IMPLEMENTATION OF MULTIWAVELET

TRANSFORM CODING FOR IMAGE

COMPRESSION

A THESIS

Submitted by

RAJAKUMAR K (Reg.No: 201008206)

In partial fulfillment for the award of the degree

Of

DOCTOR OF PHILOSOPHY

DEPARTMENT OF

ELECTRONICS AND COMMUNICATION

ENGINEERING

KALASALINGAM UNIVERSITY

(Kalasalingam Academy of Research and Education)

Anand Nagar, Krishnankoil – 626 126

SEPTEMBER 2015

iv ABSTRACT

The Multiwavelet is an advance of the well-established wavelet

theory. Today the performance of the wavelets in the field of image

processing is well known. Multiwavelet takes wavelets a step ahead in

performance. In this thesis the performance of the Integer Multiwavelet

Transform (IMWT) for Lossless and Lossy compression has been studied.

The IMWT showed good performance with reconstruction of the images. This

thesis analyses the performance of the IMWT compression with Bit plane and

Run length coding. The Transform coefficients are coded using the Run

length coding and bit plane coding techniques. Here the image is decomposed

or transformed into components that are then coded according to the

individual characteristics. The transform should have high-energy compaction

property, so as to achieve high compression ratios. The Transform coefficient

matrix is coded without taking the sign values into account, using the

Magnitude Set Variable Length Integer Representation. The sign information

of the coefficients is coded as bit plane with zero thresholds. The Bit plane so

formed can be used as it is or coded to reduce the Bits per pixels. The

Simulation was done in Matlab.

The Mean Square Error and Peak Signal to Noise Ratio and

additionally quality measures like Structural similarity and Structural

dissimilarity are tabulated for various standard test images. In this thesis,

different compression algorithms for Lossless and Lossy are simulated which

includes.

1. Magnitude Set-Variable length integer without Run length Encoding

Algorithm.

2. Magnitude Set-Variable length integer with Run length Encoding

Algorithm.

v The newer techniques such as IMWT can achieve reasonably good

image quality with higher compression ratios. The Integer Multiwavelet

transform (IMWT) has short support, symmetry, high approximation order of

two. The key concept of the thesis in image compression algorithm is the

development to determine the minimal data required to retain the necessary

information.

The IMWT image compression results in with a very low bit rate,

which results in a smaller file size. This indicates that the IMWT can be used

for wireless technology with the benefits of very low storage space, low

probability of transmission error, high security and low transmission cost.

vii TABLE OF CONTENTS

CHAPTER TITLE PAGE ABSTRACT iv

LIST OF TABLES xi

LIST OF FIGURES xii

LIST OF ABBREVIATIONS AND SYMBOLS xiv

1 INTRODUCTION 1

1.1 DATA COMPRESSION 1

1.1.1 Image Compression 1

1.2 COMPRESSION TECHNIQUES 2

1.2.1 Lossless Compression 2

1.2.1.1 Run Length Encoding 3

1.2.1.2 Huffman Encoding 4

1.2.1.3 LZW Coding 4

1.2.1.4 Area Coding 4

1.2.2 Lossy Compression 5

1.2.2.1 Transformation Coding 5

1.2.2.2 Vector Quantization 6

1.2.2.3 Fractal Coding 6

1.2.2.4 Block Truncation Coding 6

1.2.2.5 Sub band Coding 7

1.3 IMAGE COMPRESSION PERFORMANCE METRICS 7

1.3.1 Image quality 8

1.3.1.1 Distortion 8

1.3.1.2 Fidelity or Quality 8

1.3.1.3 Structural similarity 9

1.3.1.4 Structural dissimilarity 9

1.3.2 Compression Ratio (CR) 10

1.3.3 Data Compression Rate 10

viii CHAPTER TITLE PAGE 1.3.4 Speed of Compression 10

1.3.5 Power Consumption 10

1.4 THE COMPRESSION SYSTEM 11

1.5 OVERVIEW OF METHODOLOGY 14

1.5.1 Pre-Processing 16

1.5.2 Post-Processing 16

1.6 FOURIER TRANSFORMS 16

1.6.1 Fast Fourier Transform 17

1.6.2 Inverse Fast Fourier Transform 18

1.7 WAVELETS 19

1.8 WAVELET TRANSFORM 20

1.9 CONTINUOUS WAVELET TRANSFORM 22

1.10 DISCRETE WAVELET TRANSFORM (DWT) 24

1.10.1 Haar Wavelets 24

1.10.2 Daubechies Wavelets 25

1.10.3 DWT and Filter Banks 28

1.10.4 First level of Transform 30

1.10.5 Cascading and Filter Banks 31

1.11 FAST WAVELET TRANSFORM 34

1.12 2-D DISCRETE WAVELETS TRANSFORM 35

1.13 INTRODUCTION TO INTEGER MULTIWAVELET 35

TRANSFORMS

1.14 MULTIWAVELET TRANSFORMS 36

1.15 INTERGER MULTIWAVELET TRANSFORM

FUNCTION 38

1.16 MULTIWAVELET FILTER BANKS 40

1.17 MULTIWAVELET DECOMPOSITION 41

1.18 WAVELET AND MULTIWAVELET COMPARISON 42

ix 1.19 MAGNITUDE SET-VARIABLE LENGTH INTEGER

REPRESENTATION 45

1.20 ORGANIZATION OF THE THESIS 46

2 LITERATURE SURVEY 48

2.1 INTRODUCTION 48

2.2 ANALYSIS OF WAVELET AND PROCESSING 48

2.3 ENHANCED COMPRESSION ALGORITHMS 55

2.4 BEHAVIOURS OF JPEG AND JPEG 2000 57

2.5 REAL-TIME APPLICATIONS 60

2.6 PROPERTIES OF MULTIWAVELET IN FILTERS 62

2.7 MEASUREMENTS AND QUALITY METRICS 64

2.8 THE KNOWLEDGE GAP IDENTIFIED IN THE 66

EARLIER INVESTIGATIONS

2.9 RESEARCH MOTIVATION 66

2.10 AIM 68

2.11 OBJECTIVE OF THE RESEARCH WORK 68

3 IMPLEMENTATION OF IMWT 69

3.1 INTRODUCTION 69

3.2 OVERVIEW 70

3.3 INTEGER PREFILTER 71

3.4 TRANFORMATION TO OBTAIN LOW BITS 74

3.5 MS-VLI REPRESENTATION 76

3.6 LOW BIT RATE USING IMWT COMPRESSION

ALGORITHM 80

3.7 PSEUDO CODE FOR SSIM AND DSSIM 80

3.8 PERFORMANCE EVALUTION 82

x CHAPTER TITLE PAGE

4 SIMULATION RESULT AND ANALYSIS 83

4.1 LOSSLESS COMPRESSION USING IMWT 83

4.1.1 Procedure to obtain Lossless Compression 83 Using IMWT Algorithm

4.1.2 Results of Reconstructed Images 84

4.1.3 Summary of Performance for Lossless 87 Compression

4.1.4 Analysis 89

4.2 LOSSY COMPRESSION USING IMWT 89

4.2.1 Procedure to obtain Lossy Compression 90 Using IMWT Algorithm

4.2.2 Results of Reconstructed Images 91

4.2.3 Summary of Performance for Lossy 99 Compression

4.2.4 Analysis 101

4.3 COMPARISION OF EXISTING LOSSLESS WITH 102 PROPOSED LOSSY COMPRESSION TECHNIQUES 4.3.1 Analysis 106

4.4 COMPARISION OF REAL AND BINARY WAVELET WITH INTEGER MULTIWAVELET TRANSFORM 106

4.4.1 Analysis 107

4.5 SUMMARY 109

5 CONCLUSION AND FUTURE WORKS 111

5.1 Contribution of the Thesis 112

5.2 Limitation and Future works 113

APPENDIX A: PROGRAMMING CODE FOR MS-VLI REPRESENTATION 114

REFERENCES 118

LIST OF PUBLICATIONS 129

CURRICULUM VITAE 131

xi LIST OF TABLES

TABLE

NO.

TITLE PAGE

NO.

1.1 The Run Length Encoding example 3

1.2 Comparison of Scalar and Multiwavelet Transform 43

1.3 Definition of Magnitude Set Variable Length Integer Representation 46

2.1 Definition of absolute magnitude Set variable Length 52 Integer Representation

3.1 Magnitude Set Variable Length Integer Representation 76

3.2 Amplitude Intervals example for (8 to 11) 77

3.3 Amplitude Intervals example for (24 to 31) 78

3.4 SSIM and DISSM Results 78

3.5 SSIM (8x8) on Different window size 79

3.6 SSIM (32x32) on Different window size 79

3.7 SSIM and DSSIM for (512x512) Image 79

4.1 PSNR and MSE values in dB for Reconstructed images 84

4.2 The Bit Rate for Lossless Compression 85

4.3 Lossless Reconstruction and Reconstruction on LL subband 85

4.4 Comparison of PSNR and Compression ratio for Existing SPIHT and Proposed IMWT based Lossy Reconstruction 91

4.5 Reduced file size on Compression without RLE and with RLE 95

4.6 Required bits per pixels for existing and proposed Lossy compression 97

4.7 PSNR in existing and proposed reconstructed images 98

4.8 Proposed Lossy and Existing Lossless based compression 102

4.9 PSNR and MSE values in dB for Reconstructed Images 105

4.10 Existing RWT and BWT with proposed IMWT for bit Reduction 108

4.11 The Bit Reduction using IMWT Image Compression 109

xii LIST OF FIGURES

FIGURE

NO.

TITLE PAGE

NO.

1.1 The Block diagram of Compression 11

1.2 The Block diagram of Decompression 11

1.3 The Compression Process on Forward Transform 14

1.4 The Reconstruction on Reverse Transform 15

1.5 The Wavelet coefficients at II level Decomposition 24

1.6 Comparison of Sine wave and Daubechies 5 wavelet 26

1.7 Scaling and Shifting Process of the DWT 27

1.8 Comparisons of DWT and CWT example 27

1.9 The Filter Analysis 30

1.10 I-Level DWT Filter Implementation 32

1.11 IDWT Filter Implementation 32

1.12 Sub band Decomposition of Image 33

1.13 2-D Multiwavelet Decomposition of an image 42

1.14 I -level decomposition Subband structure of images 44

3.1 The Compression 70

3.2 The Reconstruction 70

3.3 Multifilter bank implementation of 1st level Multiwavelet decomposition prefiltering as Polyphase representation 71

3.4 Multiwavelet decomposition prefiltering as Equivalent nonpolyphase representation 71

3.5 2-D Process Flow of Multiwavelet decomposition 73 of an image

3.6 I-level IMWT Decomposition of Lena, Couple 74 and Man Images

3.7 Low bit required for the Information to transfer 75

xiii FIGURE

NO.

TITLE PAGE

NO.

4.1 Reconstructed images after I-level IMWT for (512x512) 86

4.2 PSNR and MSE values on Lossless Reconstruction 87

4.3 Comparison of Existing SPHIT and proposed IMWT 92 Lossy Reconstruction with Distortion of Standard Lena

4.4 Comparison of Existing SPHIT and proposed IMWT 93 Lossy Reconstruction with Distortion of Satellite Rural

4.5 Existing SPIHT and Proposed IMWT based 94 Lossy methods

4.6 I level IMWT decomposition of Lena 512 x 512 Image 96

4.7 Original and reconstructed with LL band alone 97

4.8 Bpp for the Existing and the Proposed 98 Lossy compression

4.9 Existing and Proposed Lossy compression with PSNR 99

4.10 Bits per pixels of proposed Lossy and existing lossless compression 103

4.11 Existing Lossless and proposed Lossy IMWT 104

4.12 PSNR and MSE in dB for reconstructed images 105

4.13 Comparison to obtain Bit reduction in 107

Percentage using IMWT

4.14 Comparison of Bit reduction between Existing RWT, 108 BWT and Proposed IMWT compression

xiv LIST OF ABBREVIATIONS AND SYMBOLS

DCT - Discrete Cosine Transform

CR - Compression Ratio

Bpp - Bits per Pixel

MSE - Mean Square Error

PSNR - Peak Signal to Noise Ratio

SSIM - Structural Similarity

DSSIM - Structural Dissimilarity

IMWT - Integer Multiwavelet transform

MS-VLI - Magnitude set variable length integer

DCT - Discrete Cosine Transform

RLE - Run Length Encoding

LZW - Lempel Ziff Welch

GMP - Good Multifilter Properties

DWT - Discrete wavelet transform

EZW - Embedded zero-tree wavelet

SPIHT - Set partitioning in hierarchical tree

EBCOT - Embedded block coding with optimal truncation

CCSR - Compressibility Constrained Sparse Representation

JPEG - Joint Photographic Experts Group

SVD - Singular Value Decomposition

BTC - Block Truncation Coding

DCT - Discrete Cosine Transform

DSC - Distributed Source Coding

CA - Cellular Automata

ECC - Error Correcting Codes

CADU - Collaborative Adaptive Down-Sampling Upconversion

EC - Embedded Compression

HD - High-Definition

SBT - Significant Bit Truncation

AVIRIS - Airborne visible infrared Imaging Spectrometer

xv VLC - Variable-Length Coding

DMWT - Discrete Multiwavelet Transform

STFT - Short time Fourier transform

MRA - Multiresolution analysis

CWT - Continuous Wavelet Transform

FFT - Fast Fourier Transform

DFT - Discrete Fourier Transform

DTFT - Discrete-Time Fourier Transform

WFT - Windowed Fourier Transforms

FWT - Fast Wavelet Transform

CWT - Continuous wavelet transform

DTCWT - Dual-tree complex wavelet transform

MS-VLI - Magnitude set - variable length integer representation

IMWT - Integer Multiwavelet transform

RWT - Real wavelet transform

BWT - Binary wavelet transform

SPIHT - Set Partitioning In Hierarchical Trees

CSF - Contrast Sensitivity Function

MRA - Multi Resolution Analysis

RTS - Real Time Processing

α - Attenuation factor

µ - Step size parameter

λ - Step size parameter

s(t) - Original information

n (t)/ fc - Noise/ Cutoff frequency

r(t),y(t) - Received Signal

X (n) - Digitized input

X (z) - Filter Input

Y (z) - Filter Output

H (Z) - Transfer function of filter

1

CHAPTER 1

INTRODUCTION

1.1 DATA COMPRESSION

Data Compression is an art of representing information in a compact form.

It is to reduce the number of bits required to represent a data sequence so that

storing or transmitting the data is done in an efficient manner. The basic principle

of the compression is to reduce the redundancy in the data. The data could be an

image or video or an audio, and in the present context, it is considered to be an

image. So, image compression is a type of data compression that encodes the

original image with fewer bits. The main goal is to reduce the storage size as

much as possible, and while retrieving the original image from the compressed

image, the decompressed image should be similar to the original image as much

as possible.

1.1.1 Image Compression

The image has become the most important information carrier in people’s

life or the biggest media containing information. As the need to store and transmit

images continues to increase, the field of image compression will also continue to

grow. An image contains large amount data, mostly redundant information that

occupies massive storage space and minimizes transmission bandwidth.

An image consists of pixels, which are highly correlated to each other

within a close proximity. The correlated pixels lead redundant data. There are two

types of data redundancy that are observed.

2

• Spatial Redundancy: The intensities of neighboring pixels are correlated.

So, the intensity information of an image contains unnecessarily repeated

( ie. redundant) data within one frame.

• Spectral Redundancy: Different frequencies of an image contain redundant

data due to the correlation between different color planes.

1.2 COMPRESSION TECHNIQUES

The compression algorithms are broadly classified into two categories,

namely, Lossless and Lossy compression algorithms [24]. These are briefly

explained in the following.

1.2.1 Lossless Compression

The Lossless compression techniques involve no loss of information. The

original information can be recovered exactly from the compressed data. It is

used for applications that cannot tolerate any difference between the original and

the reconstructed data. Lossless compressed image has a larger size compared

with lossy one. In a power constrained applications like wireless communication,

Lossless compression is not preferred as it consumes energy, more time for image

transfer. In the following sections focus is on the lossless compression techniques

as listed below.

Run length encoding

Huffman encoding

LZW coding

Area coding

3

1.2.1.1 Run Length Encoding

This is a very simple compression method used for sequential data. It is

very useful in case of repetitive data. This technique replaces sequences of

identical symbols (pixels) called runs by shorter symbols. The run length code for

a gray scale image is represented by a sequence (Vi, Ri) where Vi is the intensity

of pixel and Ri refers to the number of consecutive pixels with the intensity Vi as

shown in the table 1.1[63]. If both Vi and Ri are represented by one byte, this

span of 11 pixels is coded using five bytes yielding a compression ratio of 11: 5.

Table 1.1

The Run Length Encoding example

86 86 86 86 86 91 91 91 91 75 75

{86,5} {91,4} {75,2}

The Images with repeating intensities along their rows or columns can

often be compressed by representing runs of identical intensities as run-length

pairs, where each run-length pair specifies the start of a new intensity and the

number of consecutive pixels having that intensity. This technique is used for

data compression in BMP file format. The RLE is particularly effective when

compressing binary images since there are only two possible intensities as black

and white. Additionally, a variable-length coding can be applied to the run

lengths themselves. The approximate run-length entropy is

LLHHH RL

1010

+

+= (1.1)

Where H0 and H1 are entropies of the black and white runs and L0 and L1 are the

average values of black and white run lengths.

4

1.2.1.2 Huffman Encoding

This is a general technique for coding symbols based on their statistical

occurrence frequencies probabilities. The pixels in the image are treated as

symbols. The symbols that occur more frequently are assigned a smaller number

of bits, while the symbols that occur less frequently are assigned a relatively

larger number of bits. Huffman code is a prefix code. Most image coding

standards use lossy techniques in the earlier stages of compression and use

Huffman coding as the final step.

1.2.1.3 LZW Coding

LZW (Lempel-Ziv–Welch) is a dictionary based coding. This can be

static or dynamic. In static dictionary coding, dictionary is fixed during the

encoding and decoding processes. In dynamic dictionary coding, the dictionary is

updated on fly. LZW is widely used in computer industry and is implemented as

compress command on UNIX.

1.2.1.4 Area Coding

Area coding is an enhanced form of run length coding, reflecting the two

dimensional character of images. This is a significant advance over the other

lossless methods. For coding an image, it does not make too much sense to

interpret it as a sequential stream, as it is in fact an array of sequences building up

a two dimensional object. The algorithms for area coding find rectangular regions

with the same characteristics. These regions are coded in a descriptive form as an

element with two points and a certain structure. This type of coding can be highly

effective but it bears the problem of a nonlinear method, which is difficult to

implement in hardware. Therefore, the performance in terms of compression time

is not competitive.

5

1.2.2 Lossy Compression

The Lossy compression involves some loss of information. The data that

have been compressed using lossy techniques generally cannot be recovered or

reconstructed exactly. It results in higher compression ratios at the expense of

distortion in reconstruction. The benefit of lossy over lossless is high compression

ratio, less process time and low energy in case of power constrained applications.

In the following sections focus is on the lossy compression techniques [66] as

listed below.

Transformation coding

Vector quantization

Fractal coding

Block Truncation Coding

Subband coding

1.2.2.1 Transformation Coding

In this coding scheme, transforms such as DFT (Discrete Fourier

Transform) and DCT (Discrete Cosine Transform) are used to change the pixels

in the original image into frequency domain coefficients (called transform

coefficients).These coefficients have several desirable properties. One is the

energy compaction property that results in most of the energy of the original data

being concentrated in only a few of the significant transform coefficients. This is

the basis of achieving the compression. Only those few significant coefficients

are selected and the remaining is discarded. The selected coefficients are

considered for further quantization and entropy encoding. DCT coding has been

the most common approach to transform coding. It has been adopted in the JPEG

image compression standard.

6

1.2.2.2 Vector Quantization

The basic idea in this technique is to develop a dictionary of fixed-size

vectors, called code vectors. A vector is usually a block of pixel values. A given

image is then partitioned into non-overlapping blocks (vectors) called image

vectors. Then for each, vector is determined and its index in the dictionary is used

as the encoding of the original image vector. Thus, each image is represented by a

sequence of indices that can be further entropy coded.

1.2.2.3 Fractal Coding

The essential idea here is to decompose the image into segments by

using standard image processing techniques such as color separation, edge

detection, and spectrum and texture analysis. Then each segment is looked up in a

library of fractals. The library actually contains codes called iterated function

system (IFS) codes, which are compact sets of numbers. Using a systematic

procedure, a set of codes for a given image are determined, such that when the

IFS codes are applied to a suitable set of image blocks yield an image that is a

very close approximation of the original. This scheme is highly effective for

compressing images that have good regularity and self-similarity.

1.2.2.4 Block truncation coding

In this scheme, the image is divided into non overlapping blocks of

pixels. For each block, threshold and reconstruction values are determined. The

threshold is usually the mean of the pixel values in the block. Then a bitmap of

the block is derived by replacing all pixels whose values are greater than or equal

(less than) to the threshold by a 1 (0). Then for each segment (group of 1s and 0s)

in the bitmap, the reconstruction value is determined. This is the average of the

values of the corresponding pixels in the original block.

7

1.2.2.5 Sub band coding

In this scheme, the image is analyzed to produce the components

containing frequencies in well-defined bands, called sub bands. Subsequently,

quantization and coding is applied to each of the bands. The advantage of this

scheme is that the quantization and coding suitable for each sub band can be

designed separately.

Compression techniques can be applied directly to the images or to the

transformed image information (transformed domain). The transform coding

techniques are well suited for image compression. Here the image is decomposed

or transformed into components that are then coded according to the individual

characteristics. The transform should have high-energy compaction property, so

as to achieve high compression ratios. Examples: Discrete Cosine Transform

(DCT), Wavelet Transform, Multiwavelet Transform etc.

1.3 IMAGE COMPRESSION PERFORMANCE METRICS

The performance of a compression technique can be evaluated in a number

of ways- the amount of compression, the relative complexity of the technique,

memory requirement for implementation, time required for the compression on a

machine, and the distortion rate in the reconstructed image. The following are the

Performance Metrics to evaluate the compression techniques.

• Image Quality.

• Compression ratio.

• Speed of compression.

i. Computational complexity.

ii. Memory resources.

• Power consumption.

8

1.3.1 Image quality

There is a need for specifying methods to judge image quality after

reconstruction process and to measure the amount of distortion due to

compression process as minimal image distortion means better quality. There are

two types of image quality measures, subjective quality measurement and

objective quality measurements. Subjective quality measurement is established by

asking human observers to judge and report the image or video quality according

to their experience; and these measures would be relative or absolute. Absolute

measures classify image quality not regarding to any other image but according to

some criteria of television allocations study organization. On the other hand

relative measures compare image against another and choose the best one.

The quantitative measurements are discussed in the following.

1.3.1.1 Distortion

The variation between the original and reconstructed image is called as

Distortion. It is denoted using Mean Square Error (MSE) in dB.

−= ∑

=

NXN

0i

2ii10dB )YX(

NxN1log10)MSE( (1.2)

Where Xi is input uncompressed image, Yi is output compressed image.

1.3.1.2 Fidelity or Quality

It defines the resemblance between the Original and Reconstructed

image. It can be measured using Peak Signal to Noise Ratio (PSNR) in dB.

dB MSE255log 10 PSNR

2

10

= (1.3)

= 20.log10 (255)-10.log10 (MSE)

Where 255 represents maximum pixel value of gray image when pixel is

represented by using 8 bits per sample.

9

1.3.1.3 Structural Similarity

The structural similarity (SSIM) index is a method for measuring

the similarity between two images. The SSIM index is a full reference metric in

other words, the measuring of image quality based on an initial uncompressed or

distortion-free image as reference. SSIM is designed to improve on traditional

methods like peak signal-to-noise ratio (PSNR) and mean squared error (MSE),

which have proven to be inconsistent with human eye perception.

)c)(c(

)c*)(2c*(2 y)SSIM(x,

222

x12y

221

++++

++=

yx

xyyx

σσµµ

σµµ (1.4)

Where

xµ - average of x; yµ is the average of y; 2xσ is the variance of x;

2yσ - variance of y; xyσ is the covariance of x and y; 1c = (k1 L)2, 2c = (k2 L)2 two

variables to stabilize the division with weak denominator. L is the dynamic range

of the pixel-value (typically this is 256-1=255); k1= 0.01 and k2 = 0.03 by

default.

1.3.1.4 Structural Dissimilarity

Structural dissimilarity (DSSIM) is a distance metric derived from

SSIM (though the triangle inequality is not necessarily satisfied).

2

),(SSIM1),(DSSIM yxyx −= (1.5)

10

1.3.2 Compression Ratio (CR)

It is the ratio of the number of bits required to represent the image previous

to compression to the number of bits required to represent the image after

compression.

=

size file Compressedsize file edUncompress (CR) Ration Compressio (1.6)

Where CR can be used to judge how compression efficiency is, as higher

CR means better compression.

1.3.3 Data Compression Rate

It is the average number of bits required to represent a single sample. It is

represented in terms of Bits per Pixel (bpp).

⇒

=

NxNbytes ofNumber * 8

pixels ofNumber bits ofNumber (bpp) pixelper Bits

(1.7)

1.3.4 Speed of compression

Compression speed depends on the compression technique that has been

used, as well as, the nature of platform that hosts the compression process.

Compression speed is influenced by computational complexity and size of

memory. Lossy compression is a complex process that increases system

complexity, storage space and needs more computational element clock.

1.3.5 Power consumption

Power consumption is one of the main performances metric in image

compression as it is affected by the previously mentioned metrics. The nature of

11

multimedia requires massive storage space and large bandwidth that consumes

more power. Transmission power is required to manipulate visual flows, and

energy-aware compression algorithms that reduce transmission time. Therefore,

adjusting processing complexity, transmission power reduction and minimizing

data size will save energy.

1.4 THE COMPRESSION SYSTEM

The compression system model consists of two parts:

• The Compressor

• The Decompressor

Figure 1.1 The Block diagram of Compression

Figure 1.2 The Block diagram of Decompression

The compressor shown in figure 1.1 consists of a preprocessing stage that

performs data reduction and mapping [31]. The encoding stage performs

quantization and coding, whereas, the decompressor consists of a decoding stage

that performs decoding and inverse mapping followed by a post- processing

Original Image

Pre-

Processing

Encoding

Compressed Image

Compressed

Image Decoding

Post-Processing

Original Image

12

stage, as shown in figure 1.2. In compressor previous to encoding, preprocessing

is performed to prepare the image for the encoding process and consists of many

operations that are application specific. After the compressed file has been

decoded, post-processing can be performed to eliminate some of the potentially

undesirable artifacts brought about by the compression process. The compressor

can be divided into following stages:

• Data reduction: Image data can be reduced by gray level and

spatial quantization, or can undergo any desired image

improvement (for example, noise removal) process.

• Mapping: Involves mapping the original image data into another

mathematical space where it is easier to compress the data.

• Quantization: Involves taking potentially continuous data from the

mapping stage and putting it in discrete form.

• Coding: Involves mapping the discrete data from the quantized onto

a code in an optimal manner.

The mapping process is important because image data tends to be highly

correlated. If the value of one pixel is known, it is highly likely that the adjacent

pixel value is similar. On finding a mapping equation that decorrelates the data.

Such type of data redundancy can be removed.

• Differential coding: Method of reducing data redundancy by

finding the difference between adjacent pixels and encoding those

values.

• Principal components transform: It can also be used which provides

a theoretically optimal decorrelation.

As the spectral domain can also be used for image compression, so the first stage

may include mapping into the frequency or sequence domain where the energy in

13

the image is compacted mainly into the lower frequency/sequence components.

These methods are all reversible that is information preserving, although all

mapping methods are not reversible.

• Quantization may be necessary to convert the data into digital form

(BYTE data type), depending on the mapping equation used. This

is because many of these mapping methods will result in floating

point data which requires multiple bytes for representation which is

not very efficient, if the goal is data reduction.

The decompression can be divided into the following stages:

• Decoding: Takes the compressed file and reverses the original

coding by mapping the codes to the original quantized values.

• Inverse mapping: Involves reversing the original mapping process.

• Post-processing: Involves enhancing the look of the final image.

This may be done to reverse any preprocessing, for example, enlarging an

image that was shrunk in the data reduction process. In other cases the post-

processing may be used simply to enhance the image and to improve any artifacts

from the compression process itself. The development of a compression

algorithm is highly application specific. The preprocessing stage of compression

consists of processes such as enhancement, noise removal, or quantization is

applied. The goal of preprocessing is to prepare the image for the encoding

process by eliminating any irrelevant information, where irrelevant is defined by

the application. For example, many images that are only for the viewing purposes

can be preprocessed by eliminating the lower bit planes, without losing any useful

information.

14

1.5 OVERVIEW OF METHODOLOGY

The methodology [24] for the compression process flow which takes an

input image of (NxN) size is shown below.

Figure 1.3 The Compression process on forward transform

The figure 1.3 represents a compression process flow for an input image.

The compression process pre-analyzes across rows and columns and performs

encoding techniques like magnitude set and bit plane coding followed by run

length encoding. The sign data of the coefficients is coded as bit plane with zero

thresholds. This bit plane may be used as it is or coded to scale back the Bits per

Original input Image

Pre-filtration process

Pre-analysis across row

Pre-analysis across column

Performing run length encoding for decomposition

Performing magnitude set & bit plane coding

Store the compressed Image

15

pixels (Bpp).The coefficients are coded by means of magnitude set coding and

run length coding techniques which in turn results with low bits.

On reconstruction, the reverse process is done by decoding and the post-

analysis is done across columns and rows as shown in figure1.4.

Figure 1.4 The Reconstruction on Reverse transform

Compressed Image

Performing run length decoding

Performing magnitude set & bit map plane coding

Inverse for post-analysis across column

Inverse for post-filtration

Inverse post-filtration across row

Reconstructed Image

16

1.5.1 Pre-Processing

Pre-processing, also known as Pixel-level processing or low-level

processing is done on the captured image to prepare it for further analysis. Such

processing includes: grayscale or color image to a binary image, reduction of

noise to reduce extraneous data, segmentation to separate various components in

the image and thinning or boundary detection to enable easier subsequent

detection of pertinent features /objects of interest.

1.5.2 Post-Processing

Image post processing enhances the quality of a finished image to prepare

it for publication and distribution. It includes techniques to clean up images to

make them visually clearer as well as the application of filters and other

treatments to change the appearance and feel of a picture. Cleaning and

sharpening techniques can trim down noise, increase contrast, tighten the crop of

the image, and make other small changes to improve the appearance of the

picture. Image post processing can also involve removing things from the edges

when they don't belong or distract.

1.6 FOURIER TRANSFORMS

The Fourier transform’s utility lies in its ability to analyze a

signal in the time domain for its frequency content. The transform works

by first translating a function in the time domain into a function in the

frequency domain. The signal can then be analyzed for its frequency content

because the Fourier coefficients of the transformed function represent the

contribution of each sine and cosine function at each frequency. An inverse

Fourier transform just transform the data from frequency domain into time

domain. FT is represented as:

17

dtetfF tjωω −∞

∞−∫= )()( (1.8)

Where FT is the sum over all the time of signal f (t) multiplied by a complex

exponential.

1.6.1 Fast Fourier Transforms

To approximate a function by samples, and to approximate the

Fourier integral by the discrete Fourier transform, i t requires applying a

matrix whose order is the number sample points n. Since multiplying an n

× n matrix by a vector costs on the order of n2 arithmetic operations, the

problem gets quickly worse as the number of sample points increases.

However, if the samples are uniformly spaced, then the Fourier matrix can be

factored into a product of just a few sparse matrices, and the resulting factors

can be applied to a vector in a total of order n log n arithmetic operations.

This is the so-called Fast Fourier Transform [19].

FFT computes the DFT and produces exactly the same result as evaluating

the DFT definition directly. The most important difference is that FFT is much

faster. (In the presence of round-off error, many FFT algorithms are also much

more accurate than evaluating the DFT definition directly, Let x0... xN-1 be

complex numbers. The DFT is defined by the formula.

Nn

kiN

nnk exx

π21

0

−−

=∑= k=0…N-1 (1.9)

Evaluating this definition directly requires O (N2) operations, there are N

outputs Xk, and each output requires a sum of N terms. An FFT is any method to

compute the same results in O (N log N) operations. More precisely, all known

FFT algorithms require O (N log N) operations (technically, O only denotes an

18

upper bound).That is the case always when the DFT is implemented via the Fast

Fourier transform algorithm. But other common domains are [-N/2, N/2-1] (N

even) and [-(N-1)/2, (N-1)/2] (N odd), as when the left and right halves of an FFT

output sequence are swapped.

1.6.2 Inverse Fast Fourier Transform

The IFFT is a fast algorithm to perform Inverse (or backward) Fourier

Transform (IDFT), which undoes the process of DFT. IDFT of a sequence {Fn}

that can be defined as:

(1.10)

If an IFFT is performed on a complex FFT result computed by origin, this will

transform the FFT results back to its original data set.

The FT takes a signal in the so called time domain (where each sample in

the signal is associated with a time) and maps it, without loss of information, into

the frequency domain. The frequency domain representation is exactly the same

signal, in a different form. The IFT maps the signal back from the frequency

domain into the time domain. A time domain signal will usually consist of a set of

real values, where each value has an associated time (e.g., the signal consists of a

time series). The FT maps the time series into a frequency domain series, where

each value is a complex number that is associated with a given frequency. The

IFT takes the frequency series of complex values and maps them back into the

original time series. Assuming that the original time series consisted of real

values, the result of the IDFT will be complex numbers where the imaginary part

is zero.

niNnjN

nni eF

Nx

π21

0

1 ∑−

=

=

19

1. 7 WAVELETS

In Fourier analysis, the signal is analyzed using sine and cosine

components; whereas, in wavelet theory, the signal is analyzed in time for

frequency content. Wavelet generates a set of basis functions by dilating and

translating a single prototype function, Ψ(x), which is the basic wavelet. This is

some oscillatory function usually centered upon the origin, and dies out rapidly as

x→ ∞. A set of wavelet basis functions, {Ψa,b(x)},[59] can be generated by

translating and scaling the basic wavelet as,

Ψa,b(x) = (1/√a) * Ψ((x-b)/a) (1.11)

where a > 0 and b are real numbers. The variable ‘a’ reflects the scale (width of

the basis wavelet) and the variable ‘b’ specifies its translated position along the x-

axis and Ψ(x) is also called as mother wavelet. There are many Mother wavelets

like Mexican Hat, Coifflet, Biorthogonal, etc.

On having a choice among an infinite set of basis functions, the best

basis function for a given representation of a signal can be obtained. A basis

of adapted waveform is the best basis function for a given signal representation.

The chosen basis carries substantial information about the signal, and if the

basis description is efficient (that is, very few terms in the expansion are

needed to represent the signal), then that signal information has been

compressed. Some desirable properties for adapted wavelet bases are:

• Speedy computation of inner products with the other basis functions.

• Speedy superposition of the basis functions.

• Good spatial localization, so researchers can identify the position of a

signal that is contributing a large component.

• Good frequency localization, so researchers can identify signal

oscillations and Independence, so that not too many basis elements

match the same portion of the signal.

20

For adapted waveform analysis, researchers seek a basis in which the

coefficients, when rearranged in decreasing order, decrease as rapidly as

possible. To measure t h e rate of decrease, they use tools from classical

harmonic analysis including calculation of information cost functions. This is

defined a t the expense of storing the chosen representation. Examples of

such functions include the number above a threshold, concentration, entropy,

and logarithm of energy, Gauss-Markov calculations, and the theoretical

dimension of a sequence.

1.8 WAVELET TRANSFORM

The Fourier Transform has sinusoidal waves in orthonormal basis. This

transform provides a signal which is localized only in the frequency domain. For

this integral transform, the basis functions extend to infