A THESIS ON Submitted By

A THESIS ON Submitted By
A THESIS ON
DIGITAL IMAGE COMPRESSION USING DISCRETE
COSINE TRANSFORM & DISCRETE WAVELET
TRANSFORM
Submitted By
Swastik Das
Rasmi Ranjan Sethy
Roll No. : 10506008
Roll No. : 10506013
B.Tech : CSE (8th sem)
B.Tech : CSE (8th sem)
Guided By
Prof. R. Baliarsingh
Department of Computer Science and Engineering
National Institute of Technology, Rourkela
Rourkela-769008, Orissa, India
DIGITAL IMAGE COMPRESSION USING DISCRETE COSINE
TRANSFORM & DISCRETE WAVELET TRANSFORM
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Computer Science &
Engineering
By
Swastik Das
&
Rasmi Ranjan Sethy
Under the Guidance of
Prof R. Baliarsingh
Department of Computer Science and Engineering
National Institute of Technology Rourkela
Rourkela-769 008, Orissa, India
2009
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled
“IMAGE COMPRESSION USING DCT &
DWT” submitted by Sri Swastik Das(Roll No.10506008) & Sri Rasmi Ranjan Sethy(Roll NO
.10506013) in partial fulfillment of the requirements for the award of Bachelor of Technology
degree in Computer Science & Engineering
at
the
National
Institute
of
Technology,
Rourkela (Deemed University) is an authentic work carried out by them under my supervision
and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any
other University/Institute for the award of any Degree or Diploma.
Date:
Prof. R Baliarsingh
i
ACKNOWLEDGEMENT
My heart pulsates with the thrill for tendering gratitude to those persons who helped me in
completion of the project.
The most pleasant point of presenting a thesis is the opportunity to thank those who have
contributed to it. Unfortunately, the list of expressions of thank no matter how extensive is
always incomplete and inadequate. Indeed this page of acknowledgment shall never be able to
touch the horizon of generosity of those who tendered their help to me.
First and foremost, I would like to
express my gratitude and indebtedness to Prof. R.
Baliarsingh , for his kindness in allowing me for introducing the present topic and for his
inspiring guidance, constructive criticism and valuable suggestion throughout this project
work. I am sincerely thankful to him for his able guidance and pain taking effort in improving
my understanding of this project.
I am also grateful to Prof. Banshidhar Majhi (Head of the Department) for assigning me this
interesting project and for his valuable suggestions and encouragements at various stages of the
work.
An assemblage of this nature could never have been attempted without reference to and
inspiration from the works of others whose details are mentioned in reference section. I
acknowledge my indebtedness to all of them.
Last but not least, my sincere thanks to all my friends who have patiently extended all sorts of
help for accomplishing this undertaking.
DATE:
PLACE:
Swastik Das
Rasmi Ranjan Sethy
Dept. of Computer Science &
Engineering
National Institute of
Technology Rourkela – 769008
ii
ABSTRACT
Image Compression addresses the problem of reducing the amount of data required to
represent the digital image. Compression is achieved by the removal of one or more of
three basic data redundancies: (1) Coding redundancy, which is present when less than
optimal (i.e. the smallest length) code words are used; (2) Interpixel redundancy, which
results from correlations between the pixels of an image & (3) psycho visual redundancy
which is due to data that is ignored by the human visual system (i.e. visually
nonessential information). Huffman codes contain the smallest possible number of code
symbols (e.g., bits) per source symbol (e.g., grey level value) subject to the constraint that
the source symbols are coded one at a time. So, Huffman coding when combined with
technique of reducing the image redundancies using Discrete Cosine Transform (DCT)
helps in compressing the image data to a very good extent.
The Discrete Cosine Transform (DCT) is an example of transform coding. The current
JPEG standard uses the DCT as its basis. The DC relocates the highest energies to the upper
left corner of the image. The lesser energy or information is relocated into other areas. The
DCT is fast. It can be quickly calculated and is best for images with smooth edges like
photos with human subjects. The DCT coefficients are all real numbers unlike the Fourier
Transform. The Inverse Discrete Cosine Transform (IDCT) can be used to retrieve the
image from its transform representation.
The Discrete wavelet transform (DWT) has gained widespread acceptance in signal
processing and image compression. Because of their inherent multi-resolution nature,
wavelet-coding schemes are especially suitable for applications where scalability and
tolerable degradation are important. Recently the JPEG committee has released its new
image coding standard, JPEG-2000, which has been based upon DWT.
iii
List of Acronyms
Acronym
Description
DCT
Discrete Cosine Transformation
DWT
Discrete Wavelet Transformation
DFT
Discrete Fourier Transformation
FFT
Fast Fourier Transformation
FWT
Fast Wavelet Transformation
JPEG
Joint Photographic Expert Group
JPEG-2000
Joint Photographic Expert Group-2000
MPEG
Moving Pictures Experts Group
MSE
Mean Square Error
PSNR
Peak Signal to Noise Ratio
SNR
Signal-to-noise Ratio
ISO
International Standards Organization
LOT
Lapped Orthogonal Transforms
IEC
International Electro-Technical Commission
DPCM
Discrete pulse code modulation
FAX
Facsimile transmission
KLT
Karhunen Lòeve Transform
IDCT
Inverse Discrete Cosine Transform
FDCT
A Forward Discrete Cosine Transform
iv
List of Symbols
Symbol
Description
CR
Compression ratio
RD
Relative data redundancy
SNRms
Mean-square Signal-to-noise ratio
erms
Root-mean-square error
v
CONTENT
Page no
Certificate
i
Acknowledgement
Abstract
ii
iii
List of Acronyms
iv
List of Symbols
CHAPTER :1 Introduction
v
1.1 Overview
1
1.2 Objectives
1
1.3 OrganiZation of the thesis
2
CHAPTER 2.Image Compression
2.1 Preview
3
2.2 Need of compression
4
2.3 Fundamentals of image compression
5
2.4 Image Compression and Reconstruction
2.5 Different classes of compression technique
6
8
2.6 A typical image coder .
9
2.6.1 Source Encoder (or Linear Transformer)
9
2.6.2 Quantizer
9
2.6.3 Entropy Encoder
9
2.7 Motivation
10
CHAPTER 3
.
IMAGE COMPRESSION USING DISCRETE COSINE TRANSFORM
Preamble
3.1
3.2
3.3
3.4
3.5
The Process
Discrete cosine transform (DCT) equation :JPEG Compression
Quantisation
Coding
3.5.1 Huffman Coding
3.5.2 Huffman Decoding
3.6 Result
.
.
.
.
12
13
16
18
18
19
21
22
3.6.1 Original input image
22
3.6.2 Image after applying DCT
22
3.6.3 Histogram of the DCT coefficients of the upper half of the image
before quantization.
23
3.6.4 Histogram of the DCT coefficients of the lower half of the image
before quantization.
23
3.6.5 Histogram of the DCT coefficients of the upper half of the image .
after quantization.
24
3.6.6 Histogram of the DCT coefficients of the lower half of the image
after quantization.
24
3.6.7 Image after compression
25
3.6.8 Parameters associated with the output image (fig.2.4)
26
3.6.9 Parameters associated with the output image(fig.2.5)
26
CHAPTER 4. IMAGE COMPRESSION USING DISCRETE WAVELET TRANSFORM
4.1 What is a Wavelet Transform ?
27
4.2 Why Wavelet-based Compression?
27
4.3 Subband Coding
29
4.3.1 From Subband to Wavelet Coding
30
4.4 Link between Wavelet Transform and Filterbank
30
4.5 An Example of Wavelet Decomposition
31
4.6 Result
33
4.7 Conclusion
34
References
35
LIST OF TABLES
Page no
1.1
Multimedia data types and uncompressed storage space, transmission bandwidth, a
and transmission time required…………………………………………………….4
.
1.2
Huffman coding ………………………………………………………………… 20
1.3
Parameters associated with the output image at different decomposition level and .
threshold…………………………………………………………………………..33
LIST OF FIGURES
1.1
Image compression System………………………………………………………..6
1.2
Image decompression System…………………………………………………......7
1.3
A Typical Lossy Signal/Image Encoder………………………………………...…8
1.4
JPEG Encoder……………………………….16
1.5
JPEG Decoder……………………………….17
1.6
Huffman Tree……………………………………………………………………...21
1.7 Original input image(Lena image)………………………………………………...22
1.8
Image after applying DCT……………………………………………………… 22
1.9 Histogram of the DCT coefficients of the upper half of the image before
quantization………………………………………………………………………..23
2.0 Histogram of the DCT coefficients of the lower half of the image before
quantization…………………………………………………………………………23
2.1 Histogram of the DCT coefficients of the upper half of the image after
quantization………………………………………………………………………..24
2.2 Histogram of the DCT coefficients of the lower half of the image after
quantization………………………………………………………………………..24
2.3 Image after compression…………………………………………………………...25
2.4 Image of Parameters(3.6.8) associated with the output image implemented
in MATLAB…………………………………………………………………………….…26
2.5 Image of Parameters(3.6.9) associated with the output image implemented
in MATLAB………………………………………………………………………………26
2.6 (a) Original Lena Image, and (b) Reconstructed Lena with DC component only, to
show blocking artifacts…………………………………………………………………...28
2.7 Separable 4-subband Filterbank,…………………………………………………...29
2.8 Partition of the Frequency Domain……………………………………………………………..29
2.9 Three level octave-band decomposition of Lena image,………………………………….31
3.0 Spectral decomposition and ordering……………………………………………………..32
Chapter :1
Introduction
1.1 Overview
Uncompressed multimedia (graphics, audio and video) data requires considerable storage
capacity and transmission bandwidth. Despite rapid progress in mass-storage density,
processor speeds, and digital communication system performance, demand for data storage
capacity and data-transmission bandwidth continues to outstrip the capabilities of available
technologies. The recent growth of data intensive multimedia-based web applications have
not only sustained the need for more efficient ways to encode signals and images but have
made compression of such signals central to storage and communication technology.
For still image compression, the `Joint Photographic Experts Group' or JPEG [7]standard
has been established by ISO (International Standards Organization) and IEC (International
Electro-Technical Commission). The performance of these coders generally degrades at low
bit-rates mainly because of the underlying block-based Discrete Cosine Transform (DCT)
scheme. More recently, the wavelet transform has emerged as a cutting edge technology,
within the field of image compression. Wavelet-based coding provides substantial
improvements in picture quality at higher compression ratios.
Over the past few years, a variety of powerful and sophisticated wavelet-based schemes for
image compression, as discussed later, have been developed and implemented. Because of
the many advantages, the top contenders in the upcoming JPEG-2000 standard are all
wavelet-based compression algorithms.
1.2 Objectives
Image Compression addresses the problem of reducing the amount of data required
torepresent the digital image. Compression is achieved by the removal of one or more of
threebasic data redundancies: (1) Coding redundancy, which is present when less than
optimal (i.e.the smallest length) code words are used; (2) Interpixel redundancy, which
results fromcorrelations between the pixels of an image; &/or (3) psycho visual redundancy
which is due todata that is ignored by the human visual system (i.e. visually nonessential
information).Huffman codes contain the smallest possible number of code symbols (e.g.,
bits) per source symbol (e.g., grey level value) subject to the constraint that the source
symbols are coded one at a time. So, Huffman coding when combined with technique of
reducing the image redundancies using Discrete Cosine Transform (DCT) helps in
compressing the image data to a very good extent.
Page | 1
1.3
Organization of the Thesis
Rest of the thesis is organized into the following chapters :
A discussion on Fundamentals of image compression, Different classes of compression
technique ,a typical image coder are given in Chapter 2.
Image compression using discrete cosine transform is proposed in Chapter 3. The objective
is to achieve a reasonable compression ratio as well as better quality of reproduction
of image with a low power consumption. Simulation result shows that a compression
ratio of 2.0026 is achieved.
Image compression using discrete wavelet transform is proposed in Chapter 4.
Parameters associated with the compression process are analysed & the conclusion is given.
Page | 2
Chapter :2
IMAGE COMPRESSION
2.1 Preview
Image compression addresses the problem of reducing the amount of data required to
represent a digital image .The underlying basis of the reduction process is the removal of
redundant data. From a mathematical viewpoint, this amounts to transforming a 2-D pixel
array into a statistically uncorrelated data set .The transformation is applied prior to storage
or transmission of the image. At some later time, the compressed image is decompressed to
reconstruct the original image or approximation of it.
Interest in image compression dates back more than 35 years. The initial focus of research
efforts in this field was on the development of analog methods for reducing video
transmission bandwidth, a process called bandwidth compression. The advent of the digital
computer and subsequent development of advanced integrated circuits, however, caused
interest to shift from analog to digital compression approaches. with the relatively recent
adaption of several key international image compression standards ,the field has undergone
significant growth through the practical application of the theoretic work that began in the
1940s, when C.E Shannon and others first formulated the probabilistic view of information
and its representation , transmission, and compression[1].
Currently image compression is recognized as an “enabling technology”. In addition to the
areas Just mentioned ,image compression is the natural technology for handling the
increased spatial resolution of today’s imaging sensors and evolving broadcast television
standards. Furthermore image compression plays a major role in many important and
diverse applications , including televideo-conferencing ,remote sensing(the use of satellite
imagery for weather and other earth –resource applications), document and medical imaging
,facsimile transmission(FAX)[3], and the control of remotely piloted vehicles in military ,
space and hazardous waste management applications.
Page | 3
2.2 Need of compression
The figures in Table 1.1 show the qualitative transition from simple text to full-motion video
data and the disk space, transmission bandwidth, and transmission time needed to store and
transmit such uncompressed data.
Transmission
Bits/Pixel Uncompressed Transmission
Multimedia
Time (using a
or
Size
Bandwidth
Size/Duration
28.8K
Data
Bits/Sample (B for bytes)
(b for bits)
Modem)
A page of
text
11'' x 8.5''
Varying
resolution
4-8 KB
32-64
Kb/page
1.1 - 2.2 sec
Telephone
quality
speech
10 sec
8 bps
80 KB
64 Kb/sec
22.2 sec
Grayscale
Image
512 x 512
8 bpp
262 KB
Color Image
512 x 512
24 bpp
786 KB
6.29
Mb/image
3 min 39 sec
Medical
Image
2048 x 1680
12 bpp
5.16 MB
41.3
Mb/image
23 min 54 sec
SHD Image
2048 x 2048
24 bpp
12.58 MB
100
Mb/image
58 min 15 sec
Full-motion
Video
640 x 480, 1
min
(30
frames/sec)
24 bpp
1.66 GB
221 Mb/sec
5 days 8 hrs
2.1 Mb/image 1 min 13 sec
Table 1.1 Multimedia data types and uncompressed storage space, transmission
bandwidth, and transmission time required. The prefix kilo- denotes a factor of 1000
rather than 1024.
The examples above clearly illustrate the need for sufficient storage space, large
transmission bandwidth, and long transmission time for image, audio, and video data. At the
present state of technology, the only solution is to compress multimedia data before its
storage and transmission, and decompress it at the receiver for play back. For example, with
a compression ratio of 32:1, the space, bandwidth, and transmission time requirements can
be reduced by a factor of 32, with acceptable quality[2].
other hand, first transforms the image from its spatial domain representation to a different
type of representation using some well-known transform and then codes the transformed
values (coefficients). This method provides greater data compression compared to predictive
methods, although at the expense of greater computation.
Page | 4
2.3 Fundamentals of image compression
The term data compression refers to the process of reducing the amount of data required to
represent a given quantity of information. A common characteristic of most images is that
the neighboring pixels are correlated and therefore contain redundant information. The
foremost task then is to find less correlated representation of the image. Two fundamental
components of compression are redundancy and irrelevancy reduction. Redundancy
reduction aims at removing duplication from the signal source (image/video). Irrelevancy
reduction omits parts of the signal that will not be noticed by the signal receiver, namely the
Human Visual System (HVS).
It is not an abstract concept but a mathematically quantifiable entity.If n1 and n2 denote the
number of information-carrying units in the two data sets that represent the same
information , the relative data redundancy RD of the first data set (the one characterized by
n1)can be defined as
RD = 1 - 1/ CR
Where CR , commonly called the compression ratio,is
CR = n1/n2 .
n2 = n1 and RD = 0, indicating that (relative to the second data set)the first
representation of the information contains no redundant data. When n2<< n1. CR -> ∞
and RD ->1,implying significant compression and highly redundant data. When n2 >> n1,
For the case
CR ->0and RD -> -∞ ,indicating that the second data set contains much more data than the
original representation. Generally CR =10(10:1) defines that the first data set has 10
information carrying units for every 1 unit in the second or compressed data set. Thus the
corresponding redundancy of 0.9 means 90 percent of the data in the first data set is
redundant with respect to the second one.[1]
In
order
to
be
useful,
a
compression
algorithm
has
a
corresponding
decompression algorithm that reproduces the original file once the compressed file is
given. There have been many types of compression algorithms developed. These
algorithms fall into two broad types, lossless algorithms and lossy algorithms. A
lossless algorithm reproduces the data exactly same as the original one. A lossy
algorithm, as its name implies, loses some data. Data loss may be unacceptable in
Page | 5
many applications. For example, text compression must be lossless because a very
small difference can result in statements with totally different meanings. There are also
many situations where loss may be either unnoticeable or acceptable.
In image
compression, for example, the exact reconstructed value of each sample of the image
2.4 Image Compression and Reconstruction
Three basic data redundancies can be categorized in the image compression
standard.
1.
Spatial redundancy due to the correlation between neighboring pixels.
2.
Spectral redundancy due to correlation between the color components.
3.
Psycho-visual redundancy due to properties of the human visual system.
The spatial and spectral redundancies are present because certain spatial and
spectral patterns between the pixels and the color components are common to each
other, whereas the psycho-visual redundancy originates from the fact that the human
eye is insensitive to certain spatial frequencies. The principle of image compression
algorithms are (i) reducing the redundancy in the image data and (or) (ii) producing a
reconstructed image from the original image with the introduction of error that is
insignificant to the intended applications. The aim here is to obtain an acceptable
representation of digital image while preserving the essential information contained
in that particular data set.[2]
Original
Transform
Quantisation
Image
Lossless
Coding
Compressed
Image
Figure 1.1 Image
compression System
Page | 6
The
problem
faced
by
image
compression
is
very
easy
to
define,
as
demonstrated in figure 1.1. First the original digital image is usually transformed into
another domain, where it is highly de-correlated by using some transform. This decorrelation concentrates the important image information into a more compact form.
The compressor then removes the redundancy in the transformed image and stores it
into a compressed file or data stream.
In the second stage, the quantisation block
reduces the accuracy of the transformed output in accordance with some pre- established
fidelity criterion. Also this stage reduces the psycho-visual redundancy of the input
image. Quantisation operation is a reversible process and thus may be omitted when
there is a need of error free or lossless compression. In the final stage of the data
compression model the symbol coder creates a fixed or variable-length code to
represent the quantiser output and maps the output in accordance with the code.
Generally a variable-length code is used to represent the mapped and quantised data set. It
assigns the shortest code words to the most frequently occurring output values and thus
reduces coding redundancy. The operation in fact is a reversible one.
The decompression reverses the compression process to produce the
recovered image as shown in figure 1.2. The recovered image may have lost some
information due to the compression, and may have an error or distortion compared to
the original image.
Compressd
Image
Inverse
Transform
dequantisation
Lossless
Decodig
Reconstructed
Image
Figure 1.2 Image
decompression System
Page | 7
2.5
DIFFERENT CLASSES OF COMPRESSION TECHNIQUES.
Two ways of classifying compression techniques are mentioned here.
(a) Lossless vs. Lossy compression: In lossless compression schemes, the reconstructed
image, after compression, is numerically identical to the original image. However lossless
compression can only a achieve a modest amount of compression. An image reconstructed
following lossy compression contains degradation relative to the original. Often this is
because the compression scheme completely discards redundant information. However,
lossy schemes are capable of achieving much higher compression. Under normal viewing
conditions, no visible loss is perceived (visually lossless).
(b) Predictive vs. Transform coding: In predictive coding, information already sent or
available is used to predict future values, and the difference is coded. Since this is done in
the image or spatial domain, it is relatively simple to implement and is readily adapted to
local image characteristics. Differential Pulse Code Modulation (DPCM) is one particular
example of predictive coding [3]. Transform coding, on the other hand, first transforms the
image from its spatial domain representation to a different type of representation using some
well-known transform and then codes the transformed values (coefficients). This method
provides greater data compression compared to predictive methods, although at the expense
of greater computation.
2.6 A TYPICAL IMAGE CODER .
A typical lossy image compression system is shown in Fig. 1.3. It consists of three closely
connected components namely (a) Source Encoder (b) Quantizer, and (c) Entropy Encoder.
Compression is accomplished by applying a linear transform to decorrelate the image data,
quantizing the resulting transform coefficients, and entropy coding the quantized values.
Fig. 1. 3 A Typical Lossy Signal/Image Encoder
Page | 8
2.6.1 Source Encoder (or Linear Transformer)
Over the years, a variety of linear transforms have been developed which include Discrete
Fourier Transform (DFT), Discrete Cosine Transform (DCT) , Discrete Wavelet Transform
(DWT) and many more, each with its own advantages and disadvantages.
2.6.2 Quantizer
A quantizer simply reduces the number of bits needed to store the transformed coefficients
by reducing the precision of those values. Since this is a many-to-one mapping, it is a lossy
process and is the main source of compression in an encoder. Quantization can be
performed on each individual coefficient, which is known as Scalar Quantization (SQ).
Quantization can also be performed on a group of coefficients together, and this is known as
Vector Quantization (VQ) [6] . Both uniform and non-uniform quantizers can be used
depending on the problem at hand. .
2.6.3 Entropy Encoder
An entropy encoder further compresses the quantized values losslessly to give better overall
compression. It uses a model to accurately determine the probabilities for each quantized
value and produces an appropriate code based on these probabilities so that the resultant
output code stream will be smaller than the input stream. The most commonly used entropy
encoders are the Huffman encoder and the arithmetic encoder, although for applications
requiring fast execution, simple run-length encoding (RLE) has proven very effective [6].
It is important to note that a properly designed quantizer and entropy encoder are absolutely
necessary along with optimum signal transformation to get the best possible compression.
Page | 9
2.7 Motivation
Image compression is an important issue in digital image processing and
finds extensive applications in many fields. This is the basic operation performed
frequently by any digital photography technique to capture an image. For longer use
of the portable photography device it should consume less power so that battery life
will be more. To improve the Conventional techniques of image compressions using
the DCT have already been reported and sufficient literatures are available on this.
The JPEG is a lossy compression scheme, which employs the DCT as a tool and used
mainly in digital cameras for compression of images. In the recent past the demand
for low power image compression is growing. As a result various research workers
are actively engaged to evolve efficient methods of image compression using latest
digital signal processing techniques. The objective is to achieve a reasonable
compression ratio as well as better quality of reproduction of image with a low
power consumption. Keeping these objectives in mind the research work in the
present thesis has been undertaken. In sequel the following problems have been
investigated.
Page | 10
Chapter :3
IMAGE COMPRESSION USING DISCRETE COSINE TRANSFORM
Preview:
Discrete cosine transform (DCT) is widely used in image processing, especially
for
compression. Some of the applications of two-dimensional DCT involve still image
compression and compression of individual video frames, while multidimensional DCT
is mostly used for compression of video streams. DCT is also useful for transferring
multidimensional data to frequency domain, where different operations, like spreadspectrum, data compression, data watermarking, can be performed in easier and more
efficient manner. A number of papers discussing DCT algorithms is available in the
literature that signifies its importance and application.[5]
Hardware implementation of parallel DCT transform is possible, that would give higher
throughput than software solutions.
Special purpose DCT hardware decreases the
computational load from the processor and therefore improves the performance of
complete multimedia system. The throughput is directly influencing the quality of
experience of multimedia content. Another important factor that influences the
quality is the finite register length effect that affects the accuracy of the forward-inverse
transformation process.
Page | 11
Hence, the motivation for investigating hardware specific DCT algorithms is clear. As 2D DCT algorithms are the most typical for image compression, the main focus of this
chapter will be on the efficient hardware implementations of 2-D DCT based
compression by decreasing the number of computations, increasing the accuracy of
reconstruction, and reducing the chip area. This in return reduces the power
consumption of the compression technique. As the number of applications that require
higher-dimensional DCT algorithms are growing, a special attention will be paid to the
algorithms that are easily extensible to higher dimensional cases.
The JPEG standard has been around since the late 1980's and has been an effective
first solution to the standardization of image compression. Although JPEG has some
very useful strategies for DCT quantization and compression, it was only developed for
low compressions. The 8 × 8 DCT block size was chosen for speed(which is less of
an issue now, with the advent of faster processors) not for performance. The JPEG
standard will be briefly explained in this chapter to provide a basis to understand the new
DCT related work [7].
3.1 The Process:The following is the general overview of the JPEG process.Later we will go through the a
detailed tour of JPEG’s method so that a more comprehensive understanding of the process
may be acquired.
1.The image is broken into 8*8 blocks of pixels.
2. Working from left to right, top to bottom, the DCT is applies to each block.
3. Each block is compressed through quantization.
4.The array of compressed blocks that constitute the image is stored in a drastically reduced
amount of space.
5. When desired the image is constructed through decompression, a process that uses the
Inverse Discrete Cosine Transform(IDCT).
Page | 12
3.2 DISCRETE COSINE TRANSFORM (DCT) EQUATION :The DCT is a widely used transformation in transformation for data compression. It is an
orthogonal transform, which has a fixed set of (image independent) basis functions, an
efficient algorithm for computation, and good energy compaction and correlation reduction
properties. Ahmed et al found that the Karhunen Lòeve Transform (KLT) basis function of
a first order Markov image closely resemble those of the DCT [7]. They become identical as
the correlation between the adjacent pixel approaches to one.
The DCT belongs to the family of discrete trigonometric transform, which has 16 members
[44]. The 1D DCT of a 1× N vector x(n) is defined as
where k = 0,1,2,..., N −1 and
The original signal vector x(n) can be reconstructed back from the DCT coefficients
Y[k ] using the Inverse DCT (IDCT) operation and can be defined as
where n = 0,1,2,..., N −1
Page | 13
The DCT can be extended to the transformation of 2D signals or images. This can be
achieved in two steps: by computing the 1D DCT of each of the individual rows of the twodimensional image and then computing the 1D DCT of each column of the image. If
represents a 2D image of size x( n1 , n2 ) N × N , then the 2D DCT of an image is given
by:
……equ.3.1
where j, k,m, n = 0,1,2,..., N −1 and
Similarly the 2D IDCT can be defined as
…equ. 3.2
The DCT is a real valued transform and is closely related to the DFT. In particular, a N ×
N DCT of
x(n1,n2) can be expressed in terms of DFT of its even-symmetric extension,
which leads to a fast computational algorithm. Because of the even-symmetric extension
process, no artificial discontinuities are introduced at the block boundaries. Additionally the
computation of the DCT requires only real arithmetic. Because of the above properties the
DCT is popular and widely used for data compression operation.
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The DCT presented in equations (3.1) and (3.2) is orthonormal and perfectly reconstructing
provided the coefficients are represented to an infinite precision. This means that when the
coefficients are compressed it is possible to obtain a full range of compressions and image
qualities. The coefficients of the DCT are always quantized for high compression, but DCT
is very resistant to quantisation errors due to the statistics of the coefficients it produces.
The coefficients of a DCT are usually linearly quantised by dividing by a predetermined
quantisation step.
The DCT is applied to image blocks N x N pixels in size (where N is usually multiple of 2)
over the entire image. The size of the blocks used is an important factor since they
determine the effectiveness of the transform over the whole image. If the blocks are too
small then the images is not effectively decorrelated but if the blocks are too big then local
features are no longer exploited. The tiling of any transform across the image leads to
artifacts at the block boundaries. The DCT is associated with blocking artifact since the
JPEG standard suffers heavily from this at higher compressions. However the DCT is
protected against blocking artifact as effectively as possible, without interconnecting blocks,
since the DCT basis functions all have a zero gradient at the edges of their blocks. This
means that only the DC level significantly affects the blocking artifact and this can then be
targeted. Ringing is a major problem in DCT operation. When edges occur in an image DCT
relies on the high frequency components to make the image shaper. However these high
frequency components persist across the whole block and although they are effective at
improving the edge quality they tend to 'ring' in the flat areas of the block.
This ringing effect increases, when larger blocks are used, but larger blocks are better in
compression terms, so a trade off is usually established [1].
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3.3 JPEG Compression
The JPEG (Joint Photographic Experts Group) standard has been around
for some time and is the only standard for lossy still image compression. There
are quite a lot of interesting techniques used in the JPEG standard and it is
important to give an overview of how JPEG works. There are several variations of
JPEG, but only the 'baseline' method is discussed here.
8 x 8 blocks
DCT – Based Encoder
FDCT
Source Image
Data
Quantizer
Table
Specifications
Huffman
Encoder
Table
Specifications
Figure 1.4: JPEG Encoder
As shown in the figure 3.1, the image is first partitioned into non-overlapping
8 × 8 blocks. A Forward Discrete Cosine Transform (FDCT) is applied to each block
to convert the spatial domain gray levels of pixels into coefficients in frequency
domain. To improve the precision of the DCT the image is 'zero shifted', before the
DCT is applied. This converts a 0 → 255 image intensity range to a -128 → 127
range, which works more efficiently with the DCT. One of these transformed values
is referred to as the DC coefficient and the other 63 as the AC coefficients [4].
After the computation of DCT coefficients, they are normalized with
different scales according to a quantization table provided by the JPEG standard
conducted by psychovisual evidence. The quantized coefficients are rearranged in a
zigzag scan order for further compressed by an efficient lossless coding algorithm
such as runlength coding, arithmetic coding, Huffman coding. The decoding process
is simply the inverse process of encoding as shown in figure 1.5.
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The decoding process is simply the inverse process of encoding as shown in figure 1.5.
DCT – Based Decoder
Huffman
Decoder
Dequantizer
Compressed
Image Data
Table
Specifications
Table
Specifications
IDCT
Reconstruct
Image Data
Figure 1.5 : JPEG Decoder
Page | 17
3.4 QUANTISATION
DCT-based image compression relies on two techniques to reduce the data required to represent
the image. The first is quantization of the image's DCT coefficients; the second is entropy coding
of the quantized coefficients. Quantization is the process of reducing the number of possible
values of a quantity, thereby reducing the number of bits needed to represent it. In lossy image
compression the transformation decompose the image into uncorrelated parts projected on
orthogonal basis of the transformation. These basis are represented by eigenvectors which are
independent and orthogonal in nature. Taking inverse of the transformed values will result in the
retrieval of the actual image data. For compression of the image, the independent characteristic
of the transformed coefficients are considered, truncating some of these coefficients will not
affect the others. This truncation of the transformed coefficients is actually the lossy process
involved in compression and known as quantization . So we can say that DCT is perfectly
reconstructing, when all the coefficients are calculated and stored to their full resolution. For
high compression, the DCT coefficients are normalized by different scales, according to the
quantization matrix [6].
Vector quantization, (VQ) mainly used for reducing or compressing the image data . Application
VQ on images for compression started from early 1975by Hilbert mainly for the coding of
multispectral Landsat imaginary.
3.5 CODING
After the DCT coefficients have been quantized, the DC coefficients are DPCM coded and then
they are entropy coded along with the AC coefficients. The quantized AC and DC coefficient
values are entropy coded in the same way, but because of the long runs in the AC coefficient, an
additional run length process is applied to them to reduce their redundancy.
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The quantized coefficients are all rearranged in a zigzag order as shown in figure 3.4. The run
length in this zigzag order is described by a RUN-SIZE symbol. The RUN is a count of how
many zeros occurred before the quantized coefficient and the SIZE symbol is used in the same
way as it was for the DC coefficients, but on their AC counter parts. The two symbols are
combined to form a RUN-SIZE symbol and this symbol is then entropy coded. Additional bits
are also transmitted to specify the exact value of the quantized coefficient. A size of zero in the
AC coefficient is used to indicate that the rest of the 8 × 8 block is zeros (End of Block or
EOB) [7].
3.5.1 HUFFMAN CODING:
Huffman coding is an efficient source coding algorithm for source symbols that are not equally
probable. A variable length encoding algorithm was suggested by Huffman in 1952, based on
the source symbol probabilities P(xi), i=1,2…….,L . The algorithm is optimal in the sense that
the average number of bits required to represent the source symbols is a minimum provided the
prefix condition is met. The steps of Huffman coding algorithm are given below [7]:
1.
Arrange the source symbols in increasing order of heir probabilities.
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2.
Take the bottom two symbols & tie them together as shown in Figure 3. Add the
probabilities of the two symbols & write it on the combined node. Label the two branches with a
‘1’ & a ‘0’ as depicted in Figure 3
3.
Treat this sum of probabilities as a new probability associated with a new symbol. Again
pick the two smallest probabilities, tie them together to form a new probability. Each time we
perform the combination of two symbols we reduce the total number of symbols by
one. Whenever we tie together two probabilities (nodes) we label the two branches with a ‘0’ &
a ‘1’.
4.
Continue the procedure until only one procedure is left (& it should be one if
your addition is correct). This completes the construction of the Huffman Tree.
5.
To find out the prefix codeword for any symbol, follow the branches from the final node
back to the symbol. While tracing back the route read out the labels on the branches. This is the
codeword for the symbol.
The algorithm can be easily understood using the following example :
TABLE
1.2
Symbol
X1
X2
X3
X4
X5
X6
X7
Probability Codeword
0.46
1
0.30
00
0.12
010
0.06
0110
0.03
01110
0.02
011110
0.01
011111
Code length
1
2
3
4
5
6
6
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FIGURE: 1.6
Huffman Coding for Table
3.5.2 HUFFMAN DECODING
The Huffman Code in Table 1 & FIGURE 4 is an instantaneous uniquely decodable
block code. It is a block code because each source symbol is mapped into a fixed sequence of
code symbols. It is instantaneous because each codeword in a string of code symbols can
be decoded without referencing succeeding symbols. That is, in any given Huffman code,
no codeword is a prefix of any other codeword. And it is uniquely decodable because a string of
code symbols can be decoded only in one way. Thus any string of Huffman encoded symbols
can be decoded by examining the individual symbols of the string in left to right
manner. Because we are using an instantaneous uniquely decodable block code, there is no need
to insert delimiters between the encoded pixels.
For Example consider a 19 bit string 1010000111011011111 which can be decoded uniquely as
x1 x3 x2 x4 x1 x1 x7 [7].
A left to right scan of the resulting string reveals that the first valid code word is 1 which is a
code symbol for, next valid code is 010 which corresponds to x1, continuing in this manner, we
obtain a completely decoded sequence given by x1 x3 x2 x4 x1 x1 x7.
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3.6 RESULT :-
3.6.1 - Original input image(1.7)
3.6.2 - Image after applying DC (1.8)
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3.6.3 - Histogram of the DCT coefficients of the upper half of the image before
quantization(1.9)
3.6.4 Histogram of the DCT coefficients of the lower half of the image before
quantization.(2.0)
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3.6.5 Histogram of the DCT coefficients of the upper half of the image after
quantization.(2.1)
3.6.6 Histogram of the DCT coefficients of the lower half of the image after
quantization.(2.2)
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3.6.7 - Image after compression(2.3)
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3.6.8 - Parameters associated with the output image(2.4)
3.6.9 - Parameters associated with the output image(2.5)
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Chapter :4
IMAGE COMPRESSION USING DISCRETE WAVELET TRANSFORM :-
The wavelet transform (WT) has gained widespread acceptance in signal processing and image
compression.Because of their inherent multi-resolution nature, wavelet-coding schemes are
especially suitable for applications where scalability and tolerable degradation are
important.Recently the JPEG committee has released its new image coding standard, JPEG2000, which has been based upon DWT [1].
4.1 What is a Wavelet Transform ?
Wavelets are functions defined over a finite interval and having an average value of zero. The
basic idea of the wavelet transform is to represent any arbitrary function (t) as a superposition of
a set of such wavelets or basis functions. These basis functions or baby wavelets are obtained
from a single prototype wavelet called the mother wavelet, by dilations or contractions (scaling)
and translations (shifts). The Discrete Wavelet Transform of a finite length signal x(n) having N
components, for example, is expressed by an N x N matrix.
4.2 Why Wavelet-based Compression?
Despite all the advantages of JPEG compression schemes based on DCT namely simplicity,
satisfactory performance, and availability of special purpose hardware for implementation, these
are not without their shortcomings. Since the input image needs to be ``blocked,'' correlation
across the block boundaries is not eliminated. This results in noticeable and annoying ``blocking
artifacts'' particularly at low bit rates as shown in Fig 2.6 . Lapped Orthogonal Transforms (LOT)
[9] attempt to solve this problem by using smoothly overlapping blocks. Although blocking
effects are reduced in LOT compressed images, increased computational complexity of such
algorithms do not justify wide replacement of DCT by LOT.
Page | 27
(a)
(b)
Fig. 2.6 (a) Original Lena Image, and (b) Reconstructed Lena with DC component only,
to show blocking artifacts
Over the past several years, the wavelet transform has gained widespread acceptance in signal
processing in general, and in image compression research in particular. In many applications
wavelet-based schemes (also referred as subband coding) outperform other coding schemes like
the one based on DCT. Since there is no need to block the input image and its basis functions
have variable length, wavelet coding schemes at higher compression avoid blocking artifacts.
Wavelet-based coding is more robust under transmission and decoding errors, and also facilitates
progressive transmission of images. In addition, they are better matched to the HVS
characteristics. Because of their inherent multiresolution nature [9], wavelet coding schemes are
especially suitable for applications where scalability and tolerable degradation are important.
Page | 28
4.3 Subband Coding
he fundamental concept behind Subband Coding (SBC) is to split up the frequency band of a signal
(image in our case) and then to code each subband using a coder and bit rate accurately matched to
the statistics of the band. SBC has been used extensively first in speech coding and later in image
coding [12] because of its inherent advantages namely variable bit assignment among the subbands
as well as coding error confinement within the subbands.
Fig. 2.7
Separable 4-subband Filterbank,
Fig 2.8 . Partition of the Frequency Domain.
Page | 29
ods and O'Neil [12] used a separable combination of one-dimensional Quadrature Mirror
Filterbanks (QMF) to perform a 4-band decomposition by the row-column approach as shown
in Fig. 2.7 Corresponding division of the frequency spectrum is shown in Fig. 2.8 . The process
can be iterated to obtain higher band decomposition filter trees. At the decoder, the subband
signals are decoded, upsampled and passed through a bank of synthesis filters and properly
summed up to yield the reconstructed image.
4.3.1 From Subband to Wavelet Coding
Over the years, there have been many efforts leading to improved and efficient design of
filterbanks and subband coding techniques. Since 1990, methods very similar and closely related
to subband coding have been proposed by various researchers under the name of Wavelet
Coding (WC) using filters specifically designed for this purpose [12]. Such filters must meet
additional and often conflicting requirements . These include short impulse response of the
analysis filters to preserve the localization of image features as well as to have fast computation,
short impulse response of the synthesis filters to prevent spreading of artifacts (ringing around
edges) resulting from quantization errors, and linear phase of both types of filters since nonlinear
phase introduce unpleasant waveform distortions around edges. Orthogonality is another useful
requirement since orthogonal filters, in addition to preservation of energy, implement a unitary
transform between the input and the subbands. But, as in the case of 1-D, in two-band Finite
Impulse Response (FIR) systems linear phase and orthogonality are mutually exclusive, and so
orthogonality is sacrificed to achieve linear phase.
4.4 Link between Wavelet Transform and Filterbank
Construction of orthonormal families of wavelet basis functions can be carried out in continuous
time. However, the same can also be derived by starting from discrete-time filters. Daubechies
[9] was the first to discover that the discrete-time filters or QMFs can be iterated and under
certain regularity conditions will lead to continuous-time wavelets. This is a very practical and
extremely useful wavelet decomposition scheme, since FIR discrete-time filters can be used to
implement them. It follows that the orthonormal bases in correspond to a subband coding
scheme with exact reconstruction property, using the same FIR filters for reconstruction as for
decomposition. So, subband coding developed earlier is in fact a form of wavelet coding in
disguise. Wavelets did not gain popularity in image coding until Daubechies established this link
Page | 30
in late 1980s. Later a systematic way of constructing a family of compactly supported
biorthogonal wavelets was developed by Cohen, Daubechies, and Feauveau (CDF) . Although
the design and choice of various filters and the construction of different wavelets from the
iteration of such filters are very important, it is beyond the scope of this article.
4.5 An Example of Wavelet Decomposition
There are several ways wavelet transforms can decompose a signal into various subbands. These
include uniform decomposition, octave-band decomposition, and adaptive or wavelet-packet
decomposition [12]. Out of these, octave-band decomposition is the most widely used. This is a
non-uniform band splitting method that decomposes the lower frequency part into narrower
bands and the high-pass output at each level is left without any further decomposition. Fig. 2.9
shows the various subband images of a 3-level octave-band decomposed Lena using the popular
CDF-9/7 [7] biorthogonal wavelet.
Fig 2.9 Three level octave-band decomposition of Lena image,
Page | 31
Fig 3.0
Spectral decomposition and ordering.
Most of the subband and wavelet coding schemes can also be described in terms of the general
framework depicted as in Fig. 1. The main difference from the JPEG standard is the use of DWT
rather than DCT. Also, the image need not be split into 8 x 8 disjoint blocks. Of course, many
enhancements have been made to the standard quantization and encoding techniques to take
advantage of how the wavelet transforms works on an image and the properties and statistics of
transformed coefficients so generated.
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4.6 Result :-
Image size
Decomposition
Threshhold
Lavel
Compression
ratio
256 x 256
3
40
16.1759
256 x 256
3
60
21.6558
256 x 256
3
80
25.9193
256 x 256
4
40
15.7964
256 x 256
4
60
22.4193
256 x 256
4
80
28.9469
256 x 256
5
40
14.9430
256 x 256
5
60
21.3887
256 x 256
5
80
28.1812
Table 1.3 Parameters associated with the output image at different
decomposition level and threshold
Page | 33
4.7 Conclusion :Image compression is of prime importance in Real time applications like video conferencing
where data are transmitted through a channel.
Using JPEG standard DCT is used for mapping which reduces theinterpixel
redundanciesfollowed by quantization which reduces the psychovisual redundancies then
coding redundanciy is reduced by the use of optimal code word having minimum average
length.
In JPEG 2000 standard of image compression DWT is used for mapping, allother methods
remaining same.DWT is more general and efficient than DCT due to the following result:-
No need to divide the input coding into non-overlapping 2-D blocks, it has higher
compression ratios avoid blocking artifacts.
Allows good localization both in time and spatial frequency domain.
Transformation of the whole image introduces inherent scaling
Better identification of which data is relevant to human perception higher
compression ratio
Page | 34
References:
[1] R. C. Gonzalez and R. E. Woods, “Digital Image Processing”, Reading. MA:
Addison Wesley, 2004.
[2] David Salomon, Data Compression, The Complete Reference, 2nd Edition
Springer-Verlag 1998.
[3] Digital Compression and coding of Continuous-tone still images, part 1,
requirements and Guidelines. ISO/IEC JTC1 Draft International Standard
10918-1, Nov. 1991.
[4] G. K. Wallace, “The JPEG Still Picture Compression Standard”, IEEE Trans.
On Consumer Electronics, vol.38, No.1, pp. xviii – xxxiv, Feb 1992.
[5] S. Martucea, “Symmetric convolution and the discrete sine and cosine
transform”, IEEE Transaction on Signal Processing, vol. 42, p. 1038-1051,
May’ 1994.
[6] R. M. Gray, D. L. Neuhoff, “Quantization”, IEEE Trans. Inform. Theory, Vol.
44, No. 6, 1998.
[7] N. Ahmed, T. Natrajan, and K. R. Rao, “Discrete Cosine Transform”, IEEE
Transactions on Computers, vol. 23, July 1989.
[8] Pennebaker, W. B. and Mitchell, J. L. JPEG - Still Image Data Compression Standards,
Van Nostrand Reinhold, 1993.
[9] http://en.wikipedia.org/wiki/Discrete_cosine_transform
[10] Rao, K. R. and Yip, P. Discrete Cosine Transforms - Algorithms, Advantages, Applications,
Academic Press, 1990.
[11] Gersho, A. and Gray, R. M. Vector Quantization and Signal Compression, Kluwer Academic
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[12] Strang, G. and Nguyen, T. Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA,
.
1996 http://www-math.mit.edu/~gs/books/wfb.html.
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