving flexible 3D objects
(F3D) aiming at a data rate of 64 kbit/s have been presented. An OBASC consists of five parts:
image analysis, parameter coding and decoding, image synthesis, and memory for parameters.
Each object is defined by its uniform 3D motion and is described by motion, shape, and color
parameters. Moving objects are modeled by 3D model objects.
© 2001 CRC Press LLC
FIGURE 10.28
Part of the decoded frame 8 of the test sequence Miss America, to demonstrate blocking
artifacts at 64 kbit/s: (a) block-based hybrid coder H.261 (RM8), (b) F2D, (c) R3D,
(d) F3D at 56 kbit/s.
The goal of image analysis is to arrive at a compact parametric description of the current
image of a sequence, taking already transmitted parameter sets into account. Image analysis
includes shape and motion estimation as well as detection of model failures. Moving objects
are segmented using temporal change detection and motion parameters. The algorithm for
estimating these motion parameters is based on previous work on gradient-based motion estimation. A new set of equations relating the difference signal between two images to 3D
motion parameters has been established. Using robust motion estimation algorithms enables
us to track rigid as well as flexible objects such as head and shoulders throughout a video
scene, thus enabling an efficient object-based video coder. The 3D shape of a model object is
computed by applying a distance transformation, giving object depth, to the object silhouette.
Since the estimated motion parameters applied to these 3D model objects give a natural impression of motion, this distance transform is very suitable for analysis of head and shoulder
scenes.
Those areas of an image that cannot be modeled by the applied source model are referred
to as model failures and are modeled by MF objects. They are described by color and 2D
shape parameters only. Model failures are detected taking subjective criteria into account. It
is assumed that geometrical distortions such as small position and shape errors of the moving
objects do not disturb subjective image quality. Due to these subjective criteria, the average
area of model failures is less than 4% of the image area for typical videophone test sequences.
Flexible shift parameters of the source model F3D are estimated only for those parts of an
object that cannot be described using the source model R3D. This limits the additional data
© 2001 CRC Press LLC
rate required for the flexible shift parameters and increases the overall efficiency of this source
model F3D over R3D.
With respect to coding, shape and color parameters are coded for MF objects, whereas
motion and shape update parameters have to be coded for MC objects. Motion parameters are
PCM coded and shape parameters are coded using a polygon approximation. Prior to coding,
color parameters are segmented into homogeneous regions. Then a DCT for arbitrarily shaped
regions is applied.
The presented coder has been compared to OBASC based on the source model of moving
flexible 2D objects (F2D). With regard to typical head and shoulder videophone test sequences,
it is shown that the picture quality at the average bit rate of 64 kbit/s is the same regardless
of whether the source model R3D or F2D is applied. When using F3D, the data rate shrinks
from 64 to 56 kbit/s for the same picture quality. When compared to images coded according
to H.261, there are no mosquito or block artifacts because the average area for which color
parameters are transmitted is 10% of the image area for H.261 and 4% for OBASC. Therefore, OBASC allows coding of color parameters for MF objects with a data rate higher than
1.0 bit/pel. At the same time, MC objects are displayed without subjectively annoying artifacts.
This chapter has demonstrated the feasibility of OBASC using 3D models. As shown in [31],
this coder can be used as a basis for knowledge-based and semantic coders. However, image
analysis as presented here is not yet able to describe with sufficient accuracy scenes with a lot of
motion, such as gesticulating hands. In [39], the 3D motion estimation algorithm is extended
to enable the segmentation of flexibly connected rigid objects and components. In [40], an
OBASC with an image analysis is presented that enables camera motion compensation. The
concept of MF detection is viable for any coder that uses smooth motion vector fields and
knows the approximate location of object boundaries [54, 66]. It enables reduction of the data
rate of a block-based coder without decreasing the subjective image quality.
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© 2001 CRC Press LLC
Chapter 11
Rate-Distortion Techniques in Image and Video
Coding
Aggelos K. Katsaggelos and Gerry Melnikov
11.1
The Multimedia Transmission Problem
One of the central issues in multimedia communications is the efficient transmission of
information from source to destination. This broad definition includes a large number of target
applications. They may differ in the type (i.e., text, voice, image, video, etc.) and form (analog
or digital) of the data being transmitted. Applications also differ by whether lossless or lossy
data transmission is employed and by the kind of channel the transmitted data passes through
on its way from an encoder to a decoder or a storage device. Additionally, there are wide
variations in what represents acceptable loss of data. For example, whereas certain loss of
information is acceptable in compressing video signals, it is not acceptable in compressing
text signals.
In general, the goal of a communications system is the transfer of data in a way that is
resource efficient, minimally prone to errors, and without significant delay. In a particular
application we may know the target operating environment (bit rate, channel characteristics,
acceptable delay, etc.). The challenge is to arrive at the data representation satisfying these
constraints. For example, if the application at hand is transmission of digital video over a noisy
wireless channel, the task is to allocate the available bits within and across frames optimally
based on the partially known source statistics, the assumed channel model, and the desired
signal fidelity at the receiver.
In problems where some information loss is inevitable, whether caused by quantizing a
deterministic signal or through channel errors, rate distortion theory provides some theoretical
foundations for deriving performance bounds. The fundamental issue in rate distortion theory
is finding the bound for the fidelity of representation of a source of known statistical properties
under a given description length constraint. The dual problem is that in which a given fidelity
requirement is prescribed and the source description length is to be minimized.
Although there is no argument that the count of bits should be the metric for rate or the
description length, the choice of the corresponding metric for distortion is not straightforward
and may depend on the particular application. The set of all possible distortion metrics can
be broadly categorized into unquantifiable perceptual metrics and metrics for which a closedform mathematical representation exists. The latter class, particularly the mean squared error
(MSE) metric, and, to a lesser extent, the metric based on the maximum operator [34], has been
primarily used in the image and video coding community partly due to the ease of computation
and partly for historical reasons.
© 2001 CRC Press LLC
The well-known Shannon’s source and channel coding theorems [35] paved the way for separating, without loss of optimality, the processes of removing redundancy from data (source
coding) and the transmission of the resulting bitstream over a noisy channel (channel coding).
Shannon also established a lower (entropy) bound on the performance of lossless source compression algorithms and subsequent error-free transmission over a channel of limited capacity.
Hence, most practical communication systems have the structure shown in Figure 11.1.
Original
image
Source
Encoder
Channel
Encoder
Channel
Reconstructed
image
Source
Decoder
Channel
Decoder
FIGURE 11.1
High-level view of a video communications system.
Many applications demand data compression beyond entropy (i.e., with some loss of information). Wireless transmission of video is one example when the original data stream must
be compressed by factors of hundreds or thousands, due to limited channel capacity. Clearly,
with lossy compression, there is a trade-off between the transmission of fewer bits through
a channel and the quality of the reconstructed signal. The central issue in the rate-distortion
theory is how much redundancy can be removed from a given data source while satisfying a
distortion constraint. Conversely, the constraint can be imposed on the rate, in which case a
lower bound on distortion is sought. The theory is based on knowledge of the source statistics
and not on any specific source coding scheme. Hence it provides a lower bound on the rate,
which clearly may not be achieved by a particular encoder implementation. In addition to
not being constructive, this bound assumes perfect knowledge of the source statistics which,
in most cases, is not available. Source model complexity is yet another dimension in ratedistortion theory. The trade-off here is between a very simple source model (e.g., an i.i.d.
Gaussian model), which is relatively easy to treat analytically, on one hand, and a model that
is better adapted to a given source but is more complex. Clearly, any performance bounds
obtained with the help of the rate-distortion theory are only as good as the source models and
may be of little use if they are inaccurate.
Even though, traditionally, rate-distortion theory was applied only to situations where distortion was caused deterministically through quantization at the encoder, it is applied equally
well to the case when the distortion is introduced stochastically in a noisy channel. In this
case, however, the tightness of the bound also depends on the accuracy of the channel model.
The applicability of Shannon’s separation principle between source and channel coding for
video transmission, under some real-world constraints, was studied in [20] and the references
therein. The conclusion is that for applications such as the Internet, with its packetization and
delay constraints, for multicast and layered applications, where the decoding must be possible
for any subset of the received bitstream, as well as for applications requiring unequal error
protection, utilizing the separation principle yields results inferior to those obtained by joint
source–channel coding. Practical application of Shannon’s bound is also hampered by such
factors as the infinite block size (symbol length) assumption, and the corresponding time delay.
© 2001 CRC Press LLC
As discussed in Section 11.3, some of these real-world constraints can be easily incorporated
into a rate-distortion optimized coding algorithm.
Lossless data compression techniques capitalize on redundancies present in the source and
do not involve any trade-offs between rate and distortion. When it comes to designing lossy
data compression algorithms, however, the challenge is to represent the original data in such
a way that the application of coarse quantizers, which is what makes compression beyond
entropy possible, costs the least in terms of the resulting degradation in quality.
The objective of rate-distortion theory is to find a lower bound on the rate such that a known
source is represented with a given fidelity, or conversely. Thus, if the rate (R) and distortion
(D) are quantified, varying the maximum distortion or fidelity continuously, and finding the
corresponding bound R, is equivalent to tracing a curve in the R–D space. This curve is called
the rate-distortion function (RDF). Even assuming that the source model is correct, points on
this curve, however, may be difficult to achieve not only because of the infinite block length
and computational resources assumption, but also because they represent a lower bound among
all possible encoders under the sun.
11.2
The Operational Rate-Distortion Function
In the previous section we discussed the classical rate-distortion theory, the primary purpose
of which is to establish performance bounds for a given data source across all possible encoders,
operating at their “best.” In practice, however, we often deal with one fixed encoder structure
and try to optimize its free parameters. Finding performance bounds in such a setup is the
subject matter of the operational rate-distortion (ORD) theory.
Operating within the framework of a fixed encoder simplifies the problem a lot. Solving the
problem of optimal bit allocation in this restricted case, however, does not guarantee operation
at or near the bound established by the RDF. As an example, let us consider the simplest
image encoder, which approximates the image by the DC values of blocks derived from a fixed
partition. Clearly, finding the best possible quantization scheme for these coefficients, while
making the encoder optimal, will not do a good job of compressing the image or come close
to the RDF, since the encoder structure itself is very simple.
ORD theory is based on the fact that every encoder maps the input data into independent or
dependent sources of information which need to be represented efficiently. The finite number
of modes the encoder can select for each of these data subsources can be thought of as the
admissible set of quantizers. That is, a particular quantization choice for each of the subsources
in the image or video signal, for example, constitutes one admissible quantizer. A quantizer,
in this case, is defined in the most general terms. It may operate on image pixels, blocks,
or coefficients in a transformed domain. The encoder, in addition to being responsible for
selecting quantizers (reconstruction levels) for each of the subsources, must arrive at a good
partition of the original signal into subsources. Let us call a particular assignment of quantizers
to all subsources a mode of operation at the encoder. Each mode can be categorized by a rate
R and a distortion D. The set of all (R, D) pairs, whose cardinality is equal to the number
of admissible modes, constitutes the quantizer function (QF). Figure 11.2, in which the QF is
shown with the × symbol, illustrates this concept.
The ORD curve, denoted by the dotted line in Figure 11.2, is a subset of the QF and represents
modes of desirable operation of the encoder. Mathematically, the set of points on the ORD
curve is defined as follows:
ORD = i ∈ QF : Ri ≤ Rj or Di ≤ Dj , ∀j ∈ QF , i = j ,
(11.1)
© 2001 CRC Press LLC
FIGURE 11.2
The operational rate-distortion (ORD) curve.
where Ri , Di and Rj , Dj are the rate-distortion pairs associated with modes i and j , respectively. For convenience, it is customary to connect consecutive points of the ORD set, thus
forming the ORD curve. Clearly, it is desirable to operate on the ORD curve, and modes not
on the curve are not optimal in the sense that within the same encoder, a smaller distortion is
achievable for the same or smaller rate, or vice versa.
As stated, the concept of operating on the ORD curve is quite generic and can be applied
to a variety of applications. Practically, the idea of rate-distortion optimization is equivalent
to bit budget allocation among different subsources of information in a given compression or
channel transmission framework. When an allocation scheme results in a mode belonging to
the ORD set, an algorithm is operating at its optimum.
The framework of bit allocation with the goal of reaching a point on the ORD curve applies
equally well in cases when the source is stochastic in nature and is known only through the
model of its probability density function (pdf ), or in cases when transmission through a noisy
channel, rather than compression, is the problem. In these cases, E(D), the expected distortion,
is used instead of distortion, and the tightness of the bound established by the ORD curve is
sensitive to source or channel model accuracy.
11.3
Problem Formulation
The central problem in ORD optimization is to select appropriately the modes of the given
algorithm such that a point on the ORD curve is reached. This is equivalent to saying that no
other selection of parameters would lead to a better distortion performance for the same bit
rate. Following the notation of [28], if B is a code belonging to the set of all possible modes
SB generated by the given algorithm as a code for the specified data source, and R(·) and D(·)
© 2001 CRC Press LLC
are the associated rate and distortion functions, we seek a mode B ∗ , which is the solution to
the following constrained optimization problem,
min D(B),
subject to:
B∈SB
R(B) ≤ Rmax .
(11.2)
It turns out that the problem dual to that of (11.2), in which the source fidelity or distortion
is constrained, can be solved using the same tools (described in Section 11.4) as the rateconstrained problem. The dual problem can be expressed as follows:
min R(B),
B∈SB
subject to:
D(B) ≤ Dmax .
(11.3)
As stated, these problems are general enough to include many possible distortion metrics and
many parameter encoding schemes, including differential ones. Since the concept of rate and
distortion is typically associated with quantizers, it often helps to think of solutions to (11.2)
and (11.3) as optimal bit allocations among (possibly dependent) quantizers.
11.4
Mathematical Tools in RD Optimization
In this section we discuss two powerful optimization methods: the Lagrangian multiplier
method and dynamic programming (DP). These techniques are very suitable for the kind of
problems discussed here, that is, the allocation of resources among a finite number of dependent
quantizers. For an overview of optimization theory the reader is referred to [24]. In all examples
presented in this chapter, these two tools are used in conjunction with each other. First, the
Lagrangian multiplier method is used to convert a constrained optimization problem into an
unconstrained one. Then, the optimal solution is found by subdividing the whole problem into
parts with the help of DP.
11.4.1
Lagrangian Optimization
The Lagrangian multiplier method described here is the tool used to solve constrained
optimization problems. The idea behind the approach is to transfer one or more constraints
into the objective function to be minimized. In the context of image and video coding, the most
commonly used objective function is the distortion, with the bit rate being the constraint. As
stated, this problem is difficult because it provides no quantifiable measure by which an encoder
can make a local decision of selecting the best quantizer among several available quantizers.
The Lagrangian multiplier method solves this problem by adding the rate constraint to the
objective function, thereby redefining it.
Mathematically, this idea can be stated as follows. Finding the optimal solution B ∗ (λ) of
min (D(B) + λ · R(B)) ,
B∈SB
(11.4)
where λ is a positive real number, is equivalent to solving the following constrained optimization problem:
min D(B),
B∈SB
subject to:
R(B) ≤ Rmax .
(11.5)
Clearly, the optimal solution B ∗ is a function of λ, the Lagrangian multiplier. It is worth
noting that the converse is not always true. Not every solution to the constrained problem can
© 2001 CRC Press LLC
be found with the unconstrained formulation. That is, there may be values of Rmax , achievable
optimally by exhaustive search or some other method, for which the corresponding λ does not
exist and, therefore, is not achievable by minimizing (11.4).
Since, in practice, the constrained problem needs to be solved for a given Rmax , a critical
step in this method is to select λ appropriately, so that R(B ∗ (λ)) ≈ Rmax . Choosing such a λ
can be thought of as determining the appropriate trade-off between the rate and the distortion,
which is application specific.
A graphical relationship between points on the ORD curve and line segments in the first
quadrant with slope − λ1 can be established, based on the fact that the rate and distortion
components of points on the ORD curve are a nonincreasing and a nondecreasing function of
λ, respectively [26, 28]. That is, as Figure 11.3 shows, if we start with a line of that slope
passing through the origin and keep moving in the northeast direction, the sweeping line will
first intersect the ORD curve at the point(s) corresponding to the rate and the distortion that
are the optimal solutions to (11.4) when the trade-off of λ is used.
R(B)
convex hull
operational rate distortion curve
R(B*(λ))
1
R(B*(λ))
2
D(B)
D(B*(λ))
1
D(B*(λ))
2
R(B)= -1
λ
D(B) + {R(B*(λ)) + -1 D(B*(λ))}
λ
FIGURE 11.3
The line of slope − λ1 intersects the ORD curve at points having the rate-distortion tradeoff of λ.
There exist two fundamentally different approaches to finding λoptimal meeting the Rmax
budget. The first approach assumes a continuous model D(R) for the ORD function, for
example, a decaying exponential. Then, λoptimal is approximated through λ = − dD
dR , evaluated
at R = Rmax . A model-based technique, however, is only as good as the model itself.
© 2001 CRC Press LLC
The second approach, which is also based on the monotonicity of R(λ) and D(λ), uses an
iterative search to find λoptimal . The process consists of running the encoder for two different
values of λ, λl and λu , corresponding to the beginning and the end of the interval to which
λoptimal belongs. This interval is iteratively redefined either with the bisection method or based
on a fast Bezier curve search technique [28].
11.4.2
Dynamic Programming
Dynamic programming (DP) is a tool that is typically applied to optimization problems in
which the optimal solution involves a finite number of decisions, and after one decision is
made, problems of the same form, but of a smaller size, arise [2]. It is based on the principle
that the optimal solution to the overall problem consists of optimal solutions to its subparts or
subproblems. In problems of this type exhaustive search solves the same subproblems over
and over as it tries to find the global solution at once. By contrast, DP solves each subproblem
just once and their solutions are stored in memory.
In the image and video coding applications considered in this chapter, optimality is achieved
∗ ], minimizing the overall cost funcby finding the ordered sequence of quantizers [q0∗ , . . . , qN
tion,
∗
J ∗ q0∗ , . . . , qN
= min J (q0 , . . . , qN ) ,
(11.6)
q0 ,...,qN
where both the quantizers and their number, N , have to be determined by the encoder. In the
context of rate- or distortion-constrained optimization, DP is combined with the Lagrangian
multiplier method, in which case the cost function is often written as Jλ to emphasize its
dependence on λ. The DP method is applicable because the total cost function minimization
can be broken down into subproblems in a recursive manner as follows:
∗
∗
∗
J ∗ q0∗ , . . . , qN
, . . . , qN
= min J ∗ q0∗ , . . . , qi∗ + J ∗ qi+1
,
i
0≤i≤N.
(11.7)
It should be noted, however, that straightforward application of DP to problems of very large
dimensions may be impractical due to coding delay considerations. Even though DP results
in solutions of significantly lower complexity than exhaustive search, it nevertheless performs
the equivalent of exhaustive search on the local level. In such cases, a greedy suboptimal
matching pursuit approach, based on incremental return, may be called for. This was done
in [18] and [6] in the context of low-bit-rate video and fractal compression, respectively.
11.5
Applications of RD Methods
In this section we describe the application of RD-based methods to several different areas of
image and video processing. These include motion estimation, motion-compensated interpolation, object shape coding, fractal image compression, and quad-tree (QT)-based video coding.
The common theme shared by these applications is that they all can be stated as resource
allocation problems among dependent quantizers (i.e., in the forms shown in Section 11.3).
Operationally optimal solutions are obtained in each case and are shown to significantly outperform traditional heuristic approaches. In all these applications the mathematical tools
described in Section 11.4 are used in the optimization process.
© 2001 CRC Press LLC
11.5.1
QT-Based Motion Estimation and Motion-Compensated Interpolation
Most of the resources in present-day multimedia communication systems are devoted to
digital video, because its three-dimensional nature is inherently more complex than that of
speech signals and text. The function of a video codec is unique for several reasons. First, the
large amount of raw data necessitates high compression efficiency. Second, a typical video
waveform exhibits correlation in the spatial and, to a greater extent, in the temporal direction.
Furthermore, the human visual system is more sensitive to errors in the temporal direction.
Motion compensation (MC) is the technique most commonly used in video coding to capitalize on temporal correlation. In this context, some frames in a video sequence are encoded
with still image coding techniques. These frames are called I frames. The second type of
frames, P frames, are predicted from their reference frames using motion information, which
has to be estimated, usually on a block-by-block basis. A set of motion vectors, defining for
each pixel in the current frame its location in the reference frame, is called the displacement
vector field (DVF).
Due to its simplicity and easy hardware implementation, block matching is the most popular
method of motion estimation. Although the use of irregularly shaped regions potentially allows
for better local adaptivity of the DVF to a frame’s spatial segmentation, its application in video
coding is hampered by the need to transmit shape information. By allowing segmentation into
blocks of variable sizes, a compromise between a compact representation (blocks of a fixed
size) of the DVF and its local adaptivity (complex object-oriented approach) is achieved. That
is, large blocks can be used for the background and smaller blocks can be used for areas in
motion. The QT structure is an efficient way of segmenting frames into blocks of different
sizes, and it was used to represent the inhomogeneous DVF in [11, 28]. In addition, the
QT approach enables a tractable search for the joint and optimal segmentation and motion
estimation, which is not possible with the complex object-based segmentation [28].
Optimal Motion Estimation
In this section an operationally optimal motion estimator is derived [28, 32]. The overall
problem solved here can be stated as that of minimizing the displaced frame difference (DFD)
for a given maximum bit rate, and with respect to a given intra-coded reference frame. It
should be noted, however, that some efforts have been made to jointly optimize the anchor
and the motion-compensated frame encoding [10]. The QT-based frame decomposition used
here is achieved by recursively subdividing a 2N × 2N image into subimages of halved size
until the block size 2n0 × 2n0 is reached. This decomposition results in an (N − n0 + 1)-level
hierarchy, where blocks at the nth level are of size 2n × 2n .
Within this decomposition, each square block bl,i (lth level in the QT, ith block in that level)
is associated with Ml,i — the set of all admissible motion vectors, of which ml,i is a member.
Then a local state sl,i = [l, i, ml,i ] ∈ Sl,i = {l} × {i} × Ml,i can be defined for each block
bl,i . Consequently, a global state x, representing the currently chosen local state, is defined as
N−1
n0
∪4i=0 −1 Sl,i , where X is the set of all admissible global state values.
x ∈ X = ∪l=N
A complete description of the DVF in the rate-distortion framework requires that the individual block states sl,i be enumerated sequentially because the rate function may involve
arbitrary-order dependencies resulting from differential encoding of parameters. Hence, the
code for a predicted frame consists of a global state sequence x0 , . . . , xN −1 , which represents
the left-to-right ordered leaves of a valid QT .
In this context, the frame distortion is an algebraic sum of block distortions d(xj ), where
the blocks correspond to the leaves of the chosen QT decomposition . That is,
© 2001 CRC Press LLC
−1
N
D x0 , . . . , xN −1 =
d xj .
(11.8)
j =0
The individual block distortion metric chosen here is the MSE of the DFD projected on the
block.
Encoding the motion vectors of the DVF is a challenging task in itself. On the one hand, high
coding efficiency can be achieved by considering long codewords composed of many motion
vectors ml,i along the scanning path. On the other hand, as explained in Section 11.4, the
complexity of the optimal solution search is directly related to the order of dependency in the
differential encoding of parameters. As a compromise between the two, a first-order DPCM
scheme is used, allowing a first-order dependency between the leaves along the scanning path.
Another challenge is posed by the fact that a typical image exhibits intensity and motion
vector correlation in two dimensions, whereas a scanning path is inherently one dimensional.
A scan according to the Hilbert curve was shown in [28, 32] to possess certain space-filling
properties and create a representation of the 2D data, which is more correlated than that resulting
from a raster scan. It can be generated in a recursive fashion and is natural for QT-decomposed
images.
Based on the chosen first-order DPCM along a Hilbert scanning path, the overall frame rate
can be expressed as follows:
−1
N
R x0 , . . . , xN −1 =
r xj −1 , xj ,
(11.9)
j =0
where r(xj −1 , xj ) is the block bit rate, which is a function of the quantizers used for encoding
the current and the previous blocks.
Having defined the distortion and the rate, the problem of motion estimation is posed as a
constrained optimization problem as follows:
min D x0 , . . . , xN −1 , subject to: R x0 , . . . , xN −1 ≤ Rmax .
(11.10)
x0 ,...,xN −1
Since the process of approximating a block bl,i in the current frame by another block of equal
size in the reference frame, using motion vector ml,i , can be viewed as quantization of the
block bl,i , with which a certain rate and a certain distortion are associated, the problem of
rate-constrained motion estimation can be viewed as an optimal bit allocation problem among
