Analysis of Image Compression Methods Based On Transform and Fractal Coding

Analysis of Image Compression Methods Based On Transform and Fractal Coding
Analysis of Image Compression Methods Based On
Transform and Fractal Coding
Archana Deshlahra
Analysis of Image Compression Methods Based On
Transform and Fractal Coding
A Thesis Submitted in partial fulfillment
for the award degree Of
Master of Technology
In
Electronics & Communication Engineering
“Electronics & Instrumentation” Specialization
by
Archana Deshlahra (211EC3306)
Under the supervision of
Prof. Ajit Kumar Sahoo
Department of Electronic & Communication Engineering
National Institute of Technology, Rourkela.
May- 2013
National Institute Of Technology, Rourkela
Certificate
This is to certify that the thesis entitled, ―Analysis of Image Compression Methods Based
On Transform and Fractal Coding” submitted by Archana Deshlahra to the Department of
Electronic & Communication Engineering, National Institute Of Technology, Rourkela, India,
during the academic session 2012-2013 for the award of the degree of Master of Technology in
“Electronics & Instrumentation” specialization, is a bona-fide record of work carried by him
under my supervision and guidance. The thesis has fulfilled all the requirements as per the
regulations of this institute and in our opinion reached the standard for submission.
Place: Rourkela, India
Prof. Ajit Kumar Sahoo
Date:
NIT,Rourkela -769008 (India)
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Declaration
I, Archana Deshlahra, declare that:
1. The work contained in this report is original and has been done by me under the guidance of
my supervisor Prof. Ajit Kumar Sahoo
2. The work has not been submitted to any other Institute for any degree or diploma.
3. I have followed the guidelines provided by the institute in preparing the report.
4. I have conformed to the norms and guidelines given in the Ethical Code of Conduct of the
Institute.
5. Whenever I have used materials (data, theoretical analysis, figures and text) from other
sources, I have given due credit to them by citing them in the text of the report and giving their
details in the references.
Department of Electronics & Communication Engineering
NIT, Rourkela,
Rourkela-769008
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Archana Deshlahra
Acknowledgement
I avail this opportunity to extend my hearty indebtedness to my guide Prof. A.K.Sahoo,
Department of Electronics and Communication Engineering, for their valuable guidance,
constant encouragement and kind help at different stages for the execution of this work.
I also express our sincere gratitude to Dr. S. Meher, Head of the Department, Electronics
and Communication Engineering, for providing valuable departmental facilities.
My special thanks to Mr. Deepak Singh (Phd. Scholar) ,Mr. Sanand (Phd. Scholar), Mr.
Nihar (Phd. Scholar), Bijay, Ripan , Sonia for their help and valuable suggestions. I would like
to give special thanks to Sanu, Harsu di, G.S. Shirnewar those helped and supported me
throughout my research work. I must thank Mr. Saubhagya (Phd. Scholar) , Mrs. Hunny
Mehrotra (Phd. Scholar) for their support and help.
I sincerely thank to all my friends, Research scholars of ECE
Department, M.Tech
(E&I) students and all academic and non-teaching staffs in NIT Rourkela who helped me. I
would like to thank all those who made my stay in Rourkela an unforgettable and rewarding
experience.
Finally, I dedicate this thesis to my family: my dear father, my dearest mother who
supported me morally despite the distance that separates us. I thank them from the bottom of my
heart for their motivation, inspiration, love they always give me. Without their support nothing
would have been possible. I am greatly indebted to them for everything that I am.
ARCHANA DESHLAHRA
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Abstract
Image compression is process to remove the redundant information from the image so
that only essential information can be stored to reduce the storage size, transmission bandwidth
and transmission time. The essential information is extracted by various transforms techniques
such that it can be reconstructed without losing quality and information of the image. In this
thesis work comparative analysis of image compression is done by four transform method,
which are Discrete Cosine Transform (DCT), Discrete Wavelet Transform( DWT) & Hybrid
(DCT+DWT) Transform and fractal coding. MATLAB programs were written for each of the
above method and concluded based on the results obtained that hybrid DWT-DCT algorithm
performs much better than the standalone JPEG-based DCT, DWT algorithms in terms of peak
signal to noise ratio (PSNR), as well as visual perception at higher compression ratio. The
popular JPEG standard is widely used in digital cameras and web –based image delivery. The
wavelet transform, which is part of the new JPEG 2000 standard, claims to minimize some of the
visually distracting artifacts that can appear in JPEG images. For one thing, it uses much larger
blocks- selectable, but typically1024 x 1024 pixels – for compression, rather than the 8 X 8 pixel
blocks used in the original JPEG method, which often produced visible boundaries. Fractal
compression has also shown promise and claims to be able to enlarge images by inserting
―realistic‖ detail beyond the resolution limit of the original. Each method is discussed in the
thesis.
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Contents
Declaration ...................................................................................................................................ii
Acknowledgement …................................................................................................................... iii
Abstract ...................................................................................................................................... iv
Contents ........................................................................................................................................ v
List of Figures .........................................................................................................................viii
Chapter 1
Image Compression ..................................................................................................................... 1
1.1. Introduction .............................................................................................................................1
1.2 Data Compression Model......................................................................................................... 2
1.2.1 Advantages of Data Compression .............................................................................2
1.2.2 Disadvantages of Data Compression ........................................................................ 4
1.3 Image Compression Based on Entropy..................................................................................... 5
1.3.1 Lossless compression................................................................................................ 6
1.3.2 Lossy compression................................................................................................... 6
1.4 Objective of thesis……………................................................................................................ 7
1.5 Thesis Structure………………................................................................................................ 7
Chapter 2
Image Representation and Transform Coding ......................................................................... 8
2.1 Image …………..…………….................................................................................................. 8
2.2 Redundancy .............................................................................................................................12
2.2.1 Coding redundancy...................................................................................................13
2.2.2 lnterpixel redundancy…........................................................................................... 13
2.2.3 Psychovisual redundancy .........................................................................................13
2.3 Coding…………..……………................................................................................................14
2.3.1 Pixel coding ……….................................................................................................14
2.3.2 Predictive coding……..............................................................................................14
2.3.3 Transform coding……..............................................................................................15
2.4 Transform-based Image Compression.....................................................................................16
2.4.1 Linear Transform..................................................................................................... 17
2.4.2 Quantizer..................................................................................................................17
2.4.2.1 Scalar Quantization....................................................................................18
2.4.2.2 Vector Quantization...................................................................................18
2.4.2.3 Predictive Quantization..............................................................................19
2.4.3 Entropy Encoding.....................................................................................................19
2.5 Discrete Cosine Transform......................................................................................................19
2.5.1 Coding scheme.........................................................................................................20
2.5.1.1 Compression procedure.............................................................................20
2.5.1.2 Decompression procedure..........................................................................22
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2.5.2 Properties of DCT.....................................................................................................26
2.5.2.1 Decorrelation..............................................................................................26
2.5.2.2 Separability ...............................................................................................26
2.5.2.3 Energy Compaction...................................................................................26
2.5.2.4 Symmetry...................................................................................................28
2.5.2.5 Orthogonality.............................................................................................29
2.5.3 Limitations of DCT..................................................................................................29
2.5.3.1 Blocking artifacts.......................................................................................30
2.5.3.2 False contouring: …..................................................................................30
2.6 Discrete Wavelet Transform (DWT) ...................................................................................31
2.6.1 Multiresolution Concept and Analysis.....................................................................31
2.6.2 Decimator and interpolator.......................................................................................31
2.6.3 Filter bank ................................................................................................................32
2.6.3.1 Analysis bank ..........................................................................................32
2.6.3.2 Synthesis bank..........................................................................................34
2.6.4 Coding scheme.........................................................................................................36
2.6.4.1 Compression procedure............................................................................36
2.6.4.2 Decompression procedure........................................................................36
2.7 Hybrid (DCT+ DWT) Transform………............................................................................37
2.7.1 Coding scheme.........................................................................................................38
2.7.1.1Compression procedure.............................................................................38
2.7.1.2 Decompression procedure........................................................................38
2.8 Fractal Image Compression...................................................................................................40
2.8.1 Baseline Fractal Coding…………............................................................................40
2.8.2 Procedure of proposed speed up encoding................................................................41
2.8.3 Procedure for decoding method ................................................................................42
2.8.4 Advantages and Disadvantages .................................................................................42
2.9 Conclusion ..............................................................................................................................44
Chapter 3
Performance Measurement Parameters....................................................................................45
3.1 Objective evaluation parameters. ............................................................................................45
3.1.1 Mean Square Error (MSE) .........................................................................................45
3.1.2 Peak Signal to Noise Ratio (PSNR) ...........................................................................46
3.1.3 Compression ratio (CR) ...............................................................................................47
3.2 Subjective evaluation parameter..............................................................................................48
3.3 Conclusion ..............................................................................................................................48
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Chapter 4
Performance Evaluation and Simulation Results.....................................................................50
4.1 Simulation tool.........................................................................................................................50
4.2 Simulation Results...................................................................................................................50
4.3 Conclusion ..............................................................................................................................53
Chapter 5
Conclusion and Future work...................................................................................................55
Reference.....................................................................................................................................57
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List of Figures
Fig 1.1 Aspect ratio.........................................................................................................................1
Fig 1.2 A data compression model................................................................................................. 4
Fig 2.1 Digital representation of an image.....................................................................................10
Fig 2.2 Block diagram of the color image compression algorithm. ..............................................11
Fig 2.3 Lena image representation ................................................................................................11
Fig 2.4 Block diagram of a DPCM system. ..................................................................................14
Fig 2.5 Block diagram of transform based image coder .............................................................. 16
Fig 2.6 Quantization effect. ..........................................................................................................18
Fig 2.7 JPEG Quantization table .................................................................................................. 21
Fig 2.8 Zigzag ordering for DCT coefficients ............................................................................ 21
Fig 2.9 Flow chart of compression technique .............................................................................. 23
Fig 2.10 Flow chart of decompression technique ........................................................................ 24
Fig 2.11 A random value input data matrix.................................................................................. 24
Fig 2.12 Transformed coefficients after DCT of the random value input data matrix ................ 25
Fig 2.13 Quantized coefficients ................................................................................................... 25
Fig 2.14 Reconstructed output data ............................................................................................. 25
Fig 2.15 Computation of 2-D DCT using Separability property. ................................................ 26
Fig 2.16 DCT of different Images ................................................................................................ 28
Fig 2.17 Illustration of compression using DCT.......................................................................... 30
Fig 2.18 Illustration of compression using DCT.......................................................................... 30
Fig 2.19 Decimator or down sampler........................................................................................... 32
Fig 2.20 Interpolator or up sampler ............................................................................................. 32
Fig 2.21 Filter bank....................................................................................................................... 32
Fig 2.22 Finer scale and coarser scale wavelet coefficients......................................................... 33
Fig 2.23. Block diagram of 1-D forward DWT............................................................................ 33
Fig 2.24 Block diagram of 2-D forward DWT ............................................................................ 34
Fig 2.25 Block diagram of 2 dimensional inverse DWT ............................................................ 34
Fig 2.26 Illustration of 2 dimensional DWT for an image ‗Lena‘ …........................................... 37
Fig 2.27 Compression technique using Hybrid transform .......................................................... 39
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Fig 2.28 Decompression technique using Hybrid transform ....................................................... 39
Fig 2.29 Fractal coding algorithm................................................................................................. 41
Fig 3.30 A different approach for Fractal coding.........................................................................43
Fig 3.1 Original, reconstructed image using DCT, DWT, Hybrid DWT-DCT …...................... 45
Fig 3.2 CR comparison ................................................................................................................ 46
Fig 4.1 Comparison of visual image quality of reconstructed image for DCT, DWT AND Hybrid
(DCT+DWT) for test images . .................................................................................................... 49
Fig 4.2 Comparison of visual image quality of reconstructed image from the proposed method at
different threshold value. ............................................................................................................ 49
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Chapter 1
Introduction
1.1 Image Compression
The increasing demand for multimedia content such as digital images and video has led
to great interest in research into compression techniques. The development of higher quality and
less expensive image acquisition devices has produced steady increases in both image size and
resolution, and a greater consequent for the design of efficient compression systems [1].
Although storage capacity and transfer bandwidth has grown accordingly in recent years, many
applications still require compression.
In general, this thesis investigates still image compression in the transform domain.
Multidimensional, multispectral and volumetric digital images are the main topics for analysis.
The main objective is to design a compression system suitable for processing, storage and
transmission, as well as providing acceptable computational complexity suitable for practical
implementation. The basic rule of compression is to reduce the numbers of bits needed to
represent an image. In a computer an image is represented as an array of numbers, integers to be
more specific, that is called a ―digital image‖. The image array is usually two dimensional (2D),
If it is black and white (BW) and three dimensional (3D) if it is colour image [3]. Digital image
compression algorithms exploit the redundancy in an image so that it can be represented using a
smaller number of bits while still maintaining acceptable visual quality. Factors related to the
need for image compression include:

The large storage requirements for multimedia data

Low power devices such as handheld phones have small storage capacity

Network bandwidths currently available for transmission

The effect of computational complexity on practical implementation.
In the array each number represents an intensity value at a particular location in the image and is
called as a picture element or pixel. Pixel values are usually positive integers and can range
between 0 to 255. This means that each pixel of a BW image occupies 1byte in a computer
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memory. In other words , we say that the image has a grayscale resolution of 8 bits per pixel
(bpp) . On the other hand , a colour image has a triplet of values for each pixel one each for the
red, green and blue primary colours. Hence, it will need 3 bytes of storage space for each pixel.
The captured images are rectangular in shape [2]. The ratio of width to height of an image is
called the aspect ratio. In standard definition television (SDTV) the aspect ratio is 4:3, while it
is 16:9 in a high-definition television (HDTV).
Figure 1.1 aspect ratio (a) 4:3 (b) 16:9
The two aspect ratios are illustrated in Figure 1.1, where Figure 1.1(a) corresponds to an
aspect ratio of 4:3 while Figure 1.1(b) corresponds to the same picture with an aspect ratio of
16:9 . In the both pictures, the height in inches remains the same which means that the number
of rows remains the same.
If an image has 480 rows, then the number of pixels in each row will be 480X4/3 = 640
for an aspect ratio of 4:3. For HDTV, there are 1080 rows and the number of pixels in each row
will be 1080 X 16/9=1920. Thus a single SD colour image with 24 bpp will require 640 X 480
X3= 921,600 bytes of memory space, while an HD colour image with the same pixel depth will
require 19200 X 1080 X 3= 6,220,800 bytes. A video source may produce 30 or more frames
per second, in which case the raw data rate will be 221,184,000 bits per second for SDTV and
1492,992,000 bits per second for HDTV [36]. It is very clear that efficient data compression
schemes are required to bring down the huge raw video data rates to manageable values.
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1.2 Data Compression Model
A data compression system mainly consists of three major steps and that are removal or
reduction in data redundancy, reduction in entropy, and entropy encoding. A typical data
compression system can be labeled using the block diagram shown in Figure 1.2 It is performed
in steps such as image transformation, quantization and entropy coding. JPEG is one of the most
used image compression standard which uses discrete cosine transform (DCT) to transform the
image from spatial to frequency domain [2]. An image contains low visual information in its
high frequencies for which heavy quantization can be done in order to reduce the size in the
transformed representation. Entropy coding follows to further reduce the redundancy in the
transformed and quantized image data.
1.2.1 Advantages of Data Compression
The main advantage of compression is that it reduces the data storage requirements. It also offers
an attractive approach to reduce the communication cost in transmitting high volumes of data
over long-haul links via higher effective utilization of the available bandwidth in the data links.
This significantly aids in reducing the cost of communication due to the data rate reduction. Due
to the data rate reduction, data compression also increases the quality of multimedia presentation
through limited-bandwidth communication channels, Because of the reduced data rate. Offered
by the compression techniques, computer network and Internet usage is becoming more and
more image and graphic friendly, rather than being just data and text-centric phenomena. In
short, high-performance compression has created new opportunities of creative applications such
as digital library, digital archiving, video teleconferencing, telemedicine and digital
entertainment to name a few. There are many other secondary advantages in data compression.
For Example it has great implications in database access. Data compression may enhance the
database performance because more compressed records can be packed in a given buffer space in
a traditional computer implementation. This potentially increases the probability that a record
being searched will be found in the main memory. Data security can also be greatly enhanced by
encrypting the decoding parameters and transmitting them separately from the compressed
database files to restrict access of proprietary information. An extra level of security can be
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achieved by making the compression and decompression processes totally transparent to
unauthorized users. The rate of input-output operations in a computing device can be greatly
increased due to shorter representation of data. Data compression obviously reduces the cost of
backup and recovery of data in computer systems by storing the backup of large database files in
compressed form. The advantages of data compression will enable more multimedia applications
with reduced cost.
Figure 1.2 A data compression model.
1.2.2 Disadvantages of Data Compression
Data compression has some disadvantages too, depending on the application area and
sensitivity of the data. For example, the extra overhead incurred by encoding and decoding
processes is one of the most serious drawbacks of data compression, which discourages its usage
in some areas. This extra overhead is usually required in order to uniquely identify or interpret
the compressed data. Data compression generally reduces the reliability of the data records [1].
For example, a single bit error in compressed code will cause the decoder to misinterpret all
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subsequent bits, producing incorrect data. Transmission of very sensitive compressed data (e.g.,
medical information) through a noisy communication channel (such as wireless media) is risky
because the burst errors introduced by the noisy channel can destroy the transmitted data.
Another problem of data compression is the disruption of data properties, since the compressed
data is different from the original data.
In many hardware and systems implementations, the extra complexity added by data
compression can increase the system‘s cost and reduce the system‘s efficiency, especially in the
areas of applications that require very low-power VLSI implementation [3].
1.3 Image Compression Based on Entropy
The principle of digital image compression based on ―information theory‖.
Image
compression uses the concept of ‗Entropy‘ to measure the amount of information that a source
produces. The amount of information produced by a source is defined as its entropy. For each
symbol, there is a product of the symbol probability and its logarithm. The entropy is a negative
summation of the products of all the symbols in a given symbol set.
Compression algorithms are methods that reduce the number of symbols used to
represent source information, therefore reducing the amount of space needed to store the source
information or the amount of time necessary to transmit it for a given channel capacity. The
mapping from source symbols into fewer target symbols is referred to as ‗compression‘. The
transformation from the ‗target symbols‘ back into the source symbols representing a close
approximation form of the original information is called ‗decompression‘ [38]. Compression
system consist of two steps, sampling and quantization of a signal. The choice of compression
algorithm involves several conflicting considerations. These include degree of compression
required, and the speed of operation. Obviously if one is attempting to run programs direct from
their compressed state, decompression speed is paramount. The other consideration is size of
compressed file versus quality of decompressed image. Compression is also known as encoding
process and decompression is
known as decoding process [30]. Digital data compression
algorithms can be classified into two categories1. Lossless compression
2. Lossy compression
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1.3.1 Lossless compression
In lossless image compression algorithm, the original data can be recovered exactly from
the compressed data. It is used generally for discrete data such as text, computer generated data,
and certain kinds of image and video information. Lossless compression can achieve only a
modest amount of compression of the data and hence it is not useful for sufficiently high
compression ratios. GIF, Zip file format, and Tiff image format are popular examples of a
lossless compression [30, 3].
Huffman Encoding and LZW are two examples of lossless
compression algorithms. There are times when such methods of compression are unnecessarily
exact.
In other words, 'Lossless' compression works by reducing the redundancy in the data.
The decompressed data is an exact copy of the original, with no loss of data.
1.3.2 Lossy compression
Lossy compression techniques refer to the loss of information when data is compressed.
As a result of this distortion, must higher compression ratios are possible as compared to the
lossless compression in reconstruction of the image.
reproduction of data for better compression.
'Lossy' compression sacrifices exact
It both removes redundancy and creates an
approximation of the original.
The JPEG standard is currently the most popular method of lossy compression. The
degree of closeness is measured by distortion that can be defined by the amount of information
lost. Some example of lossless compression techniques are :‗CCITT T.6‘, Zip file format, and
Tiff image format. JPEG Baseline and JPEG 2000 is a example of lossy compression algorithm.
The three main criteria in the design of a lossy image compression algorithm are desired bit rate
or compression ratio, acceptable distortion, and restriction on coding and decoding time[30].
While different algorithms produce different type of distortion, the acceptability of which is
often application dependent, there is clearly an increase in distortion with decreasing bit rate.
Obviously, a lossy compression is really only suitable for graphics or sound data, where
an exact reproduction is not necessary. Lossy compression techniques are more suitable for
images, as much of the detail in an image can be discarded without greatly changing the
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appearance of the image. In practice, very fine details are lost in image compression. Though
image size is drastically reduced.
1.4 Objective of thesis
Image compression plays very important role in storing and transferring images
efficiently. With the evolution of electronic devices like Digital Cameras, Smart phones,
biometric applications, the requirement of storing images in less memory is becoming necessary.
Using different compression methods the requirement of less memory storage and efficient
transmission can be fulfilled. At the same time it should be considered that, the compression
technique must be able to reconstruct the image with low loss or without loss as compared to
original image. The objective of the thesis is to study such compression techniques and validate
the results using MATLAB programming. This thesis also aims the comparative study of various
image compression techniques and it also provides the motivations behind choosing different
compression techniques.
1.5 Thesis Structure
Chapter 2 starts with introduction of an Image and its representation in different domains.
It also presents commonly used transform techniques for the image compression. The Discrete
Cosine Transform (DCT), the Discrete Wavelets Transform (DWT), the Hybrid (DCT+DWT)
Transform and Fractal images compression technique have been described. The algorithm for
implementation these techniques using MATLAB also explained in this chapter. The advantage
and disadvantage of all these algorithms are also included in this chapter.
Chapter 3 explains the various performance measurement parameters for comparing the
compression techniques. Based on parameters explained in this chapter the analysis among the
compression techniques is done. Also the subjective and objective analysis can be done using
these parameters. The quality of reconstructed image is validating.
The simulation results and the comparisons are presented in Chapter 4. In addition, the
fast fractal encoding algorithm is also compared with the already discussed and developed
schemes.
Finally Chapter 5 provides conclusion to this thesis along with future work related to this
area.
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Chapter 2
Image Representation and Transform Coding
2.1 Image
An image is the two dimensional (2-D) picture that gives appearance to a subject usually
a physical object or a person. It is digitally represented by a rectangular matrix of dots arranged
in rows and columns, or in other words, ―An image may be defined as a two dimensional
function f(x, y), where x and y are spatial (plane) co-ordinates. The amplitude of ‗f‘ at any pair
of co-ordinate (x, y) is called the intensity or gray level of the image at that point‖ [3].
When (x ,y) and amplitude values of ‗f‘ are all finite, discrete quantities, we call the image is a
―DIGITAL IMAGE‖. A Digital image is an array of a number of picture elements called pixels.
Each pixel is represented by a real number or a set of real numbers in limited number of bits.
Based on the accuracy of the representation, we can classify image into three categories1. Black and White images
2. Grayscale images
3. Colour images
For Black and White images, each pixel is represented by one bit. These images are also
called as bi-level, binary, or bi-tonal images. In Grayscale images, each pixel is represented by
a luminance or say intensity level. For pictorial images, gray scale images are represented by 256
gray levels or 8 bits [21]. In Bit planes, grayscale images can be transformed into a sequence of
binary images by breaking them up into their bit-planes. If we consider the grey value of each
pixel of an 8-bit image as an 8-bit binary word, then the 0th bit plane consists of the last bit of
each grey value. Since this bit has the least effect in terms of the magnitude of the value, it is
called the least significant bit, and the plane consisting of those bits the least significant bit
plane. Similarly the 7th bit plane consists of the first bit in each value. This bit has the greatest
effect in terms of the magnitude of the value, so it is called the most significant bit, and the
plane consisting of those bits the most significant bit plane. In color images each pixel can be
represented by luminance and chrominance components [10]. Color images can also be
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represented in an alternative system which is also known as different color space. Some
examples of popular colour spaces are RGB, CIELAB, HSV, YIQ and YUV. Since human visual
system (HVS) are less sensitive to colour images than to luminance or brightness, RGB space
has the advantage of providing equal luminance to human vision. YIQ color spaces separate
grayscale information from colour data. This enables the same signal to be used for black and
white settings. YIQ component are luminance(Y), hue (I) and saturation (Q). Grayscale
information is expressed as luminance (Y), and colour information as chrominance, which is
both hue (I) and saturation (Q). YCbCr is another colour space that has widely been used for
digital video. Here, luminance information is stored as a single component (Y), and chrominance
information is stored as two colour-difference components (Cb) and (Cr). Cb represents the
difference between the blue component and a reference value, whereas Cr represents the
difference between the red component and a reference value. Another type of colour space is
CMYK which is used in colour printers. The primaries in this colour space are cyan (C), magenta
(M), yellow (Y) and black (K). Resolution in an image refers to the capability to represent the
finer details [2]. The RGB color space is a linear, additive, device-dependent color space. Each
value is usually represented as unsigned integers in the range from 0 to 255, giving a total color
depth of 3 x 8 = 24 bits[42]. Different color examples are shown in Table 2.1.
Table 2.1: RGB color examples
MPEG standard use luminance Y and two chrominance CB and CR to represent color.
Figure 2.1 shows the standard image ‗Einstein‘. The size of the row (M) and column (N) gives
the size (or resolution) of M X N image. A small block (8 X 8) of the image is indicated at the
lower right corner in the form of matrix. Each element in the matrix represents the dots of the
image. Each dot represents the pixel value at that position [41].
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Figure 2.1. Digital representation of an image
Intensity I in RGB model is calculated by
I= 0.3R +0.59G+0.11B
(2.1)
This is the same method used for calculating the Y value when converting from RGB to
YCBCR. RGB images are converted into more suitable YCBCR color format using the
following equations.
Y = 0.299 R + 0.587 G + 0.114 B
(2.2)
CB = −0.169 R − 0.331 G + 0.500 B = 0.564 (B – Y)
(2.3)
CR = 0.500 R − 0.419 G − 0.081 B = 0.713 (R − Y)
(2.4)
Where Y represents a monochrome compatible luminance component, and CB, CR represent
chrominance components containing color information. Most of image/video coding standards
adopt YCBCR color format as an input image signal [41]. Figure 1.3 shows a block diagram of
the color space conversion. Each of the three components (Y, Cb, and Cr) is input to the coder.
The PSNR is measured for each compressed component (Yout, Cbout, and Crout) just as we do
for gray scale images.
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Figure 2.2: Block diagram of the color image compression algorithm.
The three output components are reassembled to form a reconstructed 24-bit color image
(Imageout). It is shown in Figure 2.3[42]
Figure 2.3: (a) Lena image represented in the YCbCr color space. (b) Luminance component.
(c) Chrominance-red component. (d) Chrominance-blue component.
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Note that the weights have a total sum of 1. This color conversion has the desirable
property of packing most of the signal energy into Y and significantly less energy into the
chrominance components [3].
2.2 Redundancy
Redundancy
different amount of data might be used. If the same information can be
represented using different amounts of data, and the representations that require more data than
actual information, is referred as data redundancy. In other words, Number of bits required to
represent the information in an image can be minimized by removing the redundancy present in
it [41]. Data redundancy is of central issue in digital image compression. If n1 and n2 denote the
number of information carrying units in original and compressed image respectively, then the
compression ratio CR can be defined as
CR=n1/n2;
(2.5)
And relative data redundancy RD of the original image can be defined as
RD=1-1/CR
(2.6)
Three possibilities arise here:
(1) If n1=n2, then CR=1 and hence RD=0 which implies that original image do not contain any
redundancy between the pixels.
(2) If n1>>n1, then CR→∞ and hence RD>1 which implies considerable amount of redundancy
in the original image.
(3) If n1<<n2, then CR>0 and hence RD→-∞ which indicates that the compressed image
contains more data than original image.
There are three kinds of redundancies that may present in the image and video.
I. Coding redundancy
II. lnterpixel redundancy
III. Psychovisual redundancy
12 | P a g e
2.2.1 Coding redundancy
If the gray levels of an image are coded in a way that uses more code symbol than
absolutely necessary to represent each gray level, the resulting image is said to be code
redundancy.
In coding redundancy we assign equal number of bits for symbols of high probability
and less probability. It is better to assign fewer bits for more probable gray level and assign more
bits for less probable gray level, which will provide image compression. This method is called as
―variable length coding‖. Coding redundancy would not provide the correlation between the
pixels [37].
2.2.2 lnterpixel redundancy
In most of the images because of the value of any given pixel can be reasonably predicted
from the value of its neighbors, the information carried by individual pixel is relatively small.
That‘s why we call this type of redundancy as a interpixel redundancy. In order to reduce the
interpixel redundancy in an image is that to cade the difference between the successive pixels
and send it to the decoder side. This type of transformation is generally reversible and called
―mapping‖.
2.2.3 Psychovisual redundancy
Certain information‘s has less relative importance than other information in normal visual
processing. This information is said to be psychovisually redundancy. Its elimination is possible
only because the information itself is not essential for normal visual processing.
The elimination of psychovisual redundant data results in a loss of quantitative
information. It is commonly referred to as ―quantization‖.
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2.3 Coding
Various coding techniques are used to facilitate compression. Here are pixel coding,
predictive coding and transform coding will be discussed [10]:
2.3.1 Pixel coding
In this type of coding, each pixel in the image is coded separately. The pixel values that
occurs most frequently are assigned shorter code words (i.e. fewer bits), and those pixel values
that are more fare (i.e. less probable) are assigned longer codes. That makes the average code
word length decrease.
2.3.2 Predictive coding
As images are highly correlated from sample to sample, predictive coding technique is
relatively simple to implement [42]. Predictive coding predicts the present values of the sample
based on the past values and only encodes and transmits the difference between the predicted and
the sample value. Differential pulse code modulation (DPCM) is an example of a frequently used
predictive coding.
The DPCM system consists of two blocks as shown in Figure 2.4. The function of the
predictor is to obtain an estimate of the current sample based on the reconstructed values of the
past sample. The difference between this estimate, or prediction, and the actual value is
quantized, encoded, and transmitted to the receiver [17].
Figure 2.4 Block diagram of a DPCM system.
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The decoder generates an estimate identical to the encoder, which is then added on to
generate the reconstructed value. The requirement that the prediction algorithm use only the
reconstructed values is to ensure that the prediction at both the encoder and the decoder are
identical. The reconstructed values used by the predictor, and the prediction algorithm, are
dependent on the nature of the data being encoded.
2.3.3 Transform coding
In transform coding, an image is transformed from one domain (usually spatial or
temporal) to a different type of representation, using some well –known transform. Then the
transformed values are coded and thus provide greater data compression. In this thesis,
transforms are orthogonal so that the mapping is unique and reversible. As a result, the energy is
preserved in the transform domain that is the sum of the squares of the transformed sequence is
the same as the sum of the squares of the original sequence. Thus, the image can be completely
recovered by the inverse transform.
Transform coding (TC), is an efficient coding scheme based on utilization of interpixel
correlation. Transform coding uses frequency domain, in which the encoding system initially
converting the pixels in space domain into frequency domain via transformation function. Thus
producing a set of spectral coefficients, which are then suitably coded and transmitted [41].
The transform operation itself does not achieve any compression. It aims at decorrelating
the original data and compacting a large fraction of the signal energy into a relatively small set of
transform coefficients (energy packing property). In this way, many coefficients can be discarded
after quantization and prior to encoding. Most practical transform coding systems are based on
DCT of types II which provides good compromise between energy packing ability and
computational complexity. The energy packing property of DCT is superior to that of any other
unitary transform [45]. Transforms that redistribute or pack the most information into the fewest
coefficients provide the best sub-image approximations and, consequently, the smallest
reconstruction errors. In Transform coding, the main idea is that if the transformed version of a
signal is less correlated compared with the original signal, then quantizing and encoding the
transformed signal may lead to data compression. At the receiver, the encoded data are decoded
and transformed back to reconstruct the signal.
15 | P a g e
The purpose of the transform is to remove interpixel redundancy (or de-correlate)
from the original image representation. The image data is transformed to a new representation
where average values of transformed data are smaller than the original form. This way the
compression is achieved. The higher the correlation among the image pixels, the better is the
compression ratio achieved. There are various methods of transformations being used for data
compression as follows [42]:
i. Karhunen-Loeve Transform (KLT)
ii. Discrete Fourier Transform (DFT)
iii. Discrete Sine Transform (DST)
iv. Walsh Hadamard Transform (WHT)
v. Discrete Cosine Transform (DCT)
vi. Discrete Wavelet Transform (DWT)
2.4 Transform-based Image Compression
Transform refers to changing the coordinate basis of the original signal, such that a new
signal has the whole information in few transformed coefficients. The processing of the signals
in the transform domain is more efficient as the transformed coefficients are not correlated [41].
A popular image compression framework is the transform based image compression as shown in
below Figure 2.5
(a) Encoder
(b)Decoder
Figure 2.5 Block diagram of transform based image coder (a)Compression or Encoder
(b)Decompression or Decoder
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The first step in the encoder is to apply a linear transform to remove redundancy in the
data, followed by quantizing the transform coefficients, and finally entropy coding then we get
the quantized outputs. After the encoded input image is transmitted over the channel, the decoder
reverse all the operations that are applied in the encoder side and tries to reconstruct a decoded
image as close as to the original image [9].
2.4.1 Linear Transform
In the encoder side, the first step is to transform the image from the spatial domain to the
transformed domain (where the image information is represented in a more compact form) using
some
known
transforms
like
Discrete
Fourier
Transform(DFT),
Discrete
Cosine
Transform(DCT), Discrete Wavelet Transform(DWT), Fractal Transforms and many more.
Compression of the original image is not easy, as the energy can be concentrated in the
low frequency part of the transform domain. For conservation of energy from the spatial domain
to the transformed domain, it is necessary for the transform to be orthogonal. Here, in this thesis
Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT), Hybrid Transform
(DCT-DWT), Fractal Transforms are implemented due to their superior energy compaction and
correspondence with human visual system [9]. An ideal image transform should retain the
following two properties. These are:
(a) Maximum energy compaction
(b) Less computational complexity
2.4.2 Quantizer
Quantizer is a key component in the transform compression framework that introduces
non-linearity in the system. It maps the transformed digital image to a discrete set of levels or
discrete number.it is a lossy compression technique as it introduces an error in the image
compression process [21]. It converts sequence of floating point numbers to sequence of
integers. In other words, ―If the data symbols are real numbers, quantization may round each to
the nearest integer. If the data symbols are large numbers, quantization may convert them to
small numbers‖. Small numbers take less space than large ones. On the other hand, small
numbers convey less information than large ones, which is why quantization produces lossy
compression. Quantization is a simple approach to lossy compression [17]. It is a many-to-one
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mapping and therefore it is irreversible. The effect of Quantization can be seen in the below
Figure 2.6
Figure 2.6 Quantization effect.
Inverse quantization step restores original coefficients but with round-off error. Quantizers are of
three type, based on their working principle:
2.4.2.1 Scalar Quantization
Scalar Quantization is the simplest quantization because each input is treated separately in
producing the output. In other words, ―If the data symbols are numbers, then each is quantized to
another number in a process referred to as scalar quantization‖. Many image compression
methods are lossy, but scalar quantization is not suitable for image compression because it
creates annoying artifacts in the decompressed image. Scalar quantization produces lossy
compression, but it makes easy to control the trade-off between compression performance and
the amount of data loss [9]. Applications of scalar quantization are limited to cases where much
loss can be tolerated.
2.4.2.2 Vector Quantization
In vector quantization, the input samples are clubbed together in groups called vectors,
and then processed to give the output. If each data symbol is a vector, then vector quantization
converts a data symbol to another vector. The idea of representing groups of samples rather than
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individual samples is the main concept of vector quantization. Vector quantization is based on
the fact that adjacent data symbols in image and audio files are correlated [9].
2.4.2.3 Predictive Quantization
The quantization of the difference between the predicted value and the past samples is
called predictive quantization.
A good quantizer is one, which represents the original signal with minimum loss or
distortion.
2.4.3 Entropy Encoding
After quantization of the transformed values, entropy encoder further compresses the
quantized values to give additional compression. It is a lossless compression and is also
reversible. In the entropy encoding, the idea is to find a reversible mapping to the quantized
values such that the average number of bits or symbols is minimized. The basic principle of
entropy encoding is that we assign short codes to the letters appearing frequently whereas long
codes are assigned to the letters appearing less frequently. The two popular entropy- coding
methods are Huffman coding and Arithmetic coding [31].
2.5 Discrete Cosine Transform
DCT is an orthogonal transform, the Discrete Cosine Transform (DCT) attempts to
decorrelate the image data. After decorrelation each transform coefficient can be encoded
independently without losing compression efficiency.
The DCT transforms a signal from a spatial representation into a frequency
representation. The DCT represent an image as a sum of sinusoids of varying magnitudes and
frequencies. DCT has the property that, for a typical image most of the visually significant
information about an image is concentrated in just few coefficients of DCT. After the
computation of DCT coefficients, they are normalized according to a quantization table with
different scales provided by the JPEG standard computed by psycho visual evidence. Selection
of quantization table affects the entropy and compression ratio. The value of quantization is
inversely proportional to quality of reconstructed image, better
mean square error and better
compression ratio [42]. In a lossy compression technique, during a step called Quantization, the
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less important frequencies are discarded, and then the most important frequencies that remain are
used to retrieve the image in decomposition process. After quantization, quantized coefficients
are rearranged in a zigzag order for further compressed by an efficient lossy coding algorithm .
DCT has many advantages:
(1) It has the ability to pack most information in fewest coefficients.
(2) It minimizes the block like appearance called blocking artifact that results when
boundaries between sub-images become visible [36].
An image is represented as a two dimensional matrix, 2-D DCT is used to compute the
DCT Coefficients of an image. The 2-D DCT for an NXN input sequence can be defined as
follows [5]:
( ) ( )∑
D (i,j) =
√
∑
(
(
)
)
(
)
(
)
(
)
(2.7)
Where, P(x, y) is an input matrix image NxN, (x, y) are the coordinate of matrix elements and (i,
j) are the coordinate of coefficients, and
}
C (u) ={√
(2.8)
The reconstructed image is computed by using the inverse DCT (IDCT) according to
P(x,y)
=
√
∑
∑
() () (
)
(
(
)
)
(
(
)
)
(2.9)
The pixels of black and white image are ranged from 0 to 255, where 0 corresponds to a
pure black and 255 corresponds to a pure white. As DCT is designed to work on pixel values
ranging from -128 to 127, the original block is leveled off by 128 from every entry [21]. Step by
step procedure of getting compressed image using DCT and getting reconstructed image from
compressed image is explained in the next sections.
2.5.1 Coding scheme
2.5.1.1 Compression procedure
First the whole image is loaded to the encoder side, then we do RGB to GRAY conversion
after that whole image is divided into small NXN blocks (here N corresponds to 8) then working
from left to right, top to bottom the DCT is applied to each block. Each block‘s elements are
20 | P a g e
compressed through Quantization means dividing by some specific 8X8 matrix called Qmatrix and
rounding to the nearest integer value as shown in Eq 2.10
(
)
(
(
(
)
)
)
(2.10)
This Qmatrix is decided by the user to keep in mind that it gives Quality levels ranging
from 1 to 100, where 1 gives the poor image Quality and highest compression ratio while 100
gives best Quality of decompressed image and lowest compression ratio. The standard Qmatrix can
be shown as
Figure 2.7. JPEG Quantization table
We choose Qmatrix, with a Quality level of 50, Q50matrix gives both high
compression and
excellent decompressed image. By using Q10 we get significantly more number of 0‘s as
compared to Q90. After Quantization, all of the quantized coefficients are ordered into the
―zigzag‖ sequence. The zigzag can be done in the below manner as shown in the Figure 2.9;
21 | P a g e
Figure 2.8 Zig-zag ordering for DCT coefficients
Now encoding is done and transmitted to the receiver side in the form of one dimensional
array. This transmitted sequence saves in the text format. The array of compressed blocks that
constitute the image is stored in a drastically reduced amount of space. Further compression can
be achieved by applying appropriate scaling factor [21]. In order to reconstruct the output data,
the rescaling and the de-quantization should be performed as given in Eq. (2.11).
(
)
(
)
(
)
(2.11)
The de-quantized matrix is then transformed back using the 2-D inverse-DCT [35]. The
equation for the 2-D inverse DCT transform is given in the above mentioned Eq. (2.12)
P(x,y) =
√
∑
∑
() () (
)
(
(
)
)
(
(
)
)
(2.12)
The complete procedure of compression an image using DCT is explained in Figure 2.7 through
a flowchart [41].
2.5.1.2 Decompression procedure
To reconstruct the image, receiver decodes the quantized DCT coefficients and computes
the inverse two dimensional DCT (IDCT) of each block, then puts the blocks back together into
a single image in same manner as we done in previously. The dequantization is achieved by
multiplying each element of the received data by corresponding element in the quantization
22 | P a g e
matrix Qmatrix, then 128 added to each element for getting level shift. In this decoding process, we
have to keep block size (8X8) value same as it used in encoding process. These blocks are
merged and arranged in same order in which they were decomposed for compression to get the
decompressed image.
Figure. 2.9 Flow chart of compression technique
The following flow chart in Figure 2.10 explains whole decompression procedure step by
step. Let us take a random input data of 8 X 8 matrix representing a portion of image as shown in
Figure 2.5 The transform coefficients as shown in Figure 2.6 can be obtained by applying 2-D
DCT to a given input matrix. The element at upper left corner shown in bold is the DC
component (low frequency) having highest value. Rest of the other elements are the ac
components (high frequency). Human eye is more sensitive to the DC component and less
23 | P a g e
sensitive to AC component. Hence, the AC component can be neglected in order to achieve
higher compression- by passing the transformed data through the quantizer.
Figure 2.10. Flow chart of decompression technique
Figure 2.11 A random value input data matrix
24 | P a g e
Figure 2.12 Transformed coefficients after DCT of the random value input data matrix
The JPEG quantization table is shown in Figure 2.8. The corresponding value after
quantization is as shown in Figure 2.13. We can see that more than 70 percent of the coefficients
are quantized to zero.
Figure 2.13 Quantized coefficients
Figure 2.14 Reconstructed output data
Further compression can be achieved by the use of scaling factor (SF) to the quantized
coefficients. The reconstructed data after inverse transformation is illustrated in Figure 2.14.
There is a small deviation between the input data and output data due to the quantization.
25 | P a g e
2.5.2 Properties of DCT
There are some properties of the DCT which are of particular value to image processing
applications [31].
2.5.2.1 Decorrelation
The principle advantage of image transformation is the removal of redundancy between
neighboring pixels. This leads to uncorrelated transform coefficients which can be encoded
independently. DCT exhibits excellent decorrelation properties.
2.5.2.2 Separability
The DCT transform equation 2.7 can be expressed as,
D (i,j) =
√
( ) ( )∑
∑
(
)
(
(
)
)
(
(
)
)
for i, j = 0,1,2,…,N −1.
This property, known as ‗Separability‘, has the principle advantage that D (i, j) can be
computed in two steps by successive 1-D operations on rows and columns of an image. This idea
is graphically illustrated in Figure 2.15. The arguments presented can be identically applied for
the inverse DCT computation.
Figure 2.15 Computation of 2-D DCT using separability property.
2.5.2.3 Energy Compaction
Effectiveness of a transformation scheme can be directly evaluated by its ability to pack
input data into as few coefficients as possible. This allows the quantizer to discard coefficients
26 | P a g e
with relatively small amplitudes without introducing visual falsification in the reconstructed
image. DCT exhibits excellent energy compaction for highly correlated images.
(a) Saturn and its DCT
(b) Circuit and its DCT
(c)Baboon and its DCT
27 | P a g e
(d) sine wave and its DCT
Figure2.16. (a) Saturn and its DCT; (b) Circuit and its DCT; (c) Baboon and its DCT;
(d) sine wave and its DCT
A closer look at Figure 2.16 reveals that it comprises of four broad image classes. Figure
2.16 (a) contain large areas of slowly varying intensities. These images can be classified as low
frequency images with low spatial details. A DCT operation on these images provides very good
energy compaction in the low frequency region of the transformed image.
Figure 2.16(b) contains a number of edges (i.e., sharp intensity variations) and therefore
can be classified as a high frequency image with low spatial content. However, the image data
exhibits high correlation which is exploited by the DCT algorithm to provide good energy
compaction. Figure 2.16 (c) and is image with progressively high frequency and spatial content.
Consequently, the transform coefficients are spread over low and high frequencies. Figure
2.16(d) shows periodicity therefore the DCT contains impulses with amplitudes proportional to
the weight of a particular frequency in the original waveform. The other (relatively insignificant)
harmonics of the sine wave can also be observed by closer examination of its DCT image.
Hence, from the preceding discussion it can be inferred that DCT extracts excellent
energy compaction for correlated images. Studies have shown that the energy compaction
performance of DCT approaches optimality as image correlation approaches one i.e., DCT
provides (almost) optimal decorrelation for such images.
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2.5.2.4 Symmetry
In the above Equation (2.7), at the row and column operations reveals that these
operations are functionally identical. Such a transformation is called a symmetric transformation.
A separable and symmetric transform can be expressed in the form.
T = APA
(2.13)
where A is an N ×N symmetric transformation matrix with entries a (i, j ) given by
(2.14)
and f is the N ×N image matrix. This is an extremely useful property since it implies that the
transformation matrix can be precomputed offline and then applied to the image thereby
providing orders of magnitude improvement in computation efficiency [41].
2.5.2.5 Orthogonality
In order to extend ideas presented in the preceding section, let us denote the inverse
transformation of as
(2.15)
As discussed previously, DCT basis functions are orthogonal. Thus, the inverse
transformation matrix of A is equal to its transpose i.e.
. Therefore, and in addition
to its decorrelation characteristics, this property renders some reduction in the pre-computation
complexity [42].
2.5.3 Limitations of DCT
For the lower compression ratio, the distortion is unnoticed by human visual perception.
In order to achieve higher compression it is required to apply quantization followed by scaling to
the transformed coefficient. For such higher compression ratio DCT has following two
limitations.
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2.5.3.1 Blocking artifacts:
Blocking artifacts is a distortion that appears due to heavy compression and appears as
abnormally large pixel blocks. For the higher compression ratio, the perceptible ―blocking
artifacts‖ across the block boundaries cannot be neglected [48]. The example of appearance of
blocking artifact due to high compression is shown in Figure2.17.
Figure 2.17. Illustration of compression using DCT: (a) Original Image CR at (b) 88%, (c) 96%
2.5.3.2 False contouring:
The false contouring occurs when smoothly graded area of an image is distorted by an
deviation that looks like a contour map for specific images having gradually shaded areas [5].
Figure 2.18 Illustration of compression using DCT: (a) Original Image CR at (b) 87 %, (c) 97%
The main cause of the false contouring effect is the heavy quantization of the transform
coefficients [46]. An example of false contouring can be observed in Figure 2.18.
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2.6 Discrete Wavelet Transform (DWT)
Wavelets are a mathematical tool for changing the coordinate system in which we
represent the signal to another domain that is best suited for compression. Wavelet based coding
is more robust under transmission and decoding errors. Due to their inherent multiresolution
nature, they are suitable for applications where scalability and tolerable degradation are
important [4].
Wavelets are tool for decomposing signals such as images, into a hierarchy of increasing
resolutions. The more resolution layers, the more detailed features of the image are shown. They
are localized waves that drop to zero. They come from iteration of filters together with rescaling.
Wavelet produces a natural multi resolution of every image, including the all-important edges.
The output from the low pass channel is useful compression. Wavelet has an unconditional basis
as a result the size of the wavelet coefficients drop off rapidly. The wavelet expansion
coefficients represent a local component thereby making it easier to interpret. Wavelets are
adjustable and hence can be designed to suit the individual applications. Its generation and
calculation of DWT is well suited to the digital computer [41]. They are only multiplications and
additions in the calculations of wavelets, which are basic to a digital computer.
2.6.1 Multiresolution Concept and Analysis
The multi resolution concept is designed to represent signals, where a single event will be
decomposed into finer and finer details. A signal is represented by a coarse approximation and
finer details. The coarse and the detail subspaces are orthogonal to each other.by applying
successive approximation recursively the space of the input signal can be spanned by spaces of
successive details at all resolutions [6].
2.6.2 Decimator and interpolator
Decimator and interpolator are two operations concerned with the sampling rate.
Decimator or down sampler lowers the sampling rate whereas interpolator or the up-sampler
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raises the sampling rate [5]. The down sampler takes a signal x(n) and down samples by a factor
of two. It is shown in Figure 2.19
Figure 2.19 Decimator or down sampler
Similarly, the up-sampler takes a signal and up samples by a factor of two so as to increase the
number of samples. It means to insert zeros between the terms of the original sequence.
Figure 2.20 Interpolator or up sampler
The input sequence is stretched to twice its original length and zeros are inserted. It can be seen
in Figure 2.20.
2.6.3 Filter bank
Filter bank is a set of filters. It consists of analysis bank and synthesis bank. In this thesis
two channel filter bank is considered.it is shown in Figure 2.21 [6]
Figure 2.21 Filter bank
2.6.3.1 Analysis bank
The analysis bank has two filters: a low pass and a high pass. They separate the input
signal into frequency bands. When the input signal (image) is first passed through the analysis
filters, it decomposes the signal into four bands. They are given as LL, HL, LH, AND HH. This
32 | P a g e
is the first level of the decomposition and it represents the finer scale of the expansion
coefficients [11]. It is shown in Figure 2.22
Figure 2.22 Finer scale and coarser scale wavelet coefficients
The DWT represents an image as a sum of wavelet functions, known as wavelets, with
different location and scale. It represents the data into a set of high pass (detail) and low pass
(approximate) coefficients. The input data is passed through set of low pass and high pass filters.
used . The output of high pass and low pass filters are down sampled by 2. The output from low
pass filter is an approximate coefficient and the output from the high pass filter is a detail
coefficient. This procedure is one dimensional (1-D) DWT and Figure 2.23 shows the schematics
of this method.
Figure 2.23. Block diagram of 1-D forward DWT
The two filters double the number of coefficients but the decimator halves it. It implies
that there is a possibility of getting the original image back. The aliasing occurring in the higher
33 | P a g e
scale can be cancelled by using the signal from the lower level [11]. This is the idea behind the
perfect reconstruction filter bank.
Figure 2.24. Block diagram of 2-D forward DWT
Repeating the splitting, filtering and decimation on the scaling coefficients is called
iterating the filter bank.it is illustrated in Figure 2.24 in 2-D DWT. In case of 2-D DWT, the input
data is passed through set of both low pass and high pass filter in two directions, both rows and
columns. The outputs are then down sampled by 2 in each direction as in case of 1-D DWT [53].
As shown in Figure 2.22, output is obtained in set of four coefficients LL, HL, LH and HH. The
first alphabet represents the transform in row whereas the second alphabet represents transform
in column. The alphabet L means low pass signal and H means high pass signal. LH signal is a
low pass signal in row and a high pass in column. Hence, LH signal contain horizontal elements.
Similarly, HL and HH contains vertical and diagonal elements, respectively.
2.6.3.2 Synthesis bank
It is the inverse of the analysis bank. In digital signal processing there was decimation
and filtering in the synthesis bank [11].
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Figure 2.25. Block diagram of 2 dimensional inverse DWT
The reconstruction of the original fine scale coefficients can be made from the
combination of the scaling and the wavelet coefficients at a (lower) coarser scale [48].it is
illustrated in Figure 2.25 .In 2D, the images are considered to be matrices with N rows and M
columns. Any decomposition of an image into wavelets involves a pair of waveforms
One to represent the high frequency corresponding to the detailed part of the image
(wavelet function).

One for low frequency or smooth parts of an image (scaling function).

At every level of decomposition the horizontal data is filtered, and then the
approximation and details produced from this are filtered on columns. At every level,
four sub-images are obtained; the approximation, the vertical detail, the horizontal detail
and the diagonal detail.

Wavelet function for 2-D DWT can be obtained by multiplying wavelet functions
( (
)) and scaling function (φ(
)). After first level decomposition we get four
details of image those are,
(
Approximate details –
(
Horizontal details –
Vertical details –
Diagonal details –
(
)= φ( ) φ( )
)= φ( ) ( )
)=
(
)=
( ) ( )
( ) ( )
The approximation details can then be put through a filter bank, and this is repeated until the
required level of decomposition has been reached. The filtering step is followed by a subsampling operation that decreases the resolution from one transformation level to the other. After
35 | P a g e
applying the 2-D filler bank at a given level n, the detail coefficients are output, while the whole
filter bank is applied again upon the approximation image until the desired maximum resolution
is achieved. Figure 2.26(b) shows wavelet filter decomposition. The sub-bands are labelled by
using the following notations [12].

LLn represents the approximation image nth level of decomposition, resulting from lowpass filtering in the vertical and horizontal both directions.

LHn represents the horizontal details at nth level of decomposition and obtained from
horizontal low-pass filtering and vertical high-pass filtering.

HLn represents the extracted vertical details/edges , at nth level of decomposition and
obtained from vertical low-pass filtering and horizontal high-pass filtering.

HHn represents the diagonal details at nth level of decomposition and obtained from
high-pass filtering in both directions [33].
2.6.4Coding scheme
2.6.4.1 Compression procedure
Original image is passed through HPF and LPF by applying filter first on each row.
Output of the both image resulting from LPF and HPF is considered as L1 and H1 and they are
combine into A1, where A1= [L1, H1]. Then A1 is down sampled by 2. Again A1 is passed
through HPF and LPF by applying filter now on each column. Output of the above step is
supposed to L2 and H2 and they are combined to get A2, where A2=*
+. Now, A2 is down
sampled by 2 to get compressed image [53]. We get this compressed image by using one level of
decomposition, to get more compressed image i.e. to get more compression ratio we need to
follow above steps more number of times depending on number of decomposition level required
[51]. First level of decomposition gives four detailed version of an image those are shown in
Figure 2.20(a) and (b).
2.6.4.2 Decompression procedure
Extract LPF and HPF images from compressed image by simply taking upper half rectangle
of matrix is LPF image and down half rectangle is HPF image. Then both images are up sampled
by2. Now take the summation of both images into one image called B1.Then again extract LPF
image and HPF image by dividing vertically [50]. Two halves obtained are filtered through LPF
36 | P a g e
and HPF, summation of these halves gives the reconstructed image on each block of 32x32
block, by applying
2 D-DWT, four details are produced. Out of four sub band details,
approximation detail/sub band is further transformed again by 2 D-DWT which gives another
four sub-band of 16x16 blocks. Above step is followed to decompose the 16x16 block of
approximated detail to get new set of four sub band/ details of size 8x8. The level of
decomposition is depend on size processing block obtained initially, i.e. here we are dividing
image initially into size of 32x32, hence the level of decomposition is 2 [46]. After getting four
blocks of size 8x8, we use the approximated details for computation of discrete cosine transform
coefficients. These coefficients are then quantize and send for coding [41]. The DWT algorithm
is typically more memory intensive and time consuming compared to a DCT based coder like
JPEG. Despite this, DWT offers benefits such as :

Allowing image multiresolution representation

Allowing progressive transmission / rate scalability

Higher efficiency in term of quality of compressed image and compression ratio.
(a)
(c)
(b)
(d)
Figure 2.26. Illustration of 2 dimensional DWT for an image ‗Lena‘
37 | P a g e
2.7 HYBRID (DCT+ DWT) TRANSFORM
In section 2.5 and 2.8 we presented two different ways of achieving the goals of image
compression, which have some advantages and disadvantages, in this section we are proposing a
transform technique that will exploit advantages of DCT and DWT, to get compressed image.
Hybrid DCT-DWT transformation gives more compression ratio compared to JPEG and
JPEG2000, preserving most of the image information and create good quality of reconstructed
image. Hybrid (DCT+DWT) Transform reduces blocking artifacts, false contouring and ringing
effect.[39].
2.7.1 Coding scheme
2.7.1.1Compression procedure
The input image is first converted to gray image from colour image, after this whole
image is divided into size of 32x32 pixels blocks. Then 2D-DWT applied on each block of
32x32 blocks, by applying 2 D-DWT, four details are produced. Out of four sub band details,
approximation detail/sub band is further transformed again by 2 D-DWT which gives another
four sub-band of 16x16 blocks [52]. Above step is followed to decompose the 16x16 block of
approximated detail to get new set of four sub band/ details of size 8x8. The level of
decomposition is depend on size processing block obtained initially, i.e. here we are dividing
image initially into size of 32x32, hence the level of decomposition is 2. After getting four
blocks of size 8x8, we use the approximated details for computation of discrete cosine transform
coefficients. These coefficients are then quantize and send for coding. The complete coding
scheme is explained in Figure 2.27.
2.7.1.2 Decompression procedure
At receiver side, we decode the quantized DCT coefficients and compute the inverse
two dimensional DCT (IDCT) of each block. Then block is dequantized. Further we take inverse
wavelet transform of the dequantized block. Since the level of decomposition while compressing
was two, we take inverse wavelet transform two times to get the same block size i.e. 32x32. This
procedure followed for each block received. When all received blocks are converted to 32x32 by
38 | P a g e
following decompression procedure, explained above. We arrange all blocks to get reconstructed
image. The complete decoding procedure is explained in Figure 2.28. The hybrid DWT-DCT
algorithm has better performance as compared to stand alone DWT and DCT in terms of Peak
Signal to Noise Ratio (PSNR) and Compression Ratio (CR). In standalone DCT, the entire
image/frame is divided into 8X8 block in order to apply 8 point DCT [40].
Figure 2.27 Compression technique using Hybrid transform
Figure 2.28 Decompression technique using Hybrid transform
39 | P a g e
Whereas, in case of hybrid algorithm, image/frame is first divided into 32X32 blocks and
two level of DWT is performed for these 32X32 block image. The output after the two level of
DWT becomes 8X8 and hence the 8 point DCT is applied for that 8X8 output. This difference in
block size causes the blocking artifacts in case of standalone DCT. The contouring effect of DCT
has also been reduced by using proposed hybrid DWT-DCT algorithm.
2.8 Fractal Image Compression
Fractal coding is a new method of image compression. The main principle of the fractal
transform coding is based on the hypothesis that the image redundancies can be efficiently
exploited by means of block self-affine transformations. By removing the redundancy related to
self-similarity in an image. Fractal image compression can achieve a higher compression ratio
with high decoding quality. Fractal coding has the advantage such as resolution independence
and fast decoding as compare to other image compression methods. So fractal image
compression is a promising technique that has great potential to improve the efficiency of image
storage and image transmission [43].
2.8.1 Baseline Fractal Coding
Fractal image coding is based on partition iterated function system (PIFS),in which an
original input image is partitioned into a set of non-overlapping sub-blocks, called range block
(R) that cover up the whole image. The size of every range block is N X N. At the same time, the
original image is also partitioned into a set of other overlapping sub-blocks, called domain
blocks (D), which size is always twice the size of range blocks. The domain blocks are allowed
to be overlapping and need not cover the whole image [26]. Secondly, each of the domain blocks
is contracted by pixel averaging or down sampling to match the size of the range block. Next,
eight symmetrical transformations (rotations and flips) are applied to all contracted domain
blocks to bring out an extended domain pool, which denoted as ̂ For each range block, we
search the domain pool to get the best matched domain block D with a contractive affine
transformation. The problem with fractal coding is the highly computational complexity in the
encoding process [29].
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Figure2.29 Fractal coding algorithm
Most of the encoding time is spent on the best matching search between range blocks and
numerous domain blocks ( ̂ ) so that the fractal encoding is a time consuming process, which
limits the algorithm to practical application greatly. In order to solve this problem, lots of
researches were done earlier to speed up the block matching process. Most of these
improvements tried to restrict the search space of the domain block pool in order to reduce the
computation requirements of the best matching search, hence speeding up the fractal image
encoding procession [28].
2.8.2 Procedure of a different speed up encoding method
The different fast fractal encoding method use the variance and mean value to exclude
the searching of those domain blocks whose characteristics are inconsistent to the range block.
By reducing the computational complexity of RMSE and simplified transformation to find out
the best matched domain block would be useful [24].
The steps of the proposed algorithm are given as follows:
41 | P a g e

Partition the original image into non-overlapping range blocks(R) and overlapping
domain blocks (D).

By doing, down sampling we contract the size of the domain block to the size of the
range block and it is noted as ̂

Now, calculate the variance of range and domain blocks by the equation below. The
variance of block I is defined as,
Var (I) = √,
(∑
)
(
(∑
)) (2.16)
Where n is the size of the block and Xi is the pixel value of the range blocks.

For each range block (R), select those domain blocks that can meet the criterion
of |
( )
( ̂ )|
. Then classify those selected ̂
by the individual
mean value.

By using simplified transformation analysis and reduced RMSE to find the best matched
domain block and its transformation. Then, store this position of searched domain block
and mapped transformation.

Repeat the above steps for all other range blocks until all the range blocks are processed.
In the above method, according to variance difference between Var (R) and Var ( ̂ ), the
range blocks are classified. Then both range and domain blocks are classified into a number of
classes according to mean value. The searched domain blocks for every range block are not all
domain blocks now. The proposed method only searches the domain blocks whose mean value
classes are the same or adjacent as the class of range block to reduce the searching time [41]. The
reduction of search domain blocks decreases the decoded quality. However, searching time
decreases melodramatically.
2.8.3 Procedure for decoding method
To decompress an image, the compressor first allocates two memory buffers of equal size,
with arbitrary initial content. The iterations then begin, with buffer 1 the range image and buffer
2 the domain image. The domain image is partitioned into domain regions as specified in the FIF
file. For each domain region, its associated range region is located in the range image. Then the
42 | P a g e
Figure 3.30 A different approach for Fractal coding
corresponding affine map is applied to the content of the range region, pulling the content
toward the map's attractor. Since each of the affine maps is contractive, the range region is
contracted by the transformation. This is the reason that the range regions are required to be
larger than the domain regions during compression.
For the next iteration, the roles of the domain image and range image are switched. The
process of mapping the range regions (now in buffer 2) to their respective domain regions (in
buffer 1) is repeated, using the prescribed affine transformations. Then the entire step is repeated
again and again, with the content of buffer 1 mapped to buffer 2, then vice versa. At every step,
the content is pulled ever closer to the attractor of the IFS which forms a collage of the original
image [27]. Eventually the differences between the two images become very small, and the
content of the first buffer is the output decompressed image.
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2.8.4 Advantages and Disadvantages
Fractal image compression has the following advantages:

Fast decoding process

High compression ratio

Low performance device

Resolution independence

It can be digitally scaled to any resolution when decoded.

Image compressed in terms of self-similarity rather than pixel resolution

Lower transmission time
Disadvantage of fractal image compression
2.9

Long encoding time

Image quality
Conclusion
The algorithms for compression and decompression for various Image compressions
methods such as DCT, DWT, Hybrid, Fractals are discussed in this chapter. DCT requires less
computational resources and can achieve the energy compaction property. However, for the
higher compression ratio it introduces the blocking artifacts and the false contouring effects
while image reconstruction. DWT is the only techniques which has capacity of multi resolution
compression. However, it requires higher computational complexity as compared to other
techniques. Hence, in order to benefit from each other, hybrid DWT-DCT algorithm has been
discussed for the image compression in this chapter. Conventional Fractal method circumvents
the drawbacks of DCT, DWT, Hybrid (DCT+DWT), but it requires more computational time. A
different approach for image compression using fractals is also discussed which uses the mean
and variance of range and domain blocks. This approach speeds up the encoding time by
reducing the number range- domain comparison with remarkable amount. Each method can be
well suited with different images based on the user requirements.
44 | P a g e
Chapter 3
Performance Measurement Parameters
In this work prominence were given on the amount of compression used and how good
the reconstructed image be similar to the original. Analysis was done on the basis of the amount
of distortion, which was calculated using important distortion measures: mean square error
(MSE), peak signal-to-noise ratio (PSNR) measured in decibels (dB) and compression ratio (CR)
measures were used as performance indicators. Image having same PSNR value may have
different perceptual quality. The quality of reconstructed images can be evaluated in terms of
objective measure and subjective measure. In objective evaluation, statistical properties are
considered whereas, in subjective evaluation, viewers see and investigate image directly to
determine the image quality. A good compression algorithm would reconstruct the image with
low MSE and high PSNR [46]. Performance measurement parameters are described in the
following sub-sections.
3.1 Objective evaluation parameters.
3.1.1 Mean Square Error (MSE)
The MSE is the cumulative squared error between the compressed and the original image.
A lower value of MSE means lesser error, and it has the inverse relation with PSNR. Mean
square error is a criterion for an estimator: the choice is the one that minimizes the sum of
squared errors due to bias and due to variance. In general, it is the average of the square of the
difference between the desired response and the actual system output. As a loss function MSE is
also called squared error loss [41]. MSE measures the average of the square of the ‗error‘. The
MSE is the second moment of the error, and thus incorporates both the variance of the estimator
and its bias. For an unbiased estimator, the MSE is the variance. In an analogy to standard
deviation, taking the square root of MSE yields the root mean squared error or RMSE [43]. For
an unbiased estimator, the RMSE is the square root of the variance, known as the standard error.
45 | P a g e
MSE=
∑
∑
[ (
)
(
)]
(3.1)
Where, I(x, y) is the original image and I‘(x, y) is the reconstructed image and m, n are the
dimensions of the image. Lower the value of MSE, the lower the error and better picture quality
[49].
3.1.2 Peak Signal to Noise Ratio (PSNR)
PSNR is a measure of the peak error. Many signals have very wide dynamic range,
because of that reason PSNR is usually expressed in terms of the logarithmic decibel scale in
(dB). Normally, a higher value of PSNR is good because it means that the ratio of signal to noise
is higher [1]. Here, a signal represents original image and noise represents the error in
reconstruction. It is the ratio between the maximum possible power of a signal and the power of
the corrupting noise [9]. PSNR decreases as the compression ratio increases for an image. The
PSNR is defined as:
PSNR=10*log10,
- = 20*log10,
√
-
(3.2)
PSNR is computed by measuring the pixel difference between the original image and
compressed image [53]. Values for PSNR range between infinity for identical images, to 0 for
images that have no commonality.
(a) Original image
46 | P a g e
(b) PSNR = 25.98 dB
(c) PSNR = 23.42 dB
(d) PSNR= 30.41 dB
Figure 3.1(a) Original, reconstructed image using (b) DCT, (c) DWT, (d) Hybrid DWT-DCT
In the above Figure 3.1, the PSNR values for DCT, DWT and hybrid algorithm are 25.98
dB, 23.42 dB, and 30.41 dB respectively, as the false contouring effect is visible in the image
reconstructed by the DCT algorithm. The image reconstructed using DWT algorithm is also very
poor compared to the one with the hybrid algorithm.
3.1.3 Compression ratio (CR)
Compression ratio (CR) is a measure of the reduction of the detailed coefficient of the
data. In the process of image compression, it is important to know how much detailed
(important) coefficient one can discard from the input data in order to sanctuary critical
information of the original data. Compression ratio can be expressed as:
CR =
(3.3)
The quantization table (Q) and the scaling factor (SF) are the main controlling parameters
of the compression ratio. Each element of the transformed data is divided by corresponding
element in the quantization table (Qmatrix) and rounded to the nearest integer value by using round
function in MATLAB. This process makes some of the coefficients to be zero which can be
discarded [21].
47 | P a g e
(a) Original image
(b) CR=78.32%
(c) CR =89.28%
Figure 3.2 CR comparison (a) original image and (b), (c) reconstructed image with different compression ratio.
In order to achieve higher compression ratio, the quantizer output is then divided by some
scalar constant (SF) and rounded to nearest integer value. This process yields more zero
coefficients which can be discarded during compression [54].
The CR can be varied to get different image quality. The more the details coefficients are
discarded, the higher the CR can be achieved. Higher compression ratio means lower
reconstruction quality of the image. Compression ratio comparison can be seen in the above
Figure 3.2 .We can see that at higher compression ratio we get blurred image as compare to the
image that have less compression ratio.
3.2 Subjective evaluation parameter
The visual perception of the reconstructed image is essential. In some cases the objective
quality assessment does not give proper information about the quality of the reconstructed image.
In such scenarios, it is important to analyze the reconstructed image using subjective analysis
that means by human perceptual system [41]. When the subjective measure is considered,
viewers focus on the difference between reconstructed and original image and correlates the
differences.
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3.3 Conclusion
In this chapter, the objective and subjective evaluation parameter generally used for
image compression analysis has been presented. For the objective evaluation PSNR, CR and
variance have been discussed. The lower value of MSE indicates better picture quality. There is
an inverse relationship between MSE and PSNR. Hence, the larger PSNR value gives the better
image quality. Compression ratio indicates the efficiency of compression technique, more the
compression ratio, less memory space required. Hence, more compression ratio is always
desirable without trade off in image quality.
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Chapter 4
Performance Evaluation and Simulation Results
This chapter evaluates the performance of the various image compression algorithms.
The studied algorithms are applied on several types of images: natural images, benchmark
images such that the performance of proposed algorithm can be verified for various applications.
These benchmark images are the standard image generally used for the image processing
applications and are obtained from. The results of the meticulous simulation for all images and
are presented in this section. The results are compared with the JPEG-based DCT, DWT, Hybrid
(DCT-DWT) algorithms and using Fractals.
4.1 Simulation tool
The algorithms were implemented in MATLAB simulation tool. For the DWT and DCT,
MATLAB functions ―dwt2‖ and ―dct2‖ has been considered, respectively. The evaluation
parameters (PSNR, CR, MSE and Variance), sub-sampling, quantization and scaling routines
were manually programmed in MATLAB. The proposed algorithm is compared with DCT,
DWT and Hybrid (DCT-DWT) algorithms.
4.2 Simulation Results
Results are tabulated in table 4.1. The results are obtained for images of sizes 128x128
and 512x512. Original and reconstructed images are also shown in Figure 4.1. It can be seen
from table 4.1, the compression ratio CR is high for Hybrid transform as compare to standalone
transforms. DWT comprises between compression ratio and quality of reconstructed image, it
adds speckle noise to the image for improvement in the reconstructed image. Hence DWT
technique is useful in medical applications. DCT gives lesser compression ratio but it is
computationally efficient compared to other techniques.
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Table4.1 comparative analysis
Image
Image
Technique used
Compression Ratio
MSE
PSNR
DCT
2.6122
4.342
49.4436
DWT
3.2365
4.165
50.3181
Hybrid
10.2989
44.148
31.6817
DCT
26.5462
0.9732
48.2488
DWT
30.237
6.1613
40.2341
Hybrid
52.539
113.2097
27.5920
DCT
2.3690
0.0676
59.8335
DWT
2.9231
1.9067
45.3279
Hybrid
9.6481
35.0658
32.6820
DCT
24.5156
2.8152
43.6357
Size
128x128
Image 1
(jas.jpg)
512x512
128x128
Image 2
The proposed fractal encoding is simulated by using several images of size 512 X 512 and 128X 128 wi
(lena.jp
512x512
DWT
29.8172
29.3150
33.4599
Hybrid
49.7381
122.4307
27.2519
g)
Domain blocks are has to be same that means 8 X 8. The threshold value has to be set for
different images and it is user defined. For the image of ‗Lena‘, we set the threshold value 1800
and 1900.
51 | P a g e
Figure. 4.1 Comparison of visual image quality of reconstructed image for DCT, DWT AND HYBRID
(DCT+DWT) for test images.
Thus, analyze the difference between the decoded images. As we can see the quality of
decoded image at different threshold level in the below Figure 4.2.
(a) Original image
(b) Decoded image at threshold
Value 1800
(c ) Decoded image at threshold
Value 1900
Figure. 4.2 Comparison of visual image quality of reconstructed image from the proposed method at different
threshold value
52 | P a g e
When we implement the coding of fractal wavelet compression technique without the threshold
value in MATLAB then we got the Peak signal to noise ratio is 39.4201 which is good but we
got the encoding time very high that is 128.1167(approx. more than 2 min) and the decoding
time is 17.052 sec which is quite low then encoding time. If we set the particular threshold value
which is 1800 then we got the peak signal to noise ratio 25.9846 and encoding time is 27.529 sec
which is very low as compare to previous scheme and the decoding time is 16.4702 sec which is
little small then the previous scheme. Now further if we change the threshold value from 1800
To 1900 then we analyze the major change again in encoding time. At this threshold value the
encoding time obtained is very low i.e. 4.891 sec which is quite low. But there is no change in
the PSNR value and decoding time is very near to the previous stage. So we can say that on the
second value of threshold we obtain the best result if we concern only with encoding and
decoding time. The below table 4.2 shows comparison of the schemes i.e. ( existed and proposed
one).
Table 4.2 Comparative analysis of existed and proposed encoding approach
4.3 Conclusion
Based on the comprehensive simulation results presented above for images it can be seen
that the hybrid DWT-DCT algorithm outperforms the JPEG based DCT and DWT algorithms.
Especially, the hybrid algorithm performs better for the images that consist of detailed view,
bright colours, and gradients. Hence, it can be implemented for compressing natural and medical
images. It is observed that in case of the DWT algorithm, the reconstructed images seem to be
the worst, whereas for the DCT, it is affected by artifacts and false contouring effects. However,
for the same CR, the hybrid algorithm consistently has higher PSNR and better reconstruction
53 | P a g e
quality. It is also able to reduce the false contouring effect and artifacts for images. Fractal
encoding is computationally very expensive and hence it requires large time for encoding
process. In spite of many advancements, computational and time requirements of encoding part
is a still remains the main drawback of fractal image compression. Besides these drawbacks,
fractal technique provides more compression ratio, resolution independency; the iteration
function system provides a better quality in the images.
54 | P a g e
Chapter 5
Conclusion and Future work
5.1 Conclusion
In this thesis analysis of various Image compression techniques for different images is
done based on parameters, compression ratio(CR), mean square error (MSE), peak signal to
noise ratio (PSNR). Our simulation results from chapter 4 shows that we can achieve higher
compression ratio using Hybrid technique but loss of information is more. DWT gives better
compression ratio without losing more information of image. Pitfall of DWT is, it requires more
processing power. DCT overcomes this disadvantage since it needs less processing power, but it
gives less compression ratio. DCT based standard JPEG uses blocks of image, but there are still
correlation exits across blocks. Block boundaries are noticeable in some cases. Blocking artifacts
can be seen at low bit rates. In wavelet, there is no need to divide the image. More robust under
transmission errors. It facilitates progressive transmission of the image (scalability). Hybrid
transform gives higher compression ratio but for getting that clarity of the image is partially trade
off. It is more suitable for regular applications as it is having a good compression ratio along
with preserving most of the information.
On the other hand Fractal Image Compression gives a great improvement on the
encoding and decoding time. A weakness of the proposed design is the use of fixed size blocks
for the range and domain images. There are regions in images that are more difficult to code than
others .Therefore; there should be a mechanism to adapt the block size (R, D) depending on the
mean and variance calculated when coding the block. This type of compression can be applied in
Medical Imaging, where doctors need to focus on image details, and in Surveillance Systems,
when trying to get a clear picture of the intruder or the cause of the alarm. This is a clear
advantage over the Discrete Cosine Transform Algorithms such as that used in JPEG.
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5.2 Future Work
The result in this thesis provides a strong foundation for future work for the hardware
design. All of the analysis presented in this thesis work involved exhaustive simulations. The
algorithm can be realized in hardware implementation as a future work. It can also be a good
option for the image processor of the wireless capsule endoscopic system. The research work has
been analyzed for high compression ratio. Further research can be performed to relax high
compression ratio constraint. This work has been constrained only for the removal of the spatial
redundancy by compression of still images.
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