the blocks of a QT with leaf dependencies or dependent quantizers. Hence, the methods
of Section 11.4 (Lagrangian multiplier-based unconstrained optimization using DP within a
trellis) are applicable.
The search for the best block match under a motion vector is by far the most computationally
expensive part of the optimization. To make the search faster, a slightly suboptimal clustering
scheme is used, in which only a subset of possible motion vectors is considered [28].
Motion Estimation Results
The QT-based optimal motion estimation scheme compares favorably with TMN4 (an implementation of the H.263 standard), using the same quantizers to encode the DVF. For the
QCIF video sequence, the smallest block size of 8 × 8 was chosen and the Hilbert scan was
modified [28] to cover nonsquare frames. The predicted frame 180 of the Mother and Daughter
sequence and its corresponding scanning path are shown in Figures 11.4 and 11.5, respectively.
© 2001 CRC Press LLC
FIGURE 11.4
Segmentation and DVF of the predicted frame.
FIGURE 11.5
The overall scanning path.
When compared to the original frame 180, the DFD peak signal-to-noise ratio (PSNR), in
the case of optimal QT motion estimation, is 31.31 dB. The corresponding bit rate is 472 bits.
Compared to the TMN4 encoder, operating at the same rate and resulting in the PSNR of
30.65 dB, the optimal scheme represents a 0.67-dB improvement in PSNR. The results are
more dramatic if, rather than the rate, the PSNR is matched between the two algorithms (at
30.65 dB) (i.e., the dual problem is solved). In this case, the optimal QT-base motion estimator
encodes frame 180 with 344 bits, and that is a 26.8% improvement over the 470 bits required
by TMN4.
© 2001 CRC Press LLC
Motion-Compensated Interpolation
The problem of frame interpolation is very important in very-low-bit-rate video coding.
The transmission channel capacity, by imposing an upper limit on the bit rate, necessitates a
two-pronged approach to video compression: some frames are encoded and transmitted; other
frames are not encoded at all, or dropped, and must be interpolated at the decoder. It is common
for video codecs to operate at the rate of 7.5 or 10 frames per second (fps) by dropping every
third or fourth frame. Without frame interpolation, or with zero-hold frame interpolation, the
reconstructed video sequence will appear jerky to a human observer.
The problem of interpolation is ill posed because not enough information is given to establish
a metric by which to judge the goodness of a solution. In our context, some frame in the
future, fˆN , and some frame in the past, fˆ0 , of the interpolated frame is all that is available at
the decoder. Therefore, since the original skipped frame is not available at the decoder, the
MSE minimization-based interpolation is not possible.
What makes this problem solvable is the underlying assumption that changes in the video
scene are due to object motion, in particular linear motion. In contrast to the traditional
approach, where the motion is estimated for frame fˆN with respect to frame fˆ0 and then
projected onto the interpolated frame, here the motion is estimated directly for the interpolated
frame. Thus, the problems related to the fact that not all pels of the interpolated frame are
associated with a motion vector are avoided. Figure 11.6 demonstrates this idea.
k
0
1
2
3
4
N =3
2
N =1
1
N=4
FIGURE 11.6
Motion-compensated interpolation (of frame 1 from frame 0 and frame 4).
The problem at hand is to find the segmentation of the interpolated frame into blocks bl,i (QT
decomposition) and the associated motion vectors ml,i . The motion, however, is with respect
to both reference frames (fˆ0 and fˆN ). A block bl,i in the QT decomposition is interpolated as
© 2001 CRC Press LLC
follows:
fˆn (x, y) =
N2 fˆ0 (x − N1 mj,x , y − N1 mj,y ) + N1 fˆN (x + N2 mj,x , y + N2 mj,y )
N1 + N 2
∀(x, y) ∈ bl,i ,
(11.11)
where 0 ≤ n ≤ N , mj,x and mj,y are the x and y coordinates of the motion vector mj of
the block bl,i , and N1 , N2 are the temporal distances of the interpolated frame to frames f0
and fN , respectively. This weighted definition leads to a smooth transition of the interpolated
frame toward the closer reference frame as the distance between them decreases, causing less
jerkiness.
Let xj denote the global system state, corresponding to the interpolated block bl,i undergoing
motion mj . Consistent with the block interpolation formula defined in (11.11), the associated
block distortion is defined as follows:
2
,
fˆ0 x − N1 mj,x , y − N1 mj,y − fˆN x + N2 mj,x , y + N2 mj,y
d xj =
(x,y)∈bl,i
(11.12)
and the overall distortion is the algebraic sum of block distortions corresponding to the leaves
in the chosen QT decomposition. That is,
−1
N
D x0 , . . . , xN −1 =
d (xk ) .
(11.13)
j =0
Clearly, minimizing the frame distortion defined in (11.13) alone over all possible QT
segmentations and DVF choices would lead to the frame being segmented into blocks of the
smallest size possible. The resulting DVF would be very noisy and have little resemblance to
the underlying object motion in a video scene. It is desired for the estimated DVF to possess a
measure of smoothness present in the real DVF. It turns out that this goal can be achieved by
regularizing the objective function with the total bit rate, that is,
D x0 , . . . , xN −1 + λ · R x0 , . . . , xN −1 ,
(11.14)
min
x0 ,...,xN −1
where λ is the regularization parameter. Minimizing the above objective function leads to a
smooth DVF because, with a differential encoding scheme, there is a strong correlation between
the smoothness of the DVF and the bit rate necessary for its encoding. Hence, smoothness is
achieved for motion vectors along a scanning path. With the Hilbert scanning path employed,
this translates into DVF smoothness in all directions.
The optimal solution to the regularized optimization problem of (11.14) is then found with
the help of DP applied to a trellis structure, as explained in Section 11.4.
Interpolation Results
The described algorithm has been applied to the compressed Mother and Daughter sequence
at the frame rate of 15 fps and a constant frame PSNR of 34.0 dB. Every second frame in
this sequence is dropped, thus resulting in a 7.5 fps sequence. Then these dropped frames
are reconstructed from the two neighboring frames using the operationally optimal motioncompensated interpolation scheme described in the preceding section.
The issue of selecting the proper regularization parameter λ is addressed in [5]. Here, a λ
of 0.01 is used. Figure 11.7 shows the reconstructed frame 86, which was interpolated from
frame 84 and frame 88. The resulting DVF and QT segmentation are also overlaid on this
figure. The interpolated frame is very similar to the original frame 86 and is only 1 dB lower
in PSNR than the reconstructed version, had it been transmitted.
© 2001 CRC Press LLC
FIGURE 11.7
Segmentation and DVF of the interpolated frame.
11.5.2
QT-Based Video Encoding
In this subsection the operationally optimal bit allocation scheme among QT segmentation,
DVF, and DFD is presented. The overall problem solved here can be stated as that of minimizing
the distortion between the original and the reconstructed frames for a given bit budget Rmax ,
where motion is estimated with respect to a given reference frame. Hence, the job of the
encoder is to optimally allocate the available bit budget to segmentation, motion, and error
quantization components.
Code Structure
As in the case with optimal motion estimation and interpolation, discussed in the previous
subsection, here the QT structure is used for frame segmentation due to its being a compromise
between a fixed-block-size approach and a scene-adaptive object-based segmentation. Thus,
the original 2N × 2N image is decomposed into a hierarchy of square blocks, with the smallest
block size 2n0 × 2n0 .
Using the same notation as in Section 11.5.1, each square block bl,i is associated with
Ml,i — the set of all admissible motion vectors, of which ml,i is a member — and Ql,i —
the set of all admissible residual error quantizers, of which ql,i is a member. Then a local
state sl,i = [l, i, ql,i , ml,i ] ∈ Sl,i = {l} × {i} × Ml,i × Ql,i can be defined for block bl,i .
Consequently, a global state x, representing the currently chosen local state, is defined as
N−1
n0
∪4i=0 −1 Sl,i , where X is the set of all admissible global state values.
x ∈ X = ∪l=N
© 2001 CRC Press LLC
For completeness of description, individual block states sl,i must be enumerated sequentially
because the rate function may involve arbitrary-order dependencies resulting from differential
encoding of parameters. Hence the code for a predicted and motion-compensated frame
consists of a global state sequence x0 , . . . , xN −1 , which represents the left-to-right ordered
leaves of a valid QT .
In this context, the frame distortion is an algebraic sum of block distortions d(xj ), implemented with the MSE metric, where the blocks correspond to the leaves of the chosen QT
decomposition . That is,
−1
N
D x0 , . . . , xN −1 =
d xj .
(11.15)
j =0
Again, as a compromise between complexity and efficiency, a first-order DPCM scheme is
used for encoding motion vectors, allowing a first-order dependency between the leaves along
the scanning path. The scanning path itself is, for reasons described in Section 11.5.1, the
Hilbert curve, recursively generated to fill the frame space [28].
Based on the chosen first-order DPCM along the scanning path, the overall frame rate can
be expressed as follows:
R x0 , . . . , xN −1 =
N
−1
r xj −1 , xj ,
(11.16)
j =0
where r(xj −1 , xj ) is the block bit rate, which depends on the encoding of the current and the
previous blocks.
In (11.16) we assume that the total frame rate can be distributed among its constituent blocks.
This assumption is intuitive in the case of the rate associated with the motion vector component
r DV F (xj −1 , xj ) and the residual error quantization component r DF D (xj ). It turns out that the
QT segmentation rate can also be distributed on the block basis. With QT decomposition, only
1 bit is required to signal a splitting decision at each level. Hence, smaller blocks carry the
segmentation costs of all of their predecessors. With the sequential scanning order of blocks
belonging to the same parent in the QT, the first scanned block is arbitrarily assigned the cost
r SEG (xj ) associated with decomposition up to that level. Thus, the segmentation component
of the rate can also be defined on the block basis.
To complete the discussion of the rate, we must also take into account the fact that some
blocks in the QT decomposition are not predicted, but rather intra-coded. This may be applicable to newly appearing or uncovered objects that are not found in the reference frame
or when the motion model fails to find a good match for a block. When that happens, the
intra-coded block’s DC coefficient can be predicted from its predecessor’s along the scanning
path. Therefore, the r DC (xj −1 , xj ) component must be added to the total rate. Clearly, it
is equal to 0 for predicted blocks, and the task of deciding which blocks are coded intra and
which blocks are coded inter is a part of the optimization process at the encoder.
In summary, the block rate, corresponding to the transition from state xj −1 to state xj is
expressed as follows:
r xj −1 , xj = r SEG xj + r DF D xj + r DV F xj −1 , xj + r DC xj −1 , xj . (11.17)
Having defined the distortion and the rate, the problem of joint segmentation, motion estimation, and residual error encoding is posed as a constrained optimization problem as follows:
min D x0 , . . . , xN −1 , subject to: R x0 , . . . , xN −1 ≤ Rmax .
(11.18)
x0 ,...,xN −1
© 2001 CRC Press LLC
This problem can be viewed as the optimal bit allocation problem among the blocks of a
QT with leaf dependencies and, hence, the methods of Section 11.4 apply. In particular, it
is converted into the unconstrained minimization problem using the Lagrangian multiplier
method,
Jλ x0 , . . . , xN −1 = D x0 , . . . , xN −1 + λ · R x0 , . . . , xN −1 ,
(11.19)
and dynamic programming, also discussed in Section 11.4, is used to find the optimal solution.
∗
].
The resulting optimal solution is a sequence of states [x0∗ , . . . , xN
−1
Graphically, the DP algorithm is illustrated by Figure 11.8. In it, each node (black circle)
represents a particular state xj the corresponding block bl,i is in. The lines connecting these
states correspond to the possible scanning orders of a Hilbert curve, and weights jλ (xj −1 , xj ) =
d(xj )+λ·r(xj −1 , xj ) are associated with each transition. Then the problem of optimal resource
allocation among segmentation, motion, and DFD quantization can be stated as that of finding
the shortest path in the trellis from S to T , the two auxiliary states. In the implementation, the
FIGURE 11.8
The trellis structure.
maximum block size was 32 × 32 and the minimum block size was 8 × 8, with a consequence
that no segmentation information for blocks larger than 32 and smaller than 8 needed to be
sent to the decoder. Other implementation details can be found in [28]–[30].
Results
The encoder presented here is compared with TMN4, which is an H.263 standard implementation. Both algorithms were tested on 200 frames of the Mother and Daughter sequence,
down-sampled by a factor of 4 in the time axis, with the quantizer step size QP set to 10 in
TMN4. In the first test, the λ parameter of the optimal algorithm was adjusted for each frame
so that the resulting distortion matched that produced by TMN4 at every encoded frame. The
resulting curves are shown in Figure 11.9. In the second test, the λ parameter of the optimal
algorithm was adjusted for each frame so that the resulting rate matched that produced by
TMN4 at every encoded frame. The resulting curves are shown in Figure 11.10. Clearly,
the optimal approach significantly outperforms H.263 in both experiments. In the matched
distortion case, the average frame bit rate was reduced by about 25%, and, in the matched rate
case, the average PSNR distortion was increased by about 0.72 dB.
© 2001 CRC Press LLC
FIGURE 11.9
The matched distortion result.
FIGURE 11.10
The matched rate result.
11.5.3
Hybrid Fractal/DCT Image Compression
This section describes the application of the rate-distortion techniques to the hybrid fractal/DCT image compression. Drawing on the ability of DCT to remove interpixel redundancies
and on the ability of fractal transforms to capitalize on long-range correlations in the image,
the hybrid coder performs an optimal, in the rate-distortion sense, bit allocation among coding
parameters. An orthogonal basis framework is used within which an image segmentation and
a hybrid block-based transform are selected jointly.
Problem Formulation
Within the chosen fractal/DCT framework, the problem to be solved is that of simultaneous
segmentation of an input image into blocks of variable sizes, and, for each, to find a code
© 2001 CRC Press LLC
in such a way that any other choice of segmentation and coding parameters would result in
a greater distortion for the same rate, or vice versa. Problems of this type are discussed in
Section 11.2 and the corresponding solution tools in Section 11.4. In this context, for a given
image x, we want to solve the following optimization problem:
min D xs,c , x
subject to: R xs,c ≤ Rmax ,
(11.20)
s∈S,c∈Cs
where xs,c is the encoded image; D the distortion metric; s a member of the set of all possible
image segmentations S; c a member of Cs , the set of all possible codes given segmentation s;
R the bit rate associated with segmentation s and code c; and Rmax the target bit budget. The
distortion metric chosen here is the MSE.
Fractal Basics
Fractal image coding takes advantage of image self-similarities on different scales. That
is, instead of sending quantization indices of transformed image subblocks, a fractal coder
describes the image as a collection of nonexpansive transformations onto itself. Most fractal
algorithms, beginning with Jacquin’s implementation [8], break an image into nonoverlapping
square regions, called ranges. Each range block ri is encoded by a nonexpansive transformation
Ti∗ that operates on the entire original image x and maps a domain block di , twice the size of
the range block and located elsewhere in the image, onto ri . The job of the encoder is to find
a transformation Ti that minimizes the collage error. That is,
Ti∗ = arg min ri − Ti (x) ,
Ti ∈
(11.21)
where is the pool of available transforms. The whole transformation T is a sum of partitioned
transformations,
T (x) =
N
i=1
Ti∗ (x),
x=
N
ri ,
(11.22)
i=1
where N is the number of partitions or ranges.
The Collage Theorem establishes an upper bound on the reconstruction error of the decoded
image as a function of the collage error and s, the contractivity of T . Specifically,
d (x, xT ) ≤
1
· d(x, T x) ,
1−s
(11.23)
where xT is the decoded image under transformation T .
Transformations Ti are restricted to a set of discrete contractive affine transformations operating on x. Following the notation used in [21], each Ti has the following structure:
Ti (x) = βi Pi Di Ii F eti x + ti ,
(11.24)
where F eti is a transformation matrix that fetches the correct domain block, Ii applies one
of the standard eight isometries, Di is the decimation operator that shrinks the domain block
to the range block size, Pi is the place operator that places the result in the correct region
occupied by the range block, βi is a scalar, and ti is a constant intensity block. Hence, in
analogy with vector quantization (VQ), various permutations of F eti , Ii , and βi represent the
codebook. Implementation details on the specific choice of the variable-length coding (VLC)
for the various parameters of (11.24) can be found in [13].
Blocks for which a good approximation, under a contractive transformation, can be found
elsewhere in the image, can be efficiently encoded using the fractal transform. The selfsimilarity assumption, which is central to fractals, however, may not be justified for all blocks.
© 2001 CRC Press LLC
In this case, spending more bits on the fractal transform by employing more isometries or finer
quantizers is not efficient [4, 25].
The discrete cosine transform (DCT) has been the transform of choice for most codecs
due to its decorrelation and energy compaction properties. Complicated image features, however, require a significant number of DCT coefficients to achieve good fidelity. The coarse
quantization of these coefficients results in blocking artifacts and unsharp edges.
The coder described here is a hybrid in that it not only adaptively selects which transform
(fractal or DCT) to use on any given block, but also can use them jointly and in any proportion.
Thus it capitalizes on the ability of the fractal transform to decorrelate images on the block
level and on the ability of the DCT to decorrelate pixels within each block. The optimization
techniques of Section 11.4 are applied within the chosen framework, resulting in a code that
is optimal in the operational sense.
The decoder reconstructs an approximation to the original image by iterative application
of the transformation defined by (11.22) to any arbitrary image x0 . Although the Collage
Theorem, expressed by (11.23), guarantees the eventual convergence of this process to an
approximation with a bounded error, the number of these iterations may be quite large.
Segmentation
The goal of image segmentation in the context of compression is to adapt to local characteristics of the image. Segmenting the image into very small square blocks or into objects
of nonsquare shapes, while accomplishing this goal, is also associated with a high cost of
description. Here, the set of all possible segmentations S is restricted to be on the QT lattice
as a compromise between local adaptivity and simplicity of description. For a 256 × 256 input
image it is a three-level QT with a maximum block size of 16 × 16 pixels and a minimum size
of 4 × 4 pixels. At each level of the QT only 1 bit is required to signal a splitting decision,
with no such bit required at the lowest level.
Code Structure
As mentioned in Section 11.5.3, the transform employed for encoding range blocks consists
of the fractal and DCT components. The hybrid approach allows for more flexibility at the
encoder, and the various components of the overall transform are designed to complement each
other. This idea leads naturally to the concept of orthogonality.
The presence of the DCT component in the overall transform, as well as the fact that the
frequency domain interpretation lends itself naturally to the concept of orthogonality and
energy compactness, makes it more convenient to perform the collage error minimization
of (11.21) in the DCT domain. This is done by applying the DCT to both the range block ri
under consideration and all candidate-decimated domain blocks.
It is convenient to cast the problem of finding the overall transform as that of vector space
representation. Vectors are formed by zigzag-scanning square blocks, as shown in Figure 11.11.
Hence, minimizing the collage error with respect to a range vector r¯i is equivalent to finding
its best approximation in a subspace spanned by a combination of transformed domain vectors
and some fixed (image-independent) vectors f¯k .
In agreement with the terminology used in [23], let the vector r̄i , of size M 2 × 1, represent
the DCT coefficients of range block i, of size M × M, scanned in zigzag order. Similarly, the
vector d¯i comes from the chosen domain block, after decimation, DCT, and the application of
isometry operators. The fractal component of the overall transform can then be expressed as
follows:
d¯i = Pi Zi DCTi Di Ii F eti x ,
© 2001 CRC Press LLC
(11.25)
FIGURE 11.11
Zigzag scan of a square block.
where Pi , Di , Ii , and F eti are defined as in (11.24) and DCTi and Zi are the block DCT
and zigzag-scan operators, respectively. Let the intensity translation term be represented by a
linear combination of Ni − 1 fixed vectors f¯ik , of size M 2 × 1, where the subscript i indicates
that the pool of available fixed vectors and the total number of them may be locally adaptive
to the range vector. Mathematically, the range vector r¯i is then approximated by Ni vectors as
follows:
r̄i ≈ βi · d̄io +
N
i −2
cik · f¯ik ,
(11.26)
k=0
where d̄io is the projection of d̄i onto the orthogonal complement of the subspace spanned by
vectors f¯ik for k = 0, . . . , Ni − 2, which are themselves orthogonal to each other. Making
components of the overall transform orthogonal to each other carries many benefits [16, 22].
These include fast convergence at the decoder, noniterative determination of scaling and intensity translation parameters, no restriction on the magnitude of the scaling coefficient, and
continuity of the magnitude of the translation term between neighboring blocks.
In this implementation a bank of fixed subspaces is used to model a range vector of a given
size. To illustrate how a fixed subspace is formed from the coefficients of a block DCT, let
us, for simplicity, consider a 2 × 2 block of DCT coefficients. A subspace of dimension 3
(with the full space of dimension 4) is then formed from the low-frequency coefficients as
shown in Figure 11.12. Each f¯ik corresponds to one coefficient in the two-dimensional DCT
of size M × M. Each vector f¯ik has zeros in all positions, except the one corresponding to the
1 0 0
0 0 1
0 1 0
0 0 0
FIGURE 11.12
Mapping of selected DCT coefficients into basis vectors.
© 2001 CRC Press LLC
order in which the DCT coefficient it represents was scanned, where it has 1. Since in larger
blocks more DCT coefficients tend to be significant, the fixed space used for the encoding of a
16 × 16 range block is allowed to be of a higher dimension than that of a 4 × 4 block. With the
limited number of available subspaces for each block size, only the subspace index, and not
the positions of individual nonzero coefficients, needs to be sent to the decoder. Figure 11.13
shows the subspaces allowed for encoding range blocks of size 4 × 4 (range vectors of size
16 × 1). The banks of subspaces for range blocks of sizes 8 × 8 and 16 × 16 are defined
FIGURE 11.13
The 4 subspaces for block size of 4.
similarly and the details on this, as well as on the VLCs used, can be found in [16].
The job of the encoder is then, for each range vector r̄i , to select the fixed space dimension
Ni −1, the set of coefficients cik , the domain block to be fetched by F eti , the isometry Ii , and the
scaling coefficient βi . As a result of orthogonalization, the fixed subspace coefficients cik will
carry low-frequency information and fractal component parameters will carry high-frequency
information.
Directed Acyclic Graph (DAG) Solution
The DC coefficients of neighboring blocks in an image decomposition exhibit high correlation. In this formulation, coefficient ci0 corresponds to the quantized DC value of the
range block in question. Hence, differential encoding must be introduced between adjacent
blocks to take advantage of this redundancy. The Hilbert curve is known to satisfy certain
adjacency requirements [28] and is efficient for predictive coding. For a 256 × 256 input
image, a sixth-order Hilbert curve is used.
If we let each node represent a block in the QT decomposition of the image, and define a
transition cost gi,j as the cost of encoding range block ri with range block rj as its predecessor,
then the overall problem of finding the optimal segmentation and the hybrid fractal/DCT code
can be posed as that of finding the shortest path through the leaves of the QT decomposition or
trellis, with each leaf having one to three possible codes, corresponding to one to three possible
predecessors of a block in our Hilbert curve.
Clearly, the optimal scanning path possesses the optimal substructure property (i.e., it consists of optimal segments). This motivates the use of dynamic programming in the solution.
Refer to [28] and Section 11.4 for more details on the use of DP in problems of this type.
Fractal Compression Results
Performance of the operationally optimal hybrid fractal/DCT algorithm is compared to
JPEG, which is one of the most popular DCT-based compression schemes. Figure 11.14
shows the ORD curve obtained with this approach when compressing a 256 × 256 Lena
image. JPEG’s ORD curve is also shown on this plot. An improvement of 1.5 to 3.0 dB is
achieved across the range of bit rates.
Figures 11.15 and 11.16 demonstrate the improvement in quality, over JPEG, for the same
bit rate (0.20 bpp). Figure 11.17 shows the optimal segmentation as determined by this
encoder. Efficiency is achieved by using larger block sizes in relatively uniform areas and
smaller block sizes in edgy areas. Overall, the fractal component of the transform, representing
high-frequency information, used about 30% of the available bit budget. This, coupled with
© 2001 CRC Press LLC
FIGURE 11.14
Hybrid fractal/DCT algorithm vs. JPEG.
FIGURE 11.15
Hybrid algorithm (R = 0.20 bpp, PSNR = 26.71 dB).( © 1972 by Playboy magazine).
the rate-distortion optimized scanning path, segmentation, and code selection, resulted in
significant gains in the quality of the reconstructed image.
11.5.4
Shape Coding
In this section we show how rate-distortion operationally optimal techniques can be applied
to the problem of object shape coding. Interest in this problem is motivated by a growing
explosion in new multimedia applications, including, but not limited to, video conferencing,
© 2001 CRC Press LLC
FIGURE 11.16
JPEG (R = 0.20 bpp, PSNR = 23.19 dB).
interactive multimedia databases, film authoring, etc. They all have in common the requirement
that video information be accessible on an object-by-object basis.
Commercial video compression standards, such as MPEG-1, MPEG-2, H.261, and H.263,
are block-based codecs. They segment a video scene into fixed blocks of predetermined
sizes and achieve compression by quantizing texture and motion vectors. This format of
representation is not natural for the above-mentioned applications, since there is no clear
separation of one object from another and from the background in a generated bitstream. It
leads to an unnecessary waste of bits describing a background that carries no information.
But it is also inefficient in terms of accessing encoded information. For example, in pattern
recognition applications objects are detected through their boundaries, which are not explicitly
available in block-based bitstreams.
The emerging MPEG-4 and MPEG-7 multimedia coding standards are designed to address
these new requirements. Although it is not clear whether object-based treatment of a video
scene is justified in terms of coding efficiency, it is in some cases a requirement from the
application point of view. In an object-oriented coder, bits must be efficiently allocated among
bitstream components (segmentation, motion, shape, texture) and then within each component.
Although ORD optimal joint resource allocation among and within these components in an
object-oriented coder remains an ellusive goal, here we address the shape coding aspect of this
problem.
Shape information plays the central role in object description. Efforts at its efficient representation, which intensified as a result of the MPEG-4 standardization, can be classified into
two categories [9]. The first consists of bitmap-based coders, which can be further broken
down into context-based [1] and modified read fax-like [36] coders. The baseline-based shape
coder [12] and vertex-based polynomial coder [7, 19] belong to the second category. However,
arguably, the bitmap-based coders defeat the goal of object orientation, since in them the shape
information is not explicit.
In what follows we describe an intra-mode vertex-based boundary encoding scheme and
© 2001 CRC Press LLC
FIGURE 11.17
Optimal segmentation (R = 0.44 bpp, PSNR = 30.33 dB).
how it is optimized using the techniques of Section 11.4. Implementation details can be found
in [33]. An inter-mode vertex-based boundary encoding scheme is derived in [15].
Algorithm
The original boundary is approximated by second-order connected spline segments. A
spline segment is completely defined by three consecutive control points (pu−1 , pu , pu+1 ). It
is a parametric curve (parameterized by t), which starts at the midpoint between pu−1 and pu
and ends at the midpoint between pu and pu+1 , as t sweeps from 0 to 1. Mathematically, the
second-order spline segment used is defined as follows:


 
0.5 −1.0 0.5
pu−1,x pu−1,y
2
u (pu−1 , pu , pu+1 , t) = t t 1 ·  −1.0 1.0 0.0  ·  pu,x
pu,y  , (11.27)
Q
pu+1,x pu+1,y
0.5 0.5 0.0
where pi,x and pi,y are, respectively, the vertical and horizontal components of point pi .
Besides solving the interpolation problem at the midpoints, the definition of the spline
used makes it continuously differentiable everywhere, including the junction points. Segment
continuity is ensured because the next spline segment, (pu , pu+1 , pu+2 ), will originate at the
end of the current segment. Placing the control points appropriately, a great variety of shapes
can be approximated, including straight lines and curves, a property that makes splines a very
attractive building block for the contour approximation problem. The operationally optimal
straight line shape approximation was derived in [31].
In order to fit a continuous spline segment to the support grid of the original boundary, spline
points are quantized toward the nearest integer value. Thus the solution to our shape approximation problem is an ordered set of control points, which, based on a particular definition of
the distortion and rate (discussed below), and for a given rate-distortion trade-off λ, results in
a rate-distortion pair (R, D).
© 2001 CRC Press LLC
Theoretically, the ordered set of control points can be composed of points located anywhere
in the image. Most of them, however, are highly unlikely to belong to the solution, as they
are too far from the original boundary, and distortions of more than several pixels are not
tolerable in most applications. For this reason, and also to decrease computational complexity,
we exclude those points from consideration. What remains is the region of space, shown in
gray in Figure 11.18, centered around the original boundary and termed the admissible control
point band, to which candidate control points must belong. Pixels in this band are labeled by
FIGURE 11.18
Admissible control point band.
the index of the closest original boundary pixel (boundary pixels themselves are ordered and
labeled). Consecutive control points must be of the increasing index, thus ensuring that the
approximating curve can only go forward along the original boundary.
Distortion
Measuring the distortion between an original and approximating boundary is a nontrivial
problem. In the minimum–maximum distortion problem [31], stated in Section 11.3, the metric
is a binary function, evaluating to zero in case the approximation is inside the distortion band
of width Dmax , and evaluating to infinity in case some portion of the approximating curve
lies outside this band. This metric may be useful when fidelity of approximation must be
guaranteed for all pixels.
The minimum total distortion problem, solved optimally in [14, 17], is based on a global
distortion measure, where local errors are not explicitly constrained. This measure was used in
the MPEG-4 standardization process to compare performances of competing algorithms and
is also used here explicitly in the optimization process. It is defined as follows:
D=
© 2001 CRC Press LLC
number of pixels in error
number of interior pixels
(11.28)
A pixel is judged to be in error if it is in the interior of the original boundary but not in the
interior of the approximating boundary, or vice versa. It should be noted that the application of
this distortion metric in the optimization process is a stark departure from techniques proposed
previously in the literature, in which ad hoc algorithms were simply evaluated after their
execution, with equation (11.28). A variation of this definition was used in [14], where in the
numerator the area between the original boundary and its continuous approximation was used.
Regardless of the distortion criterion, in order to define the total boundary distortion, a segment distortion needs to be defined first. For that, the correspondence between a segment of the
approximating curve and a segment of the original boundary must be established. Figure 11.19
illustrates this concept. In it, the midpoints of the line segments (pu−1 , pu ) and (pu+1 , pu ),
FIGURE 11.19
Area between the original boundary segment and its spline approximation (circles).
l and m, respectively, are associated with the points of the boundary closest to them, l and
m . When more than one boundary pixel is a candidate, we select the one with the larger
index. This ensures that the starting boundary pixel of the next segment coincides with the last
boundary pixel of the current segment. That is, the segment of the original boundary (l , m )
is approximated by the spline segment (l, m). In order to ensure that some pixels on the edge
between two adjacent distortion areas are not counted twice, we exclude points on the line
(m, m ) from being counted among the distortion pixels (shown in circles).
Let us now define by d(pu−1 , pu , pu+1 ) the segment distortion, as shown in Figure 11.19.
Based on the segment distortions, the total boundary distortion is therefore defined by
D p0 , . . . , pNP −1 =
Np
d (pu−1 , pu , pu+1 ) ,
(11.29)
u=0
where p−1 = pNp +1 = pNp = p0 and Np is the number of control points.
Rate
Consecutive control point locations along an approximating curve are decorrelated using
a second-order prediction model [9]. Each control point is described in terms of the relative
angle α it forms with respect to the line connecting two previously encoded control points, and
by the run length β (in pixels), as shown in Figure 11.20A. The range of values of the angle
α is taken from the set {−90◦ , −45◦ , 45◦ , 90◦ }, thus requiring only 2 bits (Figure 11.20B).
The rationale for excluding the angle of 0◦ is that that orientation is unlikely, since it can be
© 2001 CRC Press LLC
FIGURE 11.20
Encoding of a spline control point.
achieved by properly placing the preceding control point, thus with fewer bits. The exception
to this scheme is the encoding of the first and second control points, for which predictive coding
does not exist. They are encoded in an absolute fashion and with a 3-bit angle, respectively.
To ensure that a closed contour is approximated by a closed contour, we force the control point
band of the last boundary pixel to collapse to just the boundary pixel itself, thereby making the
approximating curve pass through it. It should be noted, however, that, in general, this way
of encoding consecutive control points (run, angle) is somewhat arbitrary and other predictive
schemes or a different range of α could have been used without loss of generality.
If r(pu−1 , pu , pu+1 ) denotes the segment rate for representing pu+1 given control points
pu−1 , pu , then the total rate is given by
R p0 , . . . , pNP −1 =
Np −1
r (pu−1 , pu , pu+1 ) .
(11.30)
u=0
Here we do not restrict the possible locations of pu+1 , given pu−1 and pu , and let the encoder
determine the locally most efficient VLC for the vector pu+1 − pu .
DAG Solution
Having defined the distortion and the rate, we now solve the following rate-constrained
optimization problem:
min
p0 ,...,pNP −1
D p0 , . . . , pNP −1 ,
subject to:
R p0 , . . . , pNP −1 ≤ Rmax ,
(11.31)
where both the location of the control points pi and their overall number NP have to be
determined.
Let us define an incremental cost of encoding one spline segment as
w (pu−1 , pu , pu+1 ) = d (pu−1 , pu , pu+1 ) + λ · r (pu−1 , pu , pu+1 ) .
© 2001 CRC Press LLC
(11.32)
The overall Lagrangian cost function can then be written as
P −1
N
Jλ p0 , . . . , pNP −1 =
w (pu−1 , pu , pu+1 ) + w pNP −1 , pNP , pNP +1
u=1
= Jλ p0 , . . . , pNP −2 + w pNP −1 , pNP , pNP +1 .
(11.33)
This problem now can be cast as the shortest path problem in a graph with each consecutive
pair of control points playing the role of a vertex and incremental costs w() serving as the
corresponding weights [9]. The problem is then efficiently solved using the techniques of
Section 11.4.
VLC Optimization
The operationally optimal shape-coding algorithm described here can claim optimality only
with respect to the chosen representation of control points of the curve. That is, our solution
is operationally optimal when the encoding structure (run, angle) and its associated VLC are
fixed, which is equivalent to solving the following optimization problem:
∗
∗
∗
p0 , . . . , pN
=
arg
min
p
|V
LC
.
(11.34)
J
,
.
.
.
,
p
0
NP −1
λ
P −1
p0 ,...,pNP −1
Here, we take the operationally optimal approach one step further, and remove the dependency
of the ORD on an ad hoc VLC used. Our goal is to compress the source whose alphabet consists
of tuples (run, angle), exhibiting first-order dependency, close to its entropy. Therefore, the
problem can be stated as follows:
∗
∗
Jλ∗ p0 , . . . , pNP −1 ,
= arg
min
(11.35)
p0 , . . . , pN
P −1
p0 ,...,pNP −1 ;f ∈F
where the code is operationally optimal over all f belonging to the family of the probability
mass functions, F , associated with the code symbols. Hence, two problems need to be solved
jointly: the distribution model, f , and the boundary approximation based on that model.
Clearly, as f changes in the process of finding the underlying probability model, symbols
generated by the operationally optimal coder are also changed. As is typically done with such
codependent problems, the two solutions are arrived at in an iterative fashion [27]. The overall
iterative procedure is shown in Figure 11.21. Iterations begin with the optimal boundary
encoding algorithm, described in the preceding sections, compressing the input boundaries
based on some initial conditional distribution of the (runi , anglei ) symbol, conditioned on the
previously encoded runi−1 .
Having encoded the input sequence at iteration k, based on the probability mass function
f k (·), we use the frequency of the output symbols to compute f k+1 (·), and so on. It is
straightforward to show that the total Lagrangian cost is a nonincreasing function of the iteration
k. Thus each iteration brings f closer to the local minimum of Jλ (·), and f k converges to f M ,
where M is the number of the last iteration. The iterations stop when |Jλk (·) − Jλk−1 (·)| ≤ 2.
The local minimum in this context should be understood in the sense that a small perturbation
of the probability mass function f will result in increases in the cost function Jλ (·).
Shape-Encoding Results
Figure 11.22 shows the ORD curve resulting from the application of the described iterative algorithm to the SIF sequence, Kids. As shown in Figure 11.21, after convergence the
symbols were arithmetically encoded. For comparison purposes, ORD curves with no VLC
optimization and the rate-distortion performance of the baseline method, which is the most
© 2001 CRC Press LLC
FIGURE 11.21
The entropy encoder structure.
efficient method among competing algorithms in MPEG-4, are also shown. The distortion
axis represents the average of the D’s defined in (11.28) for one frame, over 100 frames. The
approach described here is by far superior to both the contour-based and pixel-based algorithms
in [9]. It also outperforms the fixed-VLC area-based approach [14] (shown with squares) and
the fixed-VLC pixel-based encoding (shown as the VLC1 and VLC3 curves).
FIGURE 11.22
Rate-distortion curves.
Figure 11.23 shows a sample frame in the sequence and the error associated with the optimal
solution. If the uncompressed boundary requires 3 bits per boundary pixel, the approximation
© 2001 CRC Press LLC
FIGURE 11.23
Shape approximation errors (white pixels).
shown is a 7.1:1 compression. It can be seen that the optimal way to encode some small objects
is not to encode them at all, as demonstrated by the space between the legs of the kid on the
left, shown in white as an erroneous area. Algorithms not optimized in the rate-distortion
sense lack the ability to discard small or noise-level objects, which is required when operating
at very low bit rates.
11.6
Conclusions
In this chapter we investigated the application of rate-distortion techniques to image and
video processing problems. The ORD theory was presented along with several useful mathematical tools commonly used to solve discrete optimization problems. We concluded by
providing several examples from different areas of multimedia research in which successful
application of RD techniques led to significant gains in performance.
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[4] Y. Fisher, Fractal Image Compression — Theory and Applications. Springer-Verlag,
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© 2001 CRC Press LLC
[5] N.P. Galatsanos, A.K. Katsaggelos, “Methods for choosing the regularization parameter
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[12] S. Lee et al., “Binary shape coding using 1-D distance values from baseline,” Proc.
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[13] G. Melnikov, “Hybrid fractal/DCT image compression algorithms using an orthogonal
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[16] G. Melnikov, A.K. Katsaggelos, “A non uniform segmentation optimal hybrid fractal/DCT image compression algorithm,” Proc. ICASSP-98, vol. 5, pp. 2573–2576, 1998.
[17] G. Melnikov, G.M. Schuster, A.K. Katsaggelos, “Simultaneous optimal boundary encoding and variable-length code selection,” Proc. ICIP-98, pp. I-256–260, Oct. 1998.
[18] R. Neff, A. Zakhor, “Matching pursuit video coding at very low bit rates,” DCC’95:
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[23] B.-B. Paul, M.H. Hayes III, “Video coding based on iterated function systems,” Proc.
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[27] D. Saupe, “Optimal piecewise linear image coding,” Proc. SPIE Conf. on Visual Communications and Image Processing, vol. 3309, pp. 747–760, 1997.
[28] G.M. Schuster, A.K. Katsaggelos, Rate-Distortion Based Video Compression, Optimal
Video Frame Compression and Object Boundary Encoding. Kluwer Academic Press,
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[30] G.M. Schuster, A.K. Katsaggelos, “A video compression scheme with optimal bit allocation between displacement vector field and displaced frame difference,” IEEE Journal
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© 2001 CRC Press LLC
Chapter 12
Transform Domain Techniques for Multimedia
Image and Video Coding
S. Suthaharan, S.W. Kim, H.R. Wu, and K.R. Rao
12.1
12.1.1
Coding Artifacts Reduction
Introduction
Block-based transform coding is used for compressing digital video that normally requires
a large bandwidth during transmission. Compression of digital video is vital for the reduction
of bandwidth for effective storage and transmission. However, this results in coding artifacts
in the decoded video, especially at low bit rates. Techniques, in either the spatial domain or
the transform domain, can be developed to reduce these artifacts. Several methods have been
proposed in the spatial domain to reduce the so-called blocking artifact. However, none of
these methods can reduce all the coding artifacts [1] at the same time. Some methods are image
enhancement techniques and others are intrinsically iterative, which makes them impossible
for real-time applications [2]. Also, they do not completely eliminate the artifact. Hence,
researchers are investigating new approaches.
The objective of the proposed approach is to present a new transform domain filtering
technique to reduce a number of coding artifacts [1] including the well-known blocking artifact.
In digital video coding, compression is achieved by first transforming the digital video from
the spatial and temporal domains into the frequency domain using the block-based discrete
cosine transform (DCT) and then applying quantization to the transform coefficients followed
by variable-length coding [3].
The DCT is a block-based transform, and at low bit rates the noise caused by the coarse
quantization of transform coefficients is visible in the form of a blocking artifact. In order
to reduce this artifact while maintaining compatibility with current video coding standards,
various spatial domain postfiltering techniques such as low-pass filtering (LPF) [4], projection
onto convex sets (POCS) [5], maximum a posteriori (MAP) [6] filters, and adaptive low-pass
filters (ALPFs) [7, 8] have been introduced.
Because the quantization of transform coefficients is the main error source for the coding
artifact at low bit rates, it would be much more effective in tackling the coding artifacts in the
transform domain than in the spatial domain. Recently, a weighted least square (WLS) [2]
method has been introduced in the transform domain to estimate the transform coefficients
from their quantized versions. It assumes the uniform probability density function for the
quantization errors and estimates their variances from the step size of the corresponding quantizer. It also estimates the variances of signal and noise separately and thus increases the
computational complexity as well as computational errors. Therefore, it is more sensible to
© 2001 CRC Press LLC
estimate the signal-to-noise ratio (SNR) as a single entity because this ratio plays a major role
in a number of image quality restoration techniques.
We have recently proposed an improved Wiener filter (IWF) [9] by estimating the SNR
in an image restoration problem. This IWF is also suitable to reduce the coding artifact in
compressed images. In this section, the IWF and the WLS is investigated and modified and
a new approach is developed in the transform domain to reduce the coding artifacts. First,
the methods WLS and approximated WLS∗ are investigated and implemented to reduce the
blocking artifacts. Second, the method IWF is further investigated and the SNR of the quantized
transform coefficients is estimated. This estimated SNR is used with the IWF to reduce the
noise in the transform coefficients. This noise is the source of coding artifacts. Reducing such
noise results in a corresponding reduction of the coding artifacts.
12.1.2
Methodology
Let us first discuss the mathematical model used in image and video transform coding
schemes. It is clear from the digital image and video coding literature that block-based transform coding can be modeled as follows:
Y = Q(T (x)) ,
(12.1)
where x, T , Q, and Y represent the input image, the discrete cosine transform, the quantization
process, and the quantized DCT coefficients, respectively.
Using this transform coding mechanism, a linear model for the quantization error can be
developed and written as follows:
n = Y − T (x) ,
(12.2)
where n is the quantization error, which introduces the signal-independent noise on the transform coefficients T (x) and results in Y . Therefore, n is often called the quantization noise. To
simplify the above linear noise model, we write it as follows:
Y =X+n,
(12.3)
where X = T (x) and n is a zero-mean additive noise introduced by the quantization process
of the DCT coefficients. Without loss of generality it can be assumed that the noise n is also
uncorrelated with transform coefficients X.
With the a priori information about n and X, the latter can be estimated from Y using the
Wiener filter technique [10, 11]. That is,
1
X̂ = I F F T 1+|n|
(12.4)
2 /2 · F F T (Y )
where F F T and I F F T represent the fast Fourier transform and inverse fast Fourier transform,
respectively. The symbol | · |2 represents the power spectrum, and the ratio |n|2 /|X|2 is called
the noise-to-signal power ratio (which is the a priori representation of the SNR).
As we can see from (12.4), computation of X-hat requires the a priori knowledge of power
spectra of the nonquantized transform coefficient (X) and the quantization noise (n). In
reality such information is rarely available. We propose two approaches to estimate the noiseto-signal power ratio as a whole and use it in (12.4) to restore the transform coefficients from
the corrupted version.
APPROACH I (IWF): Assuming Noise Power Spectrum is Known
In this approach, we use the assumption suggested by Choy et al. [2] that the quantization
noise has a uniform probability density function. Thus quantization noise variance is given
© 2001 CRC Press LLC
by σn2 = q 2 /12, where q is the known step size of the quantizer applied to the transform
coefficients X to get Y .
Since the quantization noise n is assumed zero-mean uncorrelated with X, using (12.3) we
can derive [9, 12]:
|Y |2 = |X|2 + |n|2 ,
where |n|2 = σn2 , which can be calculated from the quantization step size.
It gives
|n|2 /|X|2 = |n|2 / |Y |2 − |n|2 .
(12.5)
(12.6)
The noise-to-signal power ratio can be calculated from the power spectra of the quantized
transform coefficients Y and the quantization noise n.
Using this ratio and (12.4), we can reduce the quantization error in the transform coefficients
and in turn can reduce a number of coding artifacts including the blocking artifact in the
decompressed images.
The advantage of this proposed method over WLS is that it uses only one assumption (noise
variance), whereas WLS uses one approximation (mean of X) and one assumption (noise
variance). Because no approximation is used in our proposed approach, it gives better results
in terms of peak signal-to-noise ratio (PSNR) as well as visual quality.
APPROACH II (IWF*): Estimate the Noise-to-Signal Power Ratio
In this approach, we propose a technique to estimate the noise-to-signal power ratio as a
whole and then use it in the Wiener filter equation (12.4). We recently proposed the IWF
technique to handle the image restoration problem [9]. In this technique, we do not need the
a priori information about the noise-to-signal power ratio in order to apply the Wiener filter;
instead, it is estimated from the given degraded image. It has been successfully used in the
image restoration problem. We use this approach to remove the coding artifacts introduced by
the coarse quantization of DCT coefficients.
The IWF method needs two versions of an image so that the noise-to-signal power ratio can be
estimated. In digital video, we have a sequence of images (frames) and the consecutive frames
have very few differences, except when the scene changes. Therefore, the decoded frames can
be different because of the different quantization scalers used and thus the quantization error
(noise) can be different. Let us assume that the DCT coefficients of the ith frame are to be
restored from its quantization noise; then we can model the quantized DCT coefficients of the
ith and (i + 1)th frames as follows [refer to (12.3)]:
Y i = X + ni
Yi+1 ≈ X + ni+1
(12.7)
The restriction of the method is that the adjacent frames cannot be quantized with the same
quantization scaler. In the above equations, it has been assumed that there is no scene change
and thus the approximation of the second equation is valid. Therefore, Yi+1 cannot be used
when the scene changes. To overcome this problem, we suggest using the previous frame as
the second one when the scene changes; thus, we can write:
Yi = X + ni
Yi−1 ≈ X + ni−1
(12.8)
To implement IWF we need only two versions of an image, and in digital video coding we
can have two images using either the previous frame or the next frame for the second image as
© 2001 CRC Press LLC
shown in (12.7) and (12.8). In case of scene changes on the previous and the next frames, we
can construct the second frame (i + 1 or i − 1) from the ith frame using the methods discussed
in [10]. If we represent the quantized DCT coefficients of the frame (that is to be restored) by
Y1 and the second one by Y2 , then we have
Y1 = X + n 1
Y2 ≈ X + n 2
(12.9)
In video compression, the quantization scaler is used as a quantization parameter, and it is
valid to assume a linear relationship between the quantization noises n1 and n2 . Thus, the
linear relationship n2 = an1 is acceptable (where a is a constant).
Therefore, equation (12.9) can be written as follows:
Y1 = X + n 1
Y2 ≈ X + a · n 1
(12.10)
From these two equations and using the definition of a power spectrum [12], we can easily
derive
|Y1 − Y2 |2 ≈ (1 − a)2 |n1 |2
|Y2 − a · Y1 |2 ≈ (1 − a)2 |X|2
(12.11)
Dividing the first equation by the second, the noise-to-signal power ratio of Y1 can be approximated as follows:
|Y1 − Y2 |2
|n1 |2
≈
,
2
|X|
|Y2 − a · Y1 |2
(12.12)
where the constant a can be calculated during the encoding process as follows:
a=
V ar(Y2 − X)
,
V ar(Y1 − X)
(12.13)
and a single value for each frame can be transmitted to the decoder. Although an extra overhead
on the bit rate is created, compared to the PSNR improvement and visual quality of the images,
it is marginal. This overhead can be reduced by transmitting the differences in a.
On the other hand, to avoid this overhead, the constant a can be approximated to the corresponding ratio calculated from the decoded versions of Y1 and Y2 as follows:
a=
V ar(decoded(Y2 ))
.
V ar(decoded(Y1 ))
(12.14)
The Wiener filter in (12.4), replacing Y by Y1 and n by n1 along with the above expressions,
has been used to restore the transform coefficients X of Y1 .
The important point to note here is that the proposed methods have been used in the transform
domain to reduce the noise presented in the quantized DCT coefficients. This in turn reduces
the blocking artifacts in the decompressed images. Also, note that if we cannot assume that the
consecutive frames have very little differences, we can still use equation (12.7) by constructing
the second image from the first according to the method discussed in [9, 10] by Suthaharan,
and thus equation (12.9) is valid.
© 2001 CRC Press LLC
12.1.3
Experimental Results
In the simulation, the proposed methods (IWF and IWF∗ ) have been implemented using a
number of test video sequences, which include Flower Garden, Trevor, Footy, Calendar, and
Cameraman. We show here the effectiveness of the proposed methods with the images of
Trevor, Footy, and Cameraman. Figure 12.1 displays the original Trevor image, its MPEG
coded image, and the processed images by the WLS method and IWF∗ method, respectively.
In Figure 12.1b the coding artifacts can be clearly seen. Although the images in Figure 12.1c
and 12.1d look similar, a closer look on a region-by-region (and along the edges) basis shows
that many of the coding artifacts have been removed in Figure 12.1d, and it is supported by the
higher PSNR value. Figures 12.2 and 12.3 show similar results for the Footy and Cameraman
images. All these figures show that the proposed methods significantly reduce the coding
artifacts by removing some noise introduced by the quantization process. This improves the
PSNR by more than 1 dB and in turn improves the visual quality of the images. Note that in
our simulation we have used the MPEG quantizer with an 8 × 8 block-based DCT [13].
It is clear from [2] that Choy et al.’s proposed filter WLS gives better PSNR values than the
LPF [4] and POCS filter [5]. In our experiment, we have compared the results of IWF and
IWF∗ with those of WLS and WLS∗ , respectively (Table 12.1).
Table 12.1 PSNR Improvements of the Images Reconstructed Using
WLS and IWF
MPEG
Encoded
Images
Flower Garden
Trevor
Footy
Calendar
Cameraman
bpp
PSNR (dB)
0.3585
0.1537
0.2325
0.3398
0.2191
23.1401
31.8739
26.4812
22.4223
27.1588
PSNR Improvements over POCS [5]
WLS
WLS∗
IWF
IWF∗
0.9668
0.8074
0.9769
0.8777
0.5485
0.5317
0.6009
0.5958
0.6232
0.3568
1.1439
1.0338
1.0324
1.2363
1.1002
1.2537
1.0239
1.0253
1.1867
0.9100
From the PSNR improvements shown in Table 12.1, we can conclude that the proposed
methods give better restoration of the transform coefficients than those of the WLS and WLS∗
methods and yield a better visual quality of images.
It is evident from our simulation that our proposed methods restore the transform coefficients
from the quantized versions and, thus, they can reduce a number of coding artifacts, such as
1. Blocking artifact: This is due to coarse quantization of the transform coefficients. The
blocking artifacts can be clearly seen in all the images.
2. DCT basis images effect: This is due to the appearance of DCT basis images. For
example, it can be seen in certain blocks on the background of the Footy image and
ground of the Cameraman image. In the filtered images of Footy and Cameraman, this
effect has been reduced.
3. Ringing effect: This is due to quantization of the AC DCT coefficients along the highcontrast edges. It is prominent in the images of Trevor and Cameraman along the arm
and shoulder, respectively, and in the filtered images it has been reduced.
4. Staircase effect: This is due to the appearance of DCT basis images along the diagonal
edges. It appears due to the quantization of higher order DCT basis images and fails to
be muted with lower order basis images.
© 2001 CRC Press LLC
FIGURE 12.1
Images of Trevor. (a) Original image; (b) MPEG coded image (0.1537 bpp, 31.8739 dB);
(c) processed by WLS (32.6813 dB); (Continued).
© 2001 CRC Press LLC
FIGURE 12.1
(Cont.) Images of Trevor. (d) Processed by the proposed IWF∗ method (32.9178 dB).
12.1.4
More Comparison
In this section we have compared our proposed techniques with a recently published blocking
artifact reduction method proposed by Kim et al. [14]. In their method the artifact reduction
operation has been applied to only the neighborhood of each block boundary in the wavelet
transform at the first and second scales. The technique removes the blocking component that
reveals stepwise discontinuity at block boundaries. It is a blocking artifact reduction technique
and does not necessarily reduce the other coding artifacts mentioned above. It is evident from
the images in Figure 12.4 that this method still blurs the image (Figure 12.4c) significantly and
thus some edge details that are important for visual perception have been lost.
We have used the JPEG-coded Lena image provided by Kim et al. [14] to compare our
results. This Lena image is JPEG coded with a 40:1 compression ratio. The enlarged portion
of the original and JPEG-coded Lena images are given in Figures 12.4a and b. In Figure 12.4c,
the processed image of Figure 12.4b by the method proposed by Kim et al. is presented. As
can be seen in Figure 12.4c, their proposed method still blurs the image significantly and thus
the sharpness of the image is lost. In addition, there are a number of other obvious problems:
(1) the ringing effect along the right cheek edge, (2) the blurred stripes on the hat, and (3) the
blurred edge between the hat and the forehead, to name just a few. The image processed
by our proposed method (IWF∗ ) is presented in Figure 12.4d. In this image we can clearly
see the sharpness of the edges, while reducing a number of coding artifacts, and an overall
improvement in the visual quality of the image.
12.2
12.2.1
Image and Edge Detail Detection
Introduction
The recent interest of the Moving Pictures Expert Group (MPEG) is object-based image representation and coding. Compared to the conventional frame-based compression techniques,
the object-based coding enables MPEG-4 to cover a wide range of emerging applications including multimedia. The MPEG-4 supports new tools functionality not available in existing
standards.
© 2001 CRC Press LLC
FIGURE 12.2
Images of Footy. (a) Original image; (b) MPEG coded image (0.2325 bpp, 26.4812 dB);
(c) processed by WLS (27.4581 dB); (Continued).
© 2001 CRC Press LLC
FIGURE 12.2
(Cont.) Images of Footy. (d) Processed by the proposed IWF∗ method (27.6095 dB).
One of the important tools needed to enhance and broaden its applications is to introduce
effective methods for image segmentation. Image segmentation techniques not only enhance
the MPEG standards but also are needed for many computer vision and image processing
applications. The goal of image segmentation is to find regions that represent objects or
meaningful parts of objects. Therefore, image segmentation methods will look for objects that
either have some measure of homogeneity within them or have some measure of contrast with
the objects or their border.
In order to carry out image segmentation we need effective image and edge detail detection
and enhancement algorithms. Since edges often occur at image locations representing object
boundaries, edge detection is extensively used in image segmentation when we want to divide
the image into areas corresponding to different objects. The first stage in many edge detection
algorithms is a process of enhancement that generates an image in which ridges correspond to
statistical evidence for an edge [15]. This process is achieved using linear operators, including
Roberts, Prewitt, and Canny. These are called edge convolution enhancement techniques.
They are based on convolution and are suitable for detecting the edges for still images. These
techniques do not adapt any visual perception properties, but use only the statistical behavior
of the edges. Thus, they cannot detect the edges that might contribute to the edge fluctuations
and coding artifacts that could occur in the temporal domain [16, 17].
In this section a transform domain technique has been introduced to detect image and edge
details that are suitable for image segmentation and reduction of coding artifacts in digital
video coding. This method uses the perceptual properties and edge contrast information of
the transform coefficients and, thus, it gives meaningful edges that are correlated with human
visual perception. Also, the method allows users to select suitable edge details from different
levels of edge details detected by the method for different applications in which human visual
quality is an important factor.
12.2.2
Methodology
Let us consider an image I of size N1 n × N2 n, where I is divided into N1 × N2 blocks
with each block having n × n pixels. In transform coding, a block is transformed into the
transform domain using a two-dimensional separable unitary transform such as the DCT, and
this process can be expressed by
Y = T ∗ X ∗ T
© 2001 CRC Press LLC
(12.15)
FIGURE 12.3
Images of Cameraman. (a) Original image; (b) MPEG coded image (0.2191 bpp,
27.1588 dB); (c) processed by WLS (27.8365 dB); (Continued).
© 2001 CRC Press LLC
FIGURE 12.3
(Cont.) Images of Cameraman.
(28.0718 dB).
(d) Processed by the proposed IWF∗ method
where X and Y represent a block of I and its transform coefficients, respectively; T is the
n × n unitary transform matrix; and T represents the transpose of the matrix T . The operator
∗ represents the matrix multiplication. The 8 × 8 block-based DCT is used in the experiment,
and it is also the one used in the many international image and video coding standards.
Let a be the DC coefficient of X and U be the image that is obtained using the AC coefficients
of X and zero DC coefficient. Then we have the following:
n
a=
n
1 x(i, j )
n
(12.16)
i=1 j =1


1...1
a
X = T ∗ U ∗ T + . . . . . .
n 1...1
(12.17)
n×n
where x(i, j ) is the (i, j )th intensity value of image block X. It is known that two-dimensional
transform coefficients, in general, have different visual sensitivity and edge information. Thus,
transform coefficients in U can be decomposed into a number of regions based on frequency
level and edge structures, and they are called frequency distribution decomposition and structural decomposition, respectively [3] (see Figure 12.5). Using these regions of transform
coefficients, we treat the edge details corresponding to the low- (and medium) frequency
transform coefficients separately from the edge details corresponding to the high-frequency
coefficients.
It is clear from a visual perception viewpoint that the low- (and medium) frequency coefficients are much more sensitive than the high-frequency coefficients. In our proposed algorithm,
we use this convention and separate the edge details falling in the low- (and medium) frequency
coefficients from the edge details falling in the high-frequency coefficients. To carry out this
task, let us first define a new image block X1 from the transform coefficients of X using the
following equation, similar to equation (12.17):


1...1
α
·
a
. . . . . .
(12.18)
X1 = T ∗ (L · ∗U ) ∗ T +
n
1...1
n×n
where L is called an edge-enhancement matrix and α can be chosen to adjust the DC level of
X1 to obtain different levels of edge details with respect to the average intensity of the block
© 2001 CRC Press LLC
FIGURE 12.4
Lena image coded by JPEG with a 40:1 compression ratio and the processed images.
(a) Original image; (b) coded image; (c) processed by Kim et al. method; (Continued).
© 2001 CRC Press LLC
FIGURE 12.4
(Cont.) Lena* image coded by JPEG with a 40:1 compression ratio and the processed
images. (d) Processed by our proposed method(*Copyright © 1972 by Playboy magazine).
Vertical edges
Low Frequency
Medium
Frequency
Horizontal
Edges
(a)
Diagonal Edges
(b)
High Frequency
FIGURE 12.5
Decomposition of transform coefficients. (a) Structural decomposition based on edge
details; (b) frequency distribution decomposition.
X1 . The operator · ∗ represents the element-by-element matrix multiplication as defined in
the MATLAB package [18].
Selection of L and α is up to the user’s application and thus it forms a flexible algorithm.
Suitable selection of L and α for different applications is leading to new research. In this
approach we used the JPEG-based quantization table as the edge-enhancement matrix, and it
© 2001 CRC Press LLC
is given in the following equation:

α
 60

 70

 70
L=
 90

120

255
255
60
60
70
96
130
255
255
255
70
70
80
120
200
255
255
255
70
96
120
145
255
255
255
255
90
130
200
255
255
255
255
255
120
255
255
255
255
255
255
255
255
255
255
255
255
255
255
255

255
255

255

255

255

255

255
255
(12.19)
One can select different weighting for the edge-enhancement matrix L. During the selection
of weights L(i, j ) we treat DC and AC coefficients differently, because the DC coefficient
contains information about an average (spatial) of the intensity values of an image block
whereas the AC coefficients contain information about the edges in an image. Using the
above L and α = 1, the proposed method can easily identify vertical, horizontal, and diagonal
edges that are vital to improving image quality with respect to visual perception. Significant
separation of these two regions using appropriate L can cause intensity of certain edge details
(based on their contrast) in the low- (and medium) frequency coefficients to be pushed down
below zero while keeping the intensity of the edge details in the high-frequency coefficients
above zero.
In order to get different levels of edge details based on their contrast, the α can be adjusted
above 1. By increasing α from 1, the average edge intensity can be lifted and low- (and
medium) contrast edges can be lifted above zero while keeping the high-contrast edge details
below zero. Thus, we can obtain different levels of edge detail with respect to the average
edge details in a block.
12.2.3
Experimental Results
Experiments were carried out using a number of images to evaluate the performance of the
proposed method; however, only the results of the Cameraman and Flower Garden images are
given in this chapter. In Figures 12.6a and 12.7a the original images of the Cameraman and
Flower Garden are given, respectively.
In Figures 12.6b and 12.7b, the edge details are obtained from the images in Figures 12.6a
and 12.7a using the proposed method with the edge-enhancement matrix L in equation (12.19)
and α = 1. We see from these images that the proposed method detects the edges and also
enhances much of the image details. For example, in the Cameraman image we can see gloves
details, pockets in the jacket, windows in the tower, and so forth. Similarly, in the Flower
Garden image we can see roof textures, branches in the trees, and many other sensitive details.
The images in Figures 12.6c and d and 12.7c and d show the results with the same L in
equation (12.19) but now with different α = 25 and 50, respectively. This proves that the
increase in α leaves only high-contrast edge details. We can see from the Cameraman image
that the proposed method can even highlight the mouth edge detail.
In general, edge detection algorithms give edge details that are suitable for determining
the image boundary for segmentation. The proposed method not only gives edge details for
segmentation but also provides a flexible (adaptive to the DC level) edge detail identification
algorithm that is suitable for coding artifact reduction in the transform-coding scheme and
allows user control on the DC level to obtain different levels of edge details for different
applications.
© 2001 CRC Press LLC
FIGURE 12.6
Images of Cameraman. (a) Original image; (b) edge details with L and α = 1; (c) edge
details with L and α = 25; (Continued).
© 2001 CRC Press LLC
FIGURE 12.6
(Cont.) Images of Cameraman. (d) Edge details with L and α = 50.
FIGURE 12.7
Images of Flower Garden. (a) Original image; (b) edge details with L and α = 1;
(Continued).
© 2001 CRC Press LLC
FIGURE 12.7
(Cont.) Images of Flower Garden. (c) Edge details with L and α = 25; (d) edge details
with L and α = 50.
12.3
Summary
In Section 12.1, we introduced two approaches in the transform domain to estimate the
quantization error (noise) in DCT coefficients. The IWF method uses the Wiener filter assuming
uniform quantization noise, whereas the IWF∗ uses the improved Wiener filter in the transform
domain in order to correct the quantization error that causes the coding artifacts in the decoded
images. The advantage of these proposed methods over WLS and WLS∗ is that they use a
lesser number of approximations or assumptions compared to WLS and WLS∗ and give better
results even for low-bit-rate coded images. The IWF∗ method does not directly depend on the
noise characteristics and thus can be used for any type of noise. The proposed methods give
better results than the other two methods in terms of PSNR, bit rate, and visual image quality.
In addition, they also give better results than the method recently introduced by Kim et al. [14].
In Section 12.2, a new approach was proposed to identify image and edge details that are suitable for image segmentation and coding artifacts reduction in digital image and video coding.
In contrast to existing methods, which are mainly edge convolution enhancement techniques
and use statistical properties of the edges, the proposed method uses human perceptual information and edge details in the transform coefficients. The proposed method is suitable for many
applications, including low-level image processing, medical imaging, image/video coding and
transmission, multimedia image retrieval, and computer vision. It identifies the edge details
© 2001 CRC Press LLC
based on the human visual properties of the transform coefficients, and these edges can contribute to the temporal edge fluctuations in digital video coding, which can lead to unpleasant
artifacts. Identification of such edge details can lead to an early fix during encoding.
Acknowledgement
We thank Nam Chul Kim (Visual Communication Laboratory, Dept. of Electrical Engineering, Kyungpook National University, Korea, who has kindly given us the simulation results of
the JPEG-coded Lena images for comparison.
References
[1] M. Yuen and H.R. Wu, “A survey of hybrid MC/DPCM/DCT video coding distortions,”
Signal Processing EURASIP J., vol. 70, pp. 247–278, Oct. 1998.
[2] S.S.O. Choy, Y.-H. Chan, and W.-C. Siu, “Reduction of block-transform image coding
artifacts by using local statistics of transform coefficients,” IEEE Signal Processing Lett.,
vol. 4, no. 1, Jan. 1997.
[3] K.R. Rao and J.J. Hwang, Techniques and Standards for Image, Video, and Audio Coding,
Prentice-Hall, Englewood Cliffs, NJ, 1996.
[4] H.C. Reeves and J.S. Lim, Reduction of blocking effects in image coding, Opt. Eng.,
vol. 23, pp. 34–37, Jan./Feb. 1984.
[5] Y. Yang, N.P. Galatsanos, and A.K. Katsaggelos, “Regularized reconstruction to reduce
blocking artifacts of block discrete cosine transform compressed images,” IEEE Trans.
CSVT, vol. 3, pp. 421–432, Dec. 1993.
[6] R.L. Stevenson, “Reduction of coding artifacts in transform image coding,” Proc.
ICASSP, vol. 5, pp. 401–404, 1993.
[7] S. Suthaharan and H.R. Wu, “Adaptive-neighbourhood image filtering for MPEG-1
coded images,” Proc. ICARCV’96: Fourth Int. Conf. on Control, Automation, Robotics
and Vision, pp. 1676–1680, Singapore, Dec. 1996.
[8] S. Suthaharan, “Block-edge reduction in MPEG-1 coded images using statistical inference,” 1997 IEEE ISCAS, pp. 1329–1332, Hong Kong, June 1997.
[9] S. Suthaharan, “New SNR estimate for the Wiener filter to image restoration,” J. Electronic Imaging, vol. 3, no. 4, pp. 379–389, Oct. 1994.
[10] S. Suthaharan, “A modified Lagrange’s interpolation for image restoration,” Austr. J.
Intelligent Information Processing Systems, vol. 1, no. 2, pp. 43–52, June 1994.
[11] J.M. Blackledge, Quantitative Coherent Imaging, Academic Press, New York, 1989.
[12] J.S. Lim, Two-Dimensional Signal and Image Processing, Prentice-Hall, Englewood
Cliffs, NJ, 1990.
© 2001 CRC Press LLC
[13] Secretariat ISO/IEC JTC1/SC29, ISO CD11172-2, “Coding of moving pictures and
associated audio for digital storage media at up to about 1–5 Mbit/s,” MPEG-1, ISO,
Nov. 1991.
[14] N.C. Kim, I.H. Jang, D.H. Kim, and W.H. Hong, “Reduction of blocking artifact in blockcoded images using wavelet transform,” IEEE Trans. CSVT, vol. 8, no. 3, pp. 253–257,
June 1998.
[15] S.E. Umbaugh, Computer Vision and Image Processing: A Practical Approach Using
CVIP Tools, Prentice-Hall, Englewood Cliffs, NJ, 1998.
[16] S.M. Smith and J.M. Brady, “SUSAN — a new approach to low level image processing,”
Int. J. Computer Vision, vol. 23, no. 1, pp. 45–78, May 1997.
[17] I. Sobel, “An isotropic 3 × 3 image gradient operator.” In Machine Vison for ThreeDimensional Scenes, H. Freeman, ed., pp. 376–379, Academic Press, New York, 1990.
[18] MATLAB (version 5), “Getting started with MATLAB,” The Math Works Inc., 1996.
© 2001 CRC Press LLC
Chapter 13
Video Modeling and Retrieval
Yi Zhang and Tat-Seng Chua
13.1
Introduction
Video is the most effective media for capturing the world around us. By combining audio
and visual effects, it achieves a very high degree of reality. With the widely accepted MPEG
(Moving Pictures Expert Group) [12] digital video standard and low-cost hardware support,
digital video has gained popularity in all aspects of life.
Video plays an important role in entertainment, education, and training. The term “video” is
used extensively in the industry to represent all audiovisual recording and playback technologies. Video has been the primary concern of the movie and television industry. Over the years,
that industry has developed detailed and complete procedures and techniques to index, store,
edit, retrieve, sequence, and present video materials. The techniques, however, are mostly
manual in nature and are designed mainly to support human experts in creative moviemaking.
They are not set up to deal with the large quantity of video materials available. To manage
these video materials effectively, it is necessary to develop automated techniques to model and
manage large quantities of videos.
Conceptually, the video retrieval system should act like a library system for the users. In the
library, books are cataloged and placed on bookshelves according to well-defined classification
structures and procedures. All the particular information about a book, such as the subject area,
keywords, authors, ISBN number, etc., are stored to facilitate subsequent retrievals. At the
higher level, the classification structure acts like a conceptual map in helping users to browse
and locate related books. Video materials should be modeled and stored in a similar way for
effective retrieval. A combination of techniques developed in the library and movie industries
should be used to manage and present large quantities of videos.
This chapter discusses the modeling of video, which is the representation of video contents
and the corresponding contextual information in the form of a conceptual map. We also cover
the indexing and organization of video databases. In particular, we describe the development
of a system that can support the whole process of logging, indexing, retrieval, and virtual
editing of video materials.
© 2001 CRC Press LLC
13.2
Modeling and Representation of Video: Segmentation vs.
Stratification
There are many interesting characters, events, objects, and actions contained in a typical
video. The main purpose of video modeling is to capture such information for effective
representation and retrieval of video clips. There are two major approaches to modeling video,
namely, the segmentation [5, 15] and stratification [1] approaches.
In the segmentation approach, a video sequence is segmented into physical chunks called
shots. By definition, a video shot is the smallest sequence of frames that possesses a simple
and self-contained concept. In practice, video shots are identified and segmented using scene
change detection techniques [21]. A change in scene will mark the end of the previous shot and
the start of the next one. Figure 13.1 shows the way in which a video sequence is segmented
into shots.
Title:
Description:
Color Histogram:
Objects:
shot-1
shot-2
shot-3
... ... ... ... ...
shot-n
Segmented video shots
FocalLength:
TypeofScope:
TypeofAngle:
TypeofSpatial:
TypeofMovement:
FIGURE 13.1
Segmentation: modeling and representation of video shots.
In the segmentation approach, a video shot is an atomic unit that can be manipulated for
representation and retrieval. Figure 13.1 also shows how a video shot can be represented.
Important attributes such as title, description, audio dialogues, color histogram, and visual
objects should be extracted and logged. In addition, we can capture cinematic information [8],
such as the focal length, scope, and angle of shots, for the purpose of sequencing video shots
for presentation. By combining all the attributes mentioned above, a good representation of
video shots can be achieved.
Although the segmentation approach is easy to understand and implement, it has several
drawbacks. In such a model, the granularity of the information is a shot which typically contains
a self-contained but high-level concept. Once the video is segmented, the shot boundaries
are fixed and it is not possible for users to access or present video contents within the shot
boundaries. This inflexibility limits the users’ ability to model more complicated events in
a way not originally intended by the authors. Because of the diverse contents and meaning
inherited in the video, certain events are partially lost. For example, a shot may contain multiple
characters and actions; it is thus not possible for authors to anticipate all the users’ needs
and log all interesting events. Moreover, the segment boundaries may be event dependent.
This may result in a common story being fragmented across multiple shots, which will cause
discontinuity to users looking for the story. For these reasons, automated description of video
contents at the shot level is generally not possible.
To overcome this problem, we investigate an object-based video indexing scheme to model
the content of video based on the occurrences of simple objects and events (known as entities).
© 2001 CRC Press LLC
This is similar to the stratification approach proposed in [1]. In this scheme, an entity can be
a concept, object, event, category, or dialogue that is of interest to the users. The entities may
occur over many, possibly overlapping, time periods. A strand of an entity and its occurrences
over the video stream are known as a stratum. Figure 13.2 shows the modeling of a news video
with strata such as the object “Anchor-Person-A”; categories like “home-news,” “international
news,” “live reporting,” “finance,” “weather,” and “sports”; and audio dialogues. The meaning
of the video at any time instance is simply the union of the entities occurring during that time,
together with the available dialogue.
FIGURE 13.2
Object-based modeling of a news video.
Object-based modeling of video has several advantages. First, because strata tend to contain
only simple entities, the pseudo-object models together with relevance feedback (RF) [7] and
other learning methods may be adopted to identify and track entities automatically. Second,
the meaning of video at any instance is simply the union of strata occurring at that time.
Thus the system can flexibly compose portions of video sequences whose meanings closely
match that of the query. Finally, the strata information provides the meta-information for the
video. Such information can be used to support innovative functions such as the content-based
fast-forwarding and summarization [7] of video.
13.2.1
Practical Considerations
The main advantage of the stratification approach over segmentation is that it is possible
to automate the process of indexing. However, when the project was started several years
ago, this advantage was not realizable because of the lack of content-based analysis tools for
automated indexing. Without such tools, it is extremely tedious to index video content using
the stratification approach. Because of this, coupled with the simplicity of the segmentation
approach in supporting our initial research goals in video retrieval, virtual editing, and summarization, we adopted the segmentation approach. This project is an extension of the work
done in Chua and Ruan [5] using digital video with more advanced functionalities for video
manipulation, browsing, and retrieval. We are currently working on the development of a
system based on the stratification approach to model video in the news domain.
This chapter describes the design and implementation of our segmentation model for video
retrieval. The domain of application is documentary videos.
© 2001 CRC Press LLC
13.3
Design of a Video Retrieval System
This section considers the modeling of video using the segmentation approach. There are
several processes involved in modeling video sequences to support retrieval and browsing.
The two main steps are video parsing and indexing. The term parsing refers to the process of
segmenting and logging video content. It consists of three tasks: (1) temporal segmentation
of video material into elementary units called segments or shots; (2) extraction of content
from these segments; and (3) modeling of context in the form of a concept hierarchy. The
indexing process supports the storage of extracted segments, together with their contents and
context, in the database. Retrieval and browsing rely on effective parsing and indexing of raw
video materials. Figure 13.3 summarizes the whole process of video indexing, retrieval, and
browsing.
Raw video material
Parsing Tool
Video shots or strata
Indexing tool
Vid eo DB
Retrieval & browsing tool
FIGURE 13.3
Parsing and indexing video for retrieval and browsing.
13.3.1
Video Segmentation
The success of the segmentation approach depends largely on how well the video materials
are divided into segments or shots. This involves identifying suitable criteria to decide which
points in a video sequence constitute segment boundaries. Manual segmentation has been
done effectively in the movie industry. However, manual segmentation is time consuming and
prone to error. Moreover, its results are biased toward the authors’ intents.
With advances in image processing techniques, computer-aided segmentation is now possible [21]. The main objective is to detect the joining of two shots in the video sequence and
© 2001 CRC Press LLC
locate the exact position of these joins. These joins are made by the video editing process, and
they can be of the following types based on the techniques involved in the editing process:
• Abrupt cut
• Dissolve
• Fade in/fade out
• Curtain and circle
In an abrupt cut, the video editor does nothing but simply concatenate the two shots together.
The other three joins are generally known as gradual transitions, in which the video editor uses
a special technique to make the join appear visually smooth.
Thus, segmentation becomes a matter of finding those features that constitute transitions. A
lot of work has been done on detecting abrupt cuts, and many techniques have been developed
to handle gradual transitions in both the compressed and the decompressed domain [11, 13, 19].
We detect the cut transition by sequentially measuring successive inter-frame differences based
on features such as the color histogram. When the difference is above the global threshold, a
cut transition is declared.
When dealing with gradual transitions, Zhang et al. [21] used two thresholds. They computed accumulated differences of successive frames when the inter-frame difference was above
a lower threshold. When this accumulated difference exceeded the high threshold, a gradual
transition was declared. Other approaches employ template matching [20], model-based [10],
statistical [18], and feature-based [19] methods. Most of the methods employed for detecting
gradual transitions require careful selection of the threshold for the method to work effectively [11]. This is a difficult problem. Lin et al. [14] proposed a multi-resolution temporal
analysis approach based on wavelet theory to detect all transitions in a consistent manner.
In this chapter, we employ the Zhang et al. [21] approach of computing accumulated differences of color histogram in successive frames to detect both abrupt cut and gradual transitions.
We employ this approach because it is simple to implement and has been found to work well.
Our tests show that it could achieve a segmentation accuracy of greater than 80%. Based on
the set of segments created, a visual interface is designed to permit the authors to review and
fine-tune the segment boundaries. The interface of the Shot Editor is given in Figure 13.9.
The resulting computer-aided approach has been found to be satisfactory.
13.3.2
Logging of Shots
After the video is segmented, each shot is logged by analyzing its contents. Logging is the
process of assigning meanings to the shots. Typically, each shot is manually assigned a title
and text description, which are used in most text-based video retrieval systems. Increasingly,
speech recognition tools are being used to extract text from audio dialogues [16], and contentbased analysis tools are being used to extract pseudo-objects from visual contents [6]. These
data are logged as shown in Figure 13.1. The combination of textual and content information
enables different facets of content to be modeled. This permits more accurate retrieval of the
shots.
Because video is a temporal medium, in addition to retrieving the correct shots based on
the query, the shots retrieved must be properly sequenced for presentation. Thus, the result
of a query should be a dynamically composed video sequence, rather than a list of shots. To
do this, we need to capture cinematic information [8]. Typical information that is useful for
sequencing purposes includes the focal length and angle of shots. Such information permits
the sequencing of video by gradually showing the details (from far shots to close-up shots), a
© 2001 CRC Press LLC
typical presentation technique used in documentary video. The complete information logged
for the shot thus includes both content and cinematic information, as shown in Figure 13.1.
Video is a temporal medium that contains a large amount of other time-related information.
This includes relationships between objects and motion, the order in which main characters
appear, and camera motions/operations. However, there is still a lack of understanding of how
object motions are to be queried by users. Also, automated detection of motion and object
relationships is not feasible with the current technology. These topics are open for research
and exploration. Thus, temporal information is not included as part of the shot representation.
13.3.3
Modeling the Context between Video Shots
In the segmentation approach, although the information logged provides a good representation for individual video shots, the context information between video shots is not properly
represented. Without contextual information, it is hard to deduce the relationships between
shots. For example, in the domain of an animal, the contextual information permits different
aspects of the life of an animal, say a lynx, to be linked together (see Figure 13.10). It also
provides high-level information such as the knowledge that lion and lynx belong to the same
cat family. The context information can be captured in the structured modeling approach using
a two-layered model as shown in Figure 13.4.
Scene layer
Shot layer
FIGURE 13.4
A two-layered model for representing video context.
The layered model consists of a lower shot layer and the scene layer. The shot layer contains
all the shots indexed, whereas the scene layer models the video context information. The
context information can be modeled as a scene hierarchy as shown in Figure 13.4. Maintaining
the context information only at the scene layer permits more than one scene hierarchy to be
modeled above the same set of shots. This facilitates multiple interpretation of the shot layer
and leads to a more flexible creation of scenes.
Typical information encoded at the scene includes the title and textual descriptions, together
with parent–child relations. Because the scene hierarchy captures the knowledge of the set of
video shots, it can be used to support concept-based retrieval of the video shots. A retrieval of
a set of video shots can be viewed as the retrieval of an appropriate scene (or concept) in the
scene hierarchy. This will be illustrated in a later section.
© 2001 CRC Press LLC
13.4
13.4.1
Retrieval and Virtual Editing of Video
Video Shot Retrieval
The shots and scenes permit the video to be represented in a form that can be stored and
manipulated by the computer. Because the main semantic information captured for the shot is
the title, text descriptions, and dialogues, we store these as free text so that a free-text retrieval
technique can be employed to retrieve video shots using free-text queries. Conceptually,
each shot is stored as a free-text document in an inverted file. We employ the vector space
information retrieval (IR) model [17] to retrieve the shots. In the vector space IR model, each
shot is represented as a vector of dimension t of the form:
S i = (di1 di2 di3 . . . dit )
(13.1)
where dij is the weight of term j in the ith shot Si , and t is the number of text terms used. By
representing the query Q in the same way, that is,
Q = (q1 q2 q3 . . . qt )
(13.2)
the similarity between query Q and shot S i can be computed using the cosine similarity
formula [17], given by:
t
k=1 (dik ∗ qk )
(13.3)
Sim S i , Q = t
2 t
2
∗
(d
)
(q
)
k=1 ik
k=1 k
Equation (13.3) is used as the basis to rank all shots with respect to the query.
13.4.2
Scene Association Retrieval
To permit more effective retrieval of shots, the knowledge built into the scene structure
should be used. The scene is used to link shots semantically related to each other under the
same scene hierarchy. Many of these shots may not have common text descriptions and thus
may not be retrievable together at the same time. To improve higher retrieval accuracy, the
idea here is to retrieve higher level scenes as much as possible so that these related shots may
be retrieved by the same query. A scene association algorithm is developed for this purpose.
The algorithm is best illustrated using Figure 13.5.
After the user issues a query, the scene association algorithm proceeds as follows:
1. Shot retrieval: First we compute the similarities between the query and all the shots
in the shot layer using equation (13.3). The shots are ranked based on the similarity
values assigned to them. The list of shots whose similarity values are above a predefined
threshold is considered to be relevant to the query. This is known as the initial relevant
shot list. Shots that appear on the initial relevant shot list are assigned a value of 1;
all other shots are assigned a value of 0. In the example given in Figure 13.5, shots
S1 , S3 , S4 , S7 , and S8 are given a value of 1 because they are on the initial relevant shot
list. Shots S2 , S5 , S6 , and S9 are assigned a value of 0 because they are not.
2. Association: We then compute a similarity value for each leaf node in the scene structure.
This is done by computing the percentage of its children that are assigned the value of 1.
For example, scene leaf node C4 is given a value of 1/3.
© 2001 CRC Press LLC
C1
23/36 = 0.64
3/ 4
2/3
C2
1/2
C3
1/3
S1
1
S2
0
S3
S4
1
1
S5
C5
Scene Layer
C4
S6
S7
0
1
S8
S9
1
0
0
Shot Layer
FIGURE 13.5
Scene association to retrieve video shots.
3. Propagation: The similarity values of the scene leaf nodes are then propagated up the
scene hierarchy. Each parent node is given a value equal to the average of the similarity
values of its child nodes. For example, C1 is assigned a value that is the average of the
similarity values of all its child nodes C2 , C3 , and C5 .
4. Selection: At the end of propagation, we select the scene node with the highest value as
the most relevant scene. If the similarity value of this scene node exceeds a predefined
threshold Ts , then all shots under this scene together with the initial relevant shot list
are returned. Otherwise, only the initial relevant shot list will be returned.
In Figure 13.5, video shots S1 , S3 , S4 , S7 , and S8 are selected in the initial relevant shot list.
After the scene association step, scene C2 is selected as the most relevant scene. All the shots
under scene C2 and those in the initial relevant shot list are retrieved and returned to the users.
Notice that by using scene association, shot S2 , which is not in the initial relevant shot list, can
also be retrieved. Thus, by giving higher priority to retrieving higher level scenes, we are able
to retrieve scenes that represent more general concepts and cover a greater number of scenes
and shots. This will result in better recall in video shot retrieval.
13.4.3
Virtual Editing
Because video is a temporal medium, it has the additional problem of sequencing the retrieved video shots before presentation. In the computer industry, the automatic creation of
a video sequence from video clips is called virtual editing [3]. Cinematic rules are used as
the basis to sequence the retrieved video shots in order to form a meaningful sequence. This
process is more art than science.
Parameters for Virtual Editing
Before virtual editing can be carried out, a set of cinematic parameters must be recorded at
indexing time. Normally, the parameters logged include:
© 2001 CRC Press LLC
• Focal length
ELS — extreme long shot
LS — long shot
DFS — deep focus shot
MLS — medium long shot
MS — medium shot
CU — close-up shot
ECU — extreme close-up shot
• Type of scope
1-shot — only one main character in the shot
2-shot
3-shot
group-shot — many characters in the shot
• Type of angle
Bird view
Eye-level angle
High angle
Low angle
Oblique angle
• Type of spatial
Inclusion
Exclusion
• Type of movement
Dolly
Hand-held
Pan
Tilt
Zoom
Zoom in
Zoom out
These parameters are sufficient to generate satisfactory documentary-style movies. In fact,
the “focal length,” “type of angle,” and “type of movement” should be enough to generate most
typical presentation sequences that start from a general view before zooming into the details
or vice versa. Such styles are effective in presenting information.
Cinematic Rules for Virtual Editing
Cinematic rules have existed since the birth of film-making [2, 9]. These rules are widely
accepted by film editors and have been applied to generate a tremendous number of films.
Generally speaking, there are two classes of rules:
© 2001 CRC Press LLC
1. Basic rules that are common sense based
2. Complex rules that are normally subject matter based and rely on factors such as psychology, aesthetics, and so forth.
The rules provide only general guidelines for how a movie can be made. The making of a
memorable film relies largely on the editor’s creativity in applying the rules. Because there is
a general lack of understanding of how films are edited and sequenced, it is hard to automate
the creative aspects of film-making. Thus, for practical reasons, only the simple and easy-tounderstand rules will be automated. The rules that are commonly studied and implemented
are:
• Concentration rule: An actor, object, or topic is introduced in the sequence from long
shots, to medium shots, to close-up shots. The relationships between the objects and the
environment are also evolved through this sequence.
• Enlarge rule: This is the inverse of the concentration rule.
• General rule: This is a combination of the concentration and enlarge rules. It intends
to present an intact action in a sequence that supports better understanding.
• Parallel rule: This aims to present two different themes alternately. Normally, the two
themes are similar to or contrast with each other. Examples of such themes include the
alternating shots of two actors walking toward each other, or the typical chase scenes of
police and villains.
• Rhythm rule: The shots are chosen based on the rhythm requested. Longer duration shots
are used for slower rhythms, whereas shorter duration shots are used for faster rhythms.
The rhythm of the intra-shot action should be matched with that of the presentation.
• Sequential rule: Shots are ordered chronologically.
• Content rule: Shots in a sequence share common content attributes.
From Concept to Video Sequence
To generate a video sequence from a concept, the user needs to specify not just the query
but also presentation information such as the time constraint, the cinematic rule to apply, and
the necessary parameters for the cinematic rule. This can be achieved by using a script, as
shown in Figure 13.6. The script, as a template, is used to record necessary information for
sequencing purposes.
Script title: lynx family
Cinematic rule: sequential
Time duration: 20
Query: lynx family together
Scene (optional): lynx
FIGURE 13.6
An example of a script for virtual editing.
A general script class is defined. A specific script for each cinematic rule is implemented.
Each script has its own method to sequence the retrieved video shots. The most important
cinematic parameter used is the focal length, which is recorded for the main character. It
© 2001 CRC Press LLC
is used by the script to generate effects of the concentration rule, enlarge rule, and general
rule. Video shots are sorted based on this parameter to form the appropriate presentation
sequence. Other parameters such as the number of frames help in restricting the duration of
the presentation.
Given a script, the following steps are followed to generate a meaningful video sequence:
1. User defines the script, which contains all necessary information for sequencing, including script title, cinematic rule, query terms, time duration, etc.
2. The query is first used to retrieve the relevant video shots using, if appropriate, the scene
association method.
3. The shots retrieved are ordered using the cinematic parameter(s) determined by the
cinematic rule chosen. For example, if the enlarge rule is chosen, the shots are ordered
from long shots to close-up shots.
4. The sequence is formed by selecting a suitable number of shots in each cinematic category, subject to the time limit.
5. Before presentation, user may edit the sequence by adding new shots and/or cutting
shots from the sequence.
6. When the result is satisfactory, the final sequence is presented to the user.
13.5
Implementation
This section describes the implementation of the digital video retrieval system. The overall
design of the system is given in Figure 13.7. The three major components of the system are:
• Shot Editor: Caters to the indexing and logging of video shots.
• Scene Composer: Supports the creation and editing of scenes. It uses a retrieval tool to
help users semiautomate the creation of scenes by issuing queries. A subset of this tool
supports the browsing of scenes by end users.
• Video Presentor: Supports the retrieval and virtual editing of video materials.
Figure 13.7 also shows the interactions between the users and the system, and among different components of the system. The system is designed to support two types of users: (1)
high-level users or indexers who will use the sophisticated tools to index and edit video shots
and scenes, and (2) normal end users who will interact only with the Scene Browser to browse
through the scenes, and with the Video Presentor to query and sequence video shots. The
functionalities provided to these two types of users are quite different. Figure 13.8 gives the
main user interface and the subset of functionality provided to the indexers and end users. The
specific interfaces and functions of the components are described separately below.
Shot Editor
The Shot Editor is a tool that helps users index and log the raw video shots. Figure 13.9
shows the interface of the Shot Editor. The video is viewed on the top-left window, which
provides a VCR-like interface for users to browse the contents of the shots and/or to fine-tune
© 2001 CRC Press LLC
Video DB:
Scene Layer
Shot Layer
Scene Composer:
To construct scene
hierarchies
Shot Editor
To create, log and
index video shots.
Indexers:
They are
responsible
for the
creation of
video shots
and scenes
in the video
database.
Video Presentor:
To perform retrieval and
sequencing based on user's
requests.
End users: They are interested
in viewing video contents only.
FIGURE 13.7
System components of the digital video retrieval system.
the shot boundaries. Users may log or edit the semantic information for the shots, such as the
title, text descriptions, and dialogues, and cinematic parameters. The list of shots created is
shown in the top-right list browser.
Scene Composer and Browser
The Scene Composer is used to compose scenes that capture the context information and
domain knowledge of the video shots. Through the interface given in Figure 13.10, users may
create a new scene by clicking on the “New” button at the top menu bar. The edit panel at the
bottom permits users to view the list of existing shots and scenes that can be used as members
of the new scene. Because the possible set of shots and scenes may be large, we may limit
the set by using the shot retrieval and scene association modules (see Section 13.3) to filter
out only those that match the content of the new scene being created. This facilitates the task
of composing complex scenes and opens up the possibility of creating some of the scenes
automatically. Users may browse and edit existing scenes at any time. Users may also view
the video sequence generated from the scene by clicking on the “Play” button.
A simpler interface, without all the editing functions, is provided to end users to browse
the content of the scenes. Because the scene structures provide rich context information and
domain knowledge of the underlying interface, it is extremely beneficial for new users to have
a good understanding of the overall structure before accessing the video database.
Video Presentor
The Video Presentor is the main interface users interact with to perform video retrieval and
virtual editing. Figure 13.11 shows the interface together with its child windows. To retrieve
© 2001 CRC Press LLC
FIGURE 13.8
Main user interface of the digital video retrieval system.
the video sequence, users need to pose a free-text query. Users may optionally enter a time limit
as well as cinematic rules and parameters. Based on the query script, the system first retrieves
a relevant set of shots (given in the Retrieved List Browser) by using only the scene association
retrieval function. It then performs the virtual editing function to arrive at a final sequence
(listed in the Sequenced List Browser) that meets the users’ presentation specifications. Users
may view the generated sequence automatically by manipulating the VCR-like interface at the
top-left window. Users may also choose to view each sequenced shot individually or edit the
sequence manually. The final sequence may be saved for future viewing.
Instead of issuing a query, users may choose to view an existing sequence generated previously.
13.6
Testing and Results
The system was developed at the Multimedia Information Laboratory at the National University of Singapore. It was implemented on the Sun Solaris using C++ and employed the
MPEG-TV tool to manipulate and display digital video in MPEG-1. We chose a documentary
© 2001 CRC Press LLC
video in the domain of animal for testing. We selected more than 24 minutes of video materials
in 7 different animal categories. The video was analyzed and logged using the procedures as
outlined in Section 13.3. Altogether, 164 video shots and 45 scenes were created. A summary
of the video shots created in different categories is given in Table 13.1.
Display
List of video
FIGURE 13.9
User interface of the Shot Editor.
Table 13.1 Characteristics of Video Materials Used
Category
Koala
Lion
Lynx
Manatee
Manta ray
Marine turtle
Ostrich
Total
Duration (s)
219.9
222.9
213.2
184.4
242.2
225.7
205.7
TOTAL
1447.2
© 2001 CRC Press LLC
Number of
Number of
Shots Created Scenes Created
33
8
27
7
22
7
16
4
16
5
23
6
27
8
164
45
FIGURE 13.10
User interface of the Scene Composer.
A set of seven queries was chosen to test the retrieval effectiveness. The queries, together
with the list of relevant shots, are summarized in Table 13.2. The results of retrieval, using the
shot retrieval technique, are presented in Table 13.3 in terms of normalized recall and precision.
From Table 13.3, it can be observed that the shot retrieval technique is quite effective. Further
tests using the scene association technique demonstrated that we could achieve a further 10 to
12% improvement in retrieval performance.
To evaluate the effectiveness of virtual editing, we conducted a series of retrievals using
similar queries but with different cinematic rules. The results are shown in Figure 13.12.
Figures 13.12a–c show the results of issuing the query “koala sleeps” to retrieve the set of
relevant video shots using the scene association technique but applying different cinematic rules
to sequence the shots. The time limit chosen is 25 seconds in all three cases. The sequences
generated using concentration, enlarge, and general rules are shown in Figures 13.12a–c,
respectively. The final sequence shown in Figure 13.12d is generated by using the query
“koala scratches” with the parallel cinematic rule and longer duration.
From the results (Figures 13.12a–c), it can be seen that our system is able to generate
sequences that conform to the desired cinematic rules within the time limit. Figure 13.12(d)
also shows that the system is able to generate complicated sequences based on parallel themes.
These results clearly meet the expectations of users and demonstrate that both our retrieval and
virtual editing functions are effective.
© 2001 CRC Press LLC
Display panel
List of
sequenced
video shots
List of
retrieved
video shots
Query terms: the
user can state
what he wants.
Time duration:
limits the
presentation time.
More parameters
can be provided
FIGURE 13.11
User interface of the Video Presentor.
© 2001 CRC Press LLC
To use existing
video sequences.
FIGURE 13.12
Results of virtual editing using the same query but different cinematic rules. (a) query:
koala sleeps; rule: Concentration; duration: 25 sec (b) query: koala sleeps; rule: Enlarge; duration: 25 sec (c) query: koala sleeps; rule: General; duration: 25 sec (d) query:
koala scratches itches; rule: Parallel; Theme1: body; Theme2: head; duration: 30 sec.
13.7
Conclusion
This chapter discussed the modeling, representation, indexing, retrieval, and presentation
of digital video. It also reviewed the design and implementation of a digital video retrieval
system to accomplish the above task. The effectiveness of the system was demonstrated using
a documentary video in the domain of animals. The results showed that virtual editing can be
carried out effectively using simple cinematic rules.
With advances in automated (pseudo) object identification and transcription of audio, we are
beginning to research ways to automate the process of logging video shots. We are also developing a new system based on the stratification approach, centered on advanced functionalities
for retrieval, interaction, semantic-based fast-forwarding, and video summarization.
© 2001 CRC Press LLC
Table 13.2 Queries and Set of Relevant Video Shots
Query
Koala on a tree
Lion walking
Lynx hunting
Manatee family
Manta ray flying
Marine turtle giving birth
Ostrich running
Relevant Video Shots
koala001.mpg.txt, koala003.mpg.txt, koala007.mpg.txt,
koala008.mpg.txt, koala009.mpg.txt, koala010.mpg.txt,
koala015.mpg.txt, koala021.mpg.txt, koala022.mpg.txt,
koala024.mpg.txt
lion007.mpg.txt, lion009.mpg.txt, lion010.mpg.txt
lynx013.mpg.txt, lynx014.mpg.txt, lynx015.mpg.txt,
lynx016.mpg.txt
manatee008.mpg.txt, manatee009.mpg.txt,
manatee010.mpg.txt, manatee011.mpg.txt,
manatee013.mpg.txt
manta_ray001.mpg.txt, manta_ray010.mpg.txt,
manta_ray012.mpg.txt
marine_t012.mpg.txt, marine_t013.mpg.txt,
marine_t014.mpg.txt, marine_t015.mpg.txt,
marine_t016.mpg.txt, marine_t018.mpg.txt,
marine_t019.mpg.txt
ostrich003.mpg.txt, ostrich014.mpg.txt,
ostrich015.mpg.txt
Table 13.3 Normalized Recall and Precision of Retrieval
Query
Koala on a tree
Lion walking
Lynx hunting
Manatee family
Manta ray flying
Marine turtle giving birth
Ostrich running
Average
Recall (normalized)
0.816388
0.798734
0.882279
0.847985
0.871069
0.692742
0.753145
0.808906
Precision (normalized)
0.662276
0.616213
0.655542
0.668782
0.743911
0.514673
0.592238
0.636234
References
[1] Aguierre Smith, T.G. and Pincever, N.C. (1991): Parsing Movies in Context. USENIX,
pp. 157–168.
[2] Balazs, B. (1952): Theory of the Film, Dennis Dobson Ltd., London.
[3] Bloch, G.R. (1988): From Concepts to Film Sequences. RIAO’88, pp. 761–767.
[4] Chua, T.S. and Kankanhalli, M.S. (1998): Towards Pseudo Object Models for ContentBased Visual Information Retrieval. Proceedings of International Symposium on Multimedia Information Processing, ChengLi, Taiwan, pp. 182–192.
© 2001 CRC Press LLC
[5] Chua, T.S. and Ruan, L.Q. (1995): A Video Retrieval and Sequencing System. ACM
Transactions of Information System, vol. 13, no. 4, pp. 373–407, October.
[6] Chua, T.S. and Chu, C.X. (1998): Color-Based Pseudo Object Model for Image Retrieval with Relevance Feedback. Proceedings of International Conference on Advanced
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[7] Chua, T.S., Low, W.C., and Chu, C.X. (1998): Relevance Feedback Techniques for
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Switzerland, pp. 24–31, October.
[8] Davenport, G, Aguierre-Smith, T., and Pincever, N. (1991): Cinematic Primitives for
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[10] Hampapur, A., Jain, R., and Weymouth, T.E. (1995): Production Model Based Digital
Video Segmentation. Multimedia Tools and Applications, vol. 1, no. 1, pp. 9–46.
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[12] ISO/IEC 11172, Moving Pictures Expert Group Committee (1993): Information Technology — Coding of Moving Pictures and Associated Audio for Digital Storage Media
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[14] Lin Yi, M., Kankanhalli. S., and Chua, T.-S. (1999): Temporal Multi-Resolution Analysis for Video Segmentation. Technical report, School of Computing, National University
of Singapore.
[15] Rubin, B. and Davenport, G. (1989): Structured Content Modeling for Cinematic Information. SIGCHI Bulletin, vol. 21, no. 2, pp. 78–79.
[16] Rudnicky, A.I., Lee, K.-F., and Hauptmann, A.G. (1994): Survey of Current Speech
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Classifying Scene Breaks. Fourth ACM Multimedia Conference, pp. 189–200.
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Browsing: An Integrated and Content-Based Solution. ACM Multimedia ’95, pp. 15–23.
© 2001 CRC Press LLC
Chapter 14
Image Retrieval in Frequency Domain Using
DCT Coefficient Histograms
Jose A. Lay and Ling Guan
14.1
Introduction
As an increasing amount of multimedia data is distributed, used, and stored in the compressed format, an intuitive approach for lowering the computational complexity toward the
implementation of an efficient content-based retrieval application is to propose a scheme that
is able to perform retrieval directly in the compressed domain. In this chapter, we show how
energy histograms of the lower frequency discrete cosine transform coefficients (LF-DCT) can
be used as features for the retrieval of JPEG images and the parsing of MPEG streams. We also
demonstrate how the feature set can be designed to cope with changes in image representation
due to several common transforms by harvesting manipulation techniques known to the DCT
domain.
14.1.1
Multimedia Data Compression
Loosely speaking, multimedia data compression includes every aspect involved in using a
more economical representation to denote multimedia data. Hence, to appreciate the significance of multimedia data compression, the types and characteristics of a multimedia datum
itself need to be examined.
Multimedia is a generic term used in computing environments. In the digital information
processing domain, it refers to a data class assembled by numerous independent data types,
such as text, graphics, audio, still images, moving pictures, and other composite data types. A
composite data type is a derivative form created when instances of two or more independent
data types are combined to form a new medium. Multimedia data are used in a broad range
of applications. Sample applications include telemedicine, video telephony, defense object
tracking, and Web TV.
Multimedia data can be categorized in numerous ways. With respect to the authoring process,
multimedia data can be classified as synthesized and captured media. The classification can
also be based on the requirement of presentation where multimedia data are grouped into
discrete and continuous media.
Multimedia data have several characteristics. They are generally massive in size, which
requires considerable processing power and large storage supports. Storage requirements for
© 2001 CRC Press LLC
several uncompressed multimedia data elements are listed in Table 14.1. Where continuous
media are used, multimedia data will also be subjected to presentation constraints such as
media synchronization and the requirement for continuity in operation [1]. Consequently, the
bandwidth requirement for a synchronized media unit will be the aggregate bandwidth along
with the synchronization overhead. For instance, a musical video clip in common intermediate
format (CIF) may consist of a series of synchronized video, text, and audio data. This media
may need to be continuously played for 3 min, and the aggregate bandwidth requirement
excluding the overhead will be a steady throughput of 74.4 Mbps over the 3-min time span.
Thus, a system with a storage space of not less than 1.67 GB and a data bus of sustainable
9.3 MB/s will be required to present this media. Although the storage requirement may not
seem to be too ambitious, the data rate is certainly beyond the capability of the fastest CD-ROM
drive embedded in the current PCs.
Table 14.1 Storage Requirements for Several Uncompressed Multimedia Data
Elements
Multimedia Data Type
Stereo audio (20–20 KHz)
VGA image (640 × 480)
Digitized video (NTSC)
Web TV video (CIF)
Sampling Details
Storage
44,000 samples/s × 2 channels
× 16 bit/sample
640 × 480 pixels × 24 bit/pixel
1.41 Mbps
7.37 Mbit/image
720 × 576 pixels/frame × 24 bit/pixel
× 30 frames/s
352 × 288 pixels/frame × 24 bit/pixel
× 30 frames/s
298.59 Mbps
72.99 Mbps
Furthermore, as the Internet facilitates the multimedia data from a workstation onto the
widely networked environment, the network bandwidth will also need to match the throughput
requirement. The transmission rates for several network standards are listed in Table 14.2. It
is clear that the uncompressed multimedia data are scarcely supported by local area networks
(LANs), let alone a connection established through the public switched telephone network
(PSTN).
Table 14.2 Sample Data Rates of Several Network
Standards
Network Technology
Data Rate
Public switched telephone network (PSTN)
Integrated services digital network (ISDN)
Telecommunication T-1
Telecommunication T-3
Local area network (Ethernet)
0.3–56 Kbps
64–144 Kbps
1.5 Mbps
10 Mbps
10 Mbps/100 Mbps
To overcome the bulky storage and transmission bandwidth problems, a compressed form
of multimedia data was introduced. Extensive compression may significantly reduce the bandwidth and storage need; however, compression may also degrade the quality of multimedia
data, and often the loss is irreversible. Therefore, multimedia data compression may be viewed
as a trade-off of the efficiency and quality problem [2, 3]. The data rates of several compressed
multimedia data are listed in Table 14.3.
For the last decade, three very successful compression standards on multimedia data elements
have been JPEG, MPEG-1, and MPEG-2. JPEG has facilitated the vast distribution of images
© 2001 CRC Press LLC
Table 14.3 Sample Data Rates of Compressed Data
Compressed Data
Channel audio for MPEG
Color JPEG image
(640 × 480)
Sampling Details
64/128/192 Kbps/channel × N channel
640 × 480 pixels × 3 color components
× (0.25–2) bit/component sample
Bandwidth
MPEG-1 video
360 × 345 pixels/frame × 30
frames/s
1.5 Mbps or higher
H.261 video (CIF)
352 × 288 pixels/frame × (15–30)
frames/s
56 Kbps–2 Mbps
N × 64/128/192 Kbps
0.23–1.84
Mbit/image
on the Internet. MPEG-1 has made possible the storage of a movie title onto a couple of video
compact disc (VCD) media. MPEG-2 then extended this achievement to the forms of DVD
and HDTV. Their success stories have been prominent ones [4]. Not only have they created
many new opportunities in business, but they have also changed the computing experience by
shortening the path in multimedia content authoring and usage. Digital images are now directly
attainable from digital cameras in the JPEG format. Likewise, MPEG-1 and -2 streams can be
straightforwardly obtained from digital videocameras. In short, many more people have been
enabled to create and use multimedia contents.
14.1.2
Multimedia Data Retrieval
The various aspects concerned with the enabling of multimedia data accessing are generally
termed multimedia data retrieval (MDR). As increasing amounts of multimedia data or their
elements become available in the digital format, information technology is expected to provide
maximum usability of these data. However, the established text-based indexing schemes have
not been feasible to capture the rich content of multimedia data, as subjective annotation
may lead to undetectable similarities in the retrieval process. Consequently, content-based
retrieval (CBR) was proposed. In addition to textual descriptors, multimedia data are described
using their content information; color, texture, shape, motion vector, pitch, tone, etc., are
used as features to allow searching to be based on rich content queries. The use of textual
descriptors will still be desirable, because they are needed in identifying information that
cannot be automatically extracted from multimedia contents, such as name of the author, date
of production, etc.
Three basic modules of a CBR system are feature extraction, feature description, and proximity evaluation. Feature extraction deals with how the specific traits of content information
can be identified and extracted from the content-rich data. Feature description specifies how
those features can be described and organized for efficient retrieval processing. Lastly, proximity evaluation provides the specification in which similarities among contents can be measured
based on their features.
The advancement of CBR studies has been remarkable. The current challenge has been
the network-wide implementation of these techniques. In past years, many studies on CBR
have been conducted, notably in the image and video retrieval domain. Numerous features
and their associated description and proximity evaluation schemes have been introduced. A
handful of proprietary CBR systems have also been developed. Content-based image and
video retrieval is now considered a developed field. However, searching content information
across the Internet has not been viable, because no unified feature description scheme has
been commonly adopted. For this reason, an effort to standardize the description of content
information was initiated. The work was named Multimedia Content Description Interface
© 2001 CRC Press LLC
but is better known as MPEG-7. Surveys on CBR systems and their research issues are given
in [5]–[7].
MPEG-7 aims to extend the capabilities of the current CBR systems by normalizing a
standard set of descriptors that can be used to describe multimedia contents. MPEG-7 also
intends to standardize ways to define other descriptors and their description schemes. MPEG-7
will also standardize a language to specify description schemes [8]. Information on MPEG-7
is available through the MPEG Web site [9].
Figure 14.1 depicts the scope of MPEG-7 [8]. Because the focal interest of MPEG-7 is
on the interfacing of descriptors, the feature extraction process and how the features are used
in searching on a database will remain an open area for industry competition, since their
normalization is not required to allow interoperability.
Feature Extraction
Standard Description
Search Engine
Scope of MPEG-7
FIGURE 14.1
Diagram block of the proposed MPEG-7 scope.
14.1.3
About This Chapter
The chapter has been written to be self-contained. Background information is included to
provide newcomers with the underlying concepts. Readers familiar with those concepts may
skim through or skip them. Subjects are discussed within a general perspective. While the
primary aim of this chapter concentrates on compressed domain image and video retrieval
using the energy histogram features, efforts have been made to present the discussion under
the broader theme of MDR, which allows relation with MPEG-7 to be easily established.
More studies on retrieval and processing models are needed to optimize the applicability
of MPEG-7. Because only the interfacing of features is covered in MPEG-7, many more
research opportunities and needs are left open for feature extraction and search engine studies.
Furthermore, while feature extraction has been an active research subject in CBR studies,
works on supporting retrieval models have been scarce. Therefore, it is sensible that many
more studies on retrieval and processing models are needed before an MPEG-7 optimum search
engine can be materialized. A brief discussion on retrieval and processing models as well as
the perceived MPEG-7 search engine is consecutively presented in Sections 14.3.1, 14.3.2,
and 14.3.3.
Meanwhile, as multimedia data are distributed, used, and stored in the compressed format,
the compressed domain technique is highly relevant. The compressed domain technique deals
with data directly (or through partial decompression) in their compressed domain, hence avoiding (or reducing) the decompression overhead found in many uncompressed domain schemes.
The DCT domain features are at the center of compressed domain techniques. A noteworthy
observation on the many successful multimedia standards is that most of them are based on
transform coding using the DCT. Logically, exploitations of DCT domain features are essential
in supporting the realization of an efficient current CBR application. The underlying concepts
of DCT, the DCT coefficients in JPEG and MPEG data, and the energy histogram are presented
in Section 14.2, while an overview of the DCT domain features reported in recent studies is
given in Section 14.3.5.
© 2001 CRC Press LLC
The rest of Section 14.3 will be used to present various aspects related to the proposed retrieval scheme. An MPEG-7 optimum search engine construct is presented in Section 14.3.3.
The DCT domain manipulation techniques are covered in Section 14.3.4, while the energy
histograms of the LF-DCT coefficient features are presented in Section 14.3.5. Several proximity evaluation schemes are discussed in Section 14.3.6, and the experimental results for the
retrieval of JPEG images and the parsing of MPEG streams are provided in Section 14.3.7.
Conclusions are given in Section 14.4.
14.2
The DCT Coefficient Domain
Before we describe how the DCT coefficients can be used as potent features for retrieving
a JPEG image and/or parsing an MPEG stream, we shall first enunciate the basic concept
underlying the DCT domain. We will embark by giving an alternative explanation of the DCT
using the matrix notation, then go on to show how the DCT coefficients reside in the JPEG and
MPEG data, and finally articulate the notion of the energy histograms of the DCT coefficients.
14.2.1
A Matrix Description of the DCT
DCT was first introduced in 1974 [10]. It is now the major building block of many very
popular image and video compression standards. Together with the vast development of
semiconductors, DCT-powered compression standards have delivered a magnificent computing
environment, an internetworked world rich with multimedia contents.
The 8 × 8 block forward and inverse 2D DCTs used in JPEG and MPEG are given by
Forward 2D DCT:
1
(2i + 1)uπ
(2j + 1)vπ
f (u, v) = CuCv
s(i, j ) cos
cos
4
16
16
7
7
i=0 j =0
Inverse 2D DCT:
1 (2j + 1)vπ
(2i + 1)uπ
cos
CuCvf (u, v) cos
4
16
16
7
s(i, j ) =
7
u=0 v=0
√
where Cτ = 1/ 2 for τ = 0 and Cτ = 1 for τ = 0. The f (u, v) are the so-called DCT
coefficients and s(i, j ) are the values of the i, j input samples.
Since the 2D DCT is attainable by concatenating two 1D DCTs, we will use the latter to
convey the purpose of this section. Thus, we can reveal the concept without dealing with too
many indices in the equation. The formal definition for an 8-element 1D DCT is given by
Forward 1D DCT:
1
(2i + 1)uπ
Cu
s(i) cos
2
16
7
f (u) =
i=0
√
where Cu = 1/ 2 for u = 0 and 1 otherwise, f (u) are the 8-element 1D DCT coefficients,
and s(i) are the 8 input elements.
© 2001 CRC Press LLC
Thinking in vector terms, we can rewrite the transform using a matrix notation by arranging
f (u) and s(i) and substituting values for (2i + u):
 

 
0 0 0 0 0 0 0
0
f0
s0
 π 3π 5π 7π 9π 11π 13π 15π  s1
f 1


16
16
16
16
16
16
16
16
 

 2π 6π 10π 14π 18π 22π 26π 30π  
 



s2
f 2 

16
16
16
16
16
16
16
16
 

 
 3π 9π 15π 21π 27π 33π 39π 45π   
f 3
 16 16 16 16 16 16 16 16  s3
  1


  = Cu cos 
 .
 4π 12π 20π 28π 36π 44π 52π 60π  
f 4  2

 16 16 16 16 16 16 16 16  s4
 
 5π 15π 25π 35π 45π 55π 65π 75π   
 
 s5

f 5

 16 16 16 16 16 16 16 16  
 

 6π 18π 30π 42π 54π 66π 78π 90π  
 

 16 16 16 16 16 16 16 16  s6
f 6



7π 21π 35π 49π 63π 77π 91π 105π
16 16 16 16 16 16 16
16
f7
s7
Let us denote the f (u) vector with f, the cosine function matrix as K, and the s(i) vector
with s. We have:
1
f = CuKs .
2
Note that we have chosen to write f and s as column vectors in the equation. Intuitively,
the matrix notation shows that a DCT coefficient f (u) is simply a magnitude obtained by
multiplying a signal vector (s) with several scaled discrete cosine values distanced at certain
multiples of π/16 frequency. Therefore, calculating the DCT coefficients of a particular signal
is essentially carrying out the frequency decomposition [12] or, in a broader sense, the content
decomposition of that signal.
Each row in the cosine function matrix represents the basis function of a specific decomposition frequency set. To help visualize this concept, we will reconstruct the cosine matrix by
explicitly calculating their cosine values. Furthermore, since π/16 is a factor common to all
elements, the trigonometric rules allow the new matrix to be rewritten using only references
to the values of the first quadrant components. By doing so, we hope to communicate the
idea without getting involved with long decimal elements in the matrix. We denote the first
quadrant components of the K matrix as:
cos 0
a0
cos π/16
a1
cos 2π/16
a2
cos 3π/16
a3
cos 4π/16
a4
cos 5π/16
a5
cos 6π/16
a6
cos 7π/16
a7
The new matrix may be rewritten as:
 

 
f0
a0 a0 a0 a0 a0 a0 a0 a0
s0
f 1
 a1 a3 a5 a7 −a7 −a5 −a3 −a1  s1
 

 
f 2 
 a2 a6 −a6 −a2 −a2 −a6 a6 a2  s2
 

 
f 3 1

 
  = Cu  a3 −a7 −a1 −a5 a5 a1 a7 −a3  s3 .
f 4  2
 a4 −a4 −a4 a4 a4 −a4 −a4 a4  s4
 

 
f 5
 a5 −a1 a7 a3 −a3 −a7 a1 −a5  s5
 

 
f 6
 a6 −a2 a2 −a6 −a6 a2 −a2 a6  s6
f7
a7 −a5 a3 −a1 a1 −a3 a5 −a7
s7
Note that the occurrence of sign changes increases as we move downward along the matrix
rows. Row 0 has no sign changes, since a0 = cos 0 = 1 for every element in that row.
However, row 1 has one sign change, row 2 has two sign changes, and so on. The sign changes
within a basis function basically indicate the zero-crossings of the cosine waveform. Thus, as
© 2001 CRC Press LLC
1
1
0
0
-1
-1
K0
K4
1
1
0
0
-1
-1
K1
K5
1
1
0
0
-1
-1
K2
K6
1
1
0
0
-1
-1
K3
K7
FIGURE 14.2
Eight DCT cosine basis function waveforms.
the occurrence of the sign change intensifies, the frequency of the waveform increases. The
eight basis function waveforms associated with matrix K are shown in Figure 14.2.
Since the first cosine basis function (K0) has no alternating behavior, the DCT coefficient
associated with this basis function is usually dubbed as the DC coefficient, referring to the
abbreviation used in electrical engineering for the direct current. Consequently, the other DCT
coefficients are called AC coefficients.
© 2001 CRC Press LLC
Now that we view the DCT coefficients as the frequency domain apparatus of a time or
spatial signal, we shall expect to reconstruct the original signal from its DCT coefficients, that
is, to find the inverse DCT (IDCT). To do this, we will continue to use the matrix representation
built in the previous section.
In a matrix sense, finding the IDCT can be just a matter of solving the inverse for the
transforming operator. The matrix equivalent for the IDCT equation may be easily written as
s=
1
CuK
2
−1
f
which is basically a problem of finding an inverse for the scaled matrix. Note that we have
neglected the 1/2Cu scaling constants in the discussion so far. We do so because it really does
not inhibit us from passing on the ideas of frequency or content decomposition. However, as
we move on with formulating the IDCT equation, the importance of these scaling constants
will become relevant. In fact, a major reason for introducing the scaling constants is largely
based on the requirement for having an orthogonal transforming matrix. So one can achieve
the IDCT by merely transposing the matrix 1/2CuK.
To demonstrate the role of the scaling constants, we will need to explore some characteristics
of the basis functions. Examining the matrix K, we can see that each of the basis functions
(matrix-row) is orthogonal to the others. This means a dot product of any two rows in the
matrix will yield zero. However, none of the basis functions is orthonormal. In other words,
the basis functions are not of unit vectors. Nevertheless, the results are almost as good.

8
0

0

0
T
V = KK = 
0

0

0
0
0
4
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
4
0

0
0

0

0

0

0

0
4
Note that each of the diagonal elements in V equals the dot product of a basis function with
itself. V0,0 is directly attainable in the form of the sum of squares of the first basis function
(K0), whereas the others are computed by application of trigonometric identities.
Equally important are the zero-value elements in V, which indicate that orthogonality does
exist among the basis function vectors. However, having orthogonal row vectors alone is not
sufficient for K to become an orthogonal matrix. An orthogonal matrix also requires all of
its row vectors (column vectors) to be of unit length, so the product of the matrix with its
transpose (KKT ) can produce an identity matrix (I).
Since K is a square matrix and the orthogonal property exists among its basis function
vectors, the only task left is to turn them into unit vectors, which can be realized simply by
scaling each of the basis functions with its length:
Kiu =
Ki
Ki
(i = 0, 1, . . . , 7) and
Kiu = 1 .
√
√
√
Simple arithmetic yields k0 = 8 = 2 2 and k1 . . . k7 = 4 = 2. Therefore,
to
√ make K orthogonal, we shall scale the first basis function (K0) by a factor of 1/ 2 2 , while
the others need only to be divided by 2. Separating the 1/2 factor from the elements, Kc can
© 2001 CRC Press LLC
be devised as:

c0
 a1

 a2

1
a3
Kc = 

a4
2
 a5

 a6
a7

c0 c0 c0 c0 c0 c0 c0
a3 a5 a7 −a7 −a5 −a3 −a1 

a6 −a6 −a2 −a2 −a6 a6 a2 

−a7 −a1 −a5 a5 a1 a7 −a3 
 ,
−a4 −a4 a4 a4 −a4 −a4 a4 

−a1 a7 a3 −a3 −a7 a1 −a5 

−a2 a2 −a6 −a6 a2 −a2 a6 
−a5 a3 −a1 a1 −a3 a5 −a7
√
√
with c0 = a0/ 2 = 1/ 2 = a4. Kc is an orthogonal matrix, where Kc−1 = KcT , so
Kc KcT = KcT Kc = I. It is quite interesting to see that we have just turned the highly
sophisticated DCT into the down-to-earth Ax = b linear equation:
Forward DCT:
Inverse DCT:
f = Kc s
s = Kc−1 f ⇒ s = KcT f
Treating the DCT as linear equations not only allows us to eagerly derive the IDCT term, but
also clearly presents some of its most prominent properties by using solely the linear algebra
concept.
From the linear algebra point of view, the DCT is an orthogonal linear transformation. A
linear transformation means the DCT can preserve the vector addition and scalar multiplication
of a vector space. Thus, given any two vectors (p and q) and a scalar (α), the following relations
are true:
f (p + q) = f (p) + f (q) ,
f (αp) = αf (p) .
Linearity is useful when dealing with frequency domain image manipulation. Several simple
techniques of the DCT frequency domain image manipulation are discussed in Section 14.3.3.
Meanwhile, the orthogonal term implies that the lengths of the vectors will be preserved
subsequent to a DCT transformation. For an input vector s = [a, b, c, d, e, f, g], and the
transformed vector f, we have
s = f = a 2 + b2 + c2 + d 2 + e2 + f 2 + g 2 + h2
as f2 = Kc s2 = (Kc s)T (Kc s) = sT KcT Kc s = sT s = s2 .
This characteristic is often referred to as the energy conservation property of the DCT.
Another important property of being an orthogonal transform is that the product of two
or more orthogonal matrices will also be orthogonal. This property then enables a higher
dimensional DCT and its inverse to be performed using less complicated lower dimensional
DCT operations. An example for performing the 8 × 8 block 2D DCT and its inverse through
the 8-element DCT is given below:
Forward 2D DCT:
f2D = Kc s KcT
Inverse 2D DCT:
s2D = KcT f Kc
Now that we have described how the 2D DCT can be accomplished by successive operations
of 1D DCT, we shall turn our attention to the most interesting behavior of the DCT known as
© 2001 CRC Press LLC
the energy packing property, which indeed has brought the DCT coefficients into the heart of
several widely adapted compression techniques of the decade.
Even though the total energy of the samples remains unaffected subsequent to a DCT transform, the distribution of the energy will be immensely altered. A typical 8 × 8 block transform
will have most of the energy relocated to its upper-left region, with the DC coefficient (f00 )
representing the scaled average of the block and the other AC coefficients denoting the intensity
of edges corresponding to the frequency of the coefficients. Figure 14.3 depicts the energy
relocation that occurred in the transform of a typical 8 × 8 image data block.

10 10 10 10 10 10 10 10
 1 1 1 1 1 1 1 1


 1 1 1 1 1 1 1 1


 1 1 1 1 1 1 1 1


 1 1 1 1 1 1 1 1


 1 1 1 1 1 1 1 1


 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10


An Image Data Block
⇒
2D DCT
26
 0

 24

 0

 18

 0

 10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0

0

0

0

0

0
0
DCT Coefficients
FIGURE 14.3
Energy packing property.
Now that we have presented the fundamental idea of the DCT in the matrix dialect, we
would like to end this section by taking just another step to rewrite the IDCT in its formal vein:
Inverse 1D DCT:
s = Kc−1 f = KcT f
s(i) =
7
1
i=0
14.2.2
2
Cuf (u) cos
(2i + 1)uπ
16
where

Cu =
√1
2
Cu = 1
for
u=0
for
u>0
The DCT Coefficients in JPEG and MPEG Media
Video can be viewed as a sequence of images updated at a certain rate. This notion is also
valid in the compressed data domain. For instance, a sequence of JPEG images can be used to
constitute a motion JPEG (M-JPEG) video stream. In fact, many popular video compression
standards including MPEG-1, MPEG-2, H.261, and H.263 are built upon the DCT transform
coding techniques developed and used in JPEG. Therefore, the topic of DCT coefficients in
MPEG media will be best described after that of JPEG is presented.
The JPEG standard acknowledges two classes of encoding and decoding processes known
as lossy and lossless JPEG. Lossy JPEG is based on the energy packing characteristic of DCT
and includes mechanisms where a certain amount of information may be irreversibly lost
subsequent to its coding processes. Lossy JPEG is able to achieve substantial compression
rates. Its modes of operation are further divided into baseline, extended progressive, and
extended hierarchical. Conversely, lossless JPEG is based on predictive algorithms where
content information can be fully recovered in a reconstructed image. However, lossless JPEG
can attain only a moderate compression rate. Because lossless JPEG is based on non-DCTbased algorithms, only lossy JPEG is of interest in this chapter.
Figure 14.4 shows a block diagram of the lossy JPEG codec structure. In the encoding
process the spatial image data are grouped into a series of 8 (pixel) × 8 (pixel) blocks. Each of
these blocks is then fed into a forward 2D DCT to produce the 64 DCT coefficients. The blocks
© 2001 CRC Press LLC
are processed in a sequence from left to right and from top to bottom. The DCT coefficients
are then scalarly quantized using a quantization factor set in a quantization table [12]:
f (u, v)
fq (u, v) = round
Q(u, v)
where f (u, v), fq (u, v), and Q(u, v) are the DCT coefficients being quantized, their quantized
values, and the quantization factors provided in the quantization table, respectively.
Quantization
Entropy Coding
Tables
Tables
Dequantization
Entropy Decoding
Tables
Tables
2D F-DCT
Spatial
Image
2D I-DCT
JPEG
Image
FIGURE 14.4
A JPEG codec structure.
The quantization step is the lossy part of lossy JPEG. Quantization is primarily employed to
prune the higher frequency coefficients by dividing them with larger factors. Thus, variations
of quantization tables can be used to tune the desirable compression ratio. However, because
a rounding-off operation is involved in every quantization process, quantized coefficients may
be subjected to irreversible loss of information. Therefore, quantization tables need to be
specifically designed so that the quality degradation is still in the tolerable range. Separate
quantization tables are used for the luminance and chrominance components of JPEG data.
Two quantization tables provided in the JPEG standard are tabulated in Figure 14.5.

16
 12

 14

 14

 18

 24

 49
72
11
12
13
17
22
35
64
92
10
14
16
22
37
55
78
95
16 24 40
19 26 58
24 40 57
29 51 87
56 68 109
64 81 104
87 103 121
98 112 100

51 61
60 55 

69 56 

80 62 

103 77 

113 92 

120 101 
103 99
Luminance Quantization Table

17
 18

 24

 47

 99

 99

 99
99
18
21
26
66
99
99
99
99
24
26
56
99
99
99
99
99
47
66
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99
99

99
99 

99 

99 

99 

99 

99 
99
Chrominance Quantization Table
FIGURE 14.5
Quantization tables.
Upon quantization, the 8 × 8 DCT coefficients within a block are arranged in a zigzag
order. Since the DC coefficients tend to be highly correlated among the adjacent data blocks,
the difference of two consecutive DC coefficients (rather than an actual DC value) is coded
to enhance the compression ratio, whereas the other AC coefficients are run-length coded to
remove the “zeros” redundancy. These semicoded coefficients are then further entropy coded
using Huffman or arithmetic coding techniques to produce the final JPEG bits. Conversely, on
the decoding side, inverse operations of the encoding processes are performed.
© 2001 CRC Press LLC
In addition to the DCT-based transform coding mechanisms, color subsampling is used in
JPEG to further enhance the compression rate. It is understood that human eyes are more
sensitive to brightness (luminance) than to color (chrominance). Therefore, certain color
information may be arbitrarily reduced from a color image without generating significant
quality losses to human perceptions. Consequently, the YUV or YCb Cr (rather than RGB)
color representation system is adopted in JPEG and MPEG. The luminance (Y) component
represents a gray-scale version of the image. The chrominance components (UV) are used
to add color to the gray-scale image. One commonly used subsampling ratio is 4:2:2, which
means that the luminance of each pixel is sampled while the chrominance of every two pixels
is sampled. Several useful JPEG resources are provided in [13]–[15].
Because the updating rate of a video sequence is normally not less than tens of images per
second, adjacent images in a video stream may be expected to be in high correlation. Therefore,
temporal coding techniques can be used on top of the spatial coding to further enhance the
compression performance.
Relying on both, MPEG adopted the intra- and inter-coding schemes for its data. An MPEG
stream consists of I (intra), P (predictive), and B (bidirectional) coded frames. An I frame
is an intra-coded independent frame. Spatial redundancy on independent frames is removed
by the DCT coding where a coded image can be independently decoded. P and B frames are
inter-coded reference frames. Temporal redundancy on reference frames is detached by the
means of motion estimation. A P frame is coded based on its preceding I or P frame, while a
B frame is coded using both of the preceding and the following I and/or P frames. Therefore,
decoding a reference frame may depend on one or more related frames. A fine survey of the
current and emerging image and video coding standards is presented in [16].
M-JPEG is an extension of JPEG to cope with moving pictures where each frame of a video
stream is compressed individually using the JPEG compression technique. The independent
compression allows easy random access to be performed on an M-JPEG stream, thus enabling
M-JPEG to enjoy much popularity in the nonlinear video editing application.
14.2.3
Energy Histograms of the DCT Coefficients
Histogram techniques were originally introduced into the field of image retrieval in the
form of color histograms [17]. A color (gray-level) histogram of a digital image is formed by
counting the number of times a particular color (intensity) occurs in that image.
h[i] = ni
h[i] = color histogram of color i
ni = number of times color i occurs in the image
Since color images are normally presented in a multidimensional color space (e.g., RGB or
YUV), color histograms can be defined using either a multidimensional vector or several
one-dimensional vectors.
The color histogram of an image is a compelling feature. As a global property of color
distribution, color histograms are generally invariant to translation and perpendicular rotations.
They can also sustain modest alterations of viewing angle, changes in scale, and occlusion [17].
Their versatility may also be extended to include scaling invariance through the means of
normalization [18]. However, color histograms are intolerant to the changes of illumination.
A small perturbation in the illumination may contribute to considerable differences in histogram
data.
Similar to color histograms, an energy histogram of the DCT coefficients is obtained by
counting the number of times an energy level appears in the DCT coefficient blocks of a DCT
compressed image. Thus, the energy histograms for a particular color component (hc ) in an 8
© 2001 CRC Press LLC
× 8 DCT data block can be written as
hc [t] =
7
7 u=0 v=0
1
0
if E(f [u, v]) = t
otherwise
where E(f [u, v]) denotes the energy level of the coefficient at the (u, v) location and t is the
particular energy bin.
As with color histograms, energy histograms are generally tolerant to rotations and modest
object translations.
14.3
Frequency Domain Image/Video Retrieval Using DCT Coefficients
Frequency domain CBR offers twofold advantages. Computational complexity issues introduced by the discrepancy of spatial domain (uncompressed) feature schemes and frequency
domain (compressed) data can be a hindrance in implementing an efficient CBR application,
especially on the real-time platform. The frequency domain CBR approach is able to reduce
the complexity by processing the compressed data directly or through partial decompression
in their frequency domain. Furthermore, direct processing of compressed data allows the retrieval system to operate on rather compact data, which are beneficial in terms of computation
resources and network-wide processing.
Compressed
Image
Decompression
Uncompressed Image
Features
Extraction
Image & Feature
Database
FIGURE 14.6
Extracting uncompressed domain features from a compressed image.
Nevertheless, many more studies are needed before a full-fledged frequency domain CBR
system can be materialized. Because conventional features and processing techniques may
not be directly accessible in the compressed domain, exploration of new frequency domain
features and processing techniques is becoming mandatory.
Repetition and modified variations of data are common in network environments. Since
data sharing is likely in network computing, modified copies of a content are expected across
network-based databases. For instance, image databases may contain many images that may
differ only in their visual representation. These images are often of basic transformed operations (e.g., mirroring, transposing, or rotating). Therefore, detecting similarities on those
transformed images is pertinent to a network-based CBR system.
Later in this section, the energy histograms of LF-DCT coefficients are used as features
for retrieval of JPEG images (based on the query by model method) as well as for parsing of
MPEG videos. The targeted retrieval scheme is desired to be able to support network- and
MPEG-7-based implementation. Therefore, current content-based retrieval and processing
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models and their requirements are studied. An MPEG-7 optimum search engine construct is
also presented. Image manipulation techniques in the DCT domain are examined with regard
to the building of limited transformed variant proof features.
FIGURE 14.7
Transformed variants are common in network databases.
In video applications, studies have shown that the DC coefficients can be used to detect abrupt
scene changes. However, the use of DC coefficients alone does not provide a robust method
for parsing of more complex video sequences such as ones with luminance changes and/or
dissolving transitions. The energy histogram features are used to enhance the segmentation
of DCT-based video. Experimental results for video parsing of MPEG streams along with the
retrieval of JPEG images are presented in Section 14.3.7, while a CBR model for content-based
video retrieval is briefly described in Section 14.3.1.
14.3.1
Content-Based Retrieval Model
The current CBR model is characterized by a separate feature database. To avoid the high
computational cost posted by the uncompressed feature techniques, many of the current CBR
systems are built on a dual-database model where a pair of independent databases are used
to catalogue features and data [18, 19]. Figure 14.8 shows the dual-database CBR model
used in image retrieval applications. The independent feature database is built in addition
to the image database itself during the setup phase. Proximity evaluation can be performed
by contrasting the extracted features of a query with the records maintained in the feature
database. When matches are obtained, the associated image data are returned from the image
database. Therefore, the dual-database model is also known as the off-line or indexing model,
because off-line feature extraction and pre-indexing processing are required during a database
formation.
The dual-database model is advantageous from several perspectives. Since the structure
of the dual-database model is comparable to the general indexing system used in text-based
databases, this model may enjoy the support of many established techniques and developed
tools. Furthermore, because features are pre-extracted during database creation, conventional
spatial domain techniques may be used without causing high computational complexities at run
time. The dual-database model also fits well with the needs of the video retrieval application,
where features from key frames representing the segmented video shots are extracted for
indexing use. The content-based video retrieval model is discussed later in this section.
Nevertheless, there are also drawbacks attached to the dual-database model. Because searching in a dual-database CBR system is performed on the pre-extracted feature sets, the query’s
features have to conform with the feature scheme used by the feature database. Consequently,
choices of features are determined by the in-search feature database. Moreover, universal
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Creation of Image Database
Feature
Extraction
Image
Database
Output
Feature
Database
Proximity Measure
Query
Feature
Extraction
Extracted
Features
FIGURE 14.8
The dual-database content-based image retrieval model.
Image Database
Output
Image
Image Database Creation
On the fly Feature
Extraction and
Similarity Measure
Query Image
Searching and Retrieval
Query Image
FIGURE 14.9
The single-database content-based image retrieval model.
searching across the Internet would also be impracticable until the unified description sought
by MPEG-7 is widely implemented.
Alternatively, the single-database CBR model [20] can be employed. The single-database
CBR model used in image retrieval applications is illustrated in Figure 14.9. In such a model,
no preprocessing is required during database construction. Features are extracted on the fly
within a retrieval cycle directly from data. Therefore, rapid feature extraction and proximity evaluation are obligatory to the single-database systems. Because feature extraction and
proximity evaluation are executed on the fly within a retrieval cycle, the single-database CBR
model is also known as the online CBR model.
As with the dual-database model, the single-database model also has upsides and downsides.
It is practical for compressed domain-based retrieval (pull application) and filtering (push
application), especially when content-based coding such as that of MPEG-4 is used. It also
supports ad hoc Internet-wide retrieval implementations because raw compressed data can be
© 2001 CRC Press LLC
read and processed locally at the searcher machine. This local processing of feature extraction
will unlock the restriction of the choices of features imposed by feature databases. However,
sending raw compressed data across a network is disadvantageous, because it tends to generate
high traffic loads.
Video retrieval is generally more efficient to implement with the dual-database model. Video
streams are segmented into a number of independent shots. An independent shot is a sequence
of image frames representing a continuous action in time and space. Subsequent to the segmentation, one or more representative frames of each of the segmented sequences are extracted
for use as key frames in indexing the video streams. Proximity evaluations can then be performed as that of a dual-database image retrieval system (i.e., by contrasting the query frame
with each of the key frames). When matches are obtained, relevant video shots are returned
from the video database. The structure of a simplified content-based video database is shown
in Figure 14.10.
Video Database
Video Stream
Parsing
Video Shots
Video Stream
Database
Feature Database
Temporal
Features
Indexing
Key Frames
Key Frame
Features
FIGURE 14.10
Structure of a simplified content-based video database.
14.3.2
Content-Based Search Processing Model
From the search processing perspective, two fundamental models can be associated with the
dual-database and single-database CBR applications. For the dual-database systems, search
processing is normally performed on and controlled by the database-in-search. Thus, the
associated search processing model is termed the archivist processing model, since proximity
evaluation is executed on the archivist environments. The model is also known as the client–
server model, because all the processing and know-how is owned and offered by the archivist
(server) to the searcher (client). Conversely, on a single-database system, search processing is
normally performed on and controlled by the search initiator. Therefore, the associated search
processing model is termed the searcher processing model. The archivist processing model
and the searcher processing model are illustrated in Figures 14.11a and b, respectively.
The current search processing models are unsatisfactory. The client–server model is undesirable because all the knowledge on how a search is performed is owned and controlled by
the archivist server. The searcher processing model is impractical because its operation may
involve high network traffic.
Alternatively, a paradigm called the search agent processing model (SAPM) [22] can be
employed. The SAPM is a hybrid model built upon the mobile agent technology. Under
© 2001 CRC Press LLC
(a)
Searching
submit query
Query Source
Query Source
data request
Database in Search
search hits
Database in Search
transmit data
Searching
(b)
FIGURE 14.11
(a) Archivist processing model; (b) searcher processing model.
Query, codes,
Data
Mobile SE
SAPM H
Mobile SE
Client
SAPM Enabled
Server
FIGURE 14.12
The search agent processing model (SAPM).
the SAPM, an agent (a traveling program) can be sent to perform feature extraction and/or
proximity evaluation on remote databases. Figure 14.12 illustrates the sending of a mobile
search engine to an SAPM-enabled database host.
14.3.3
Perceiving the MPEG-7 Search Engine
One way to perceive the characteristics of an MPEG-7 optimum search engine is to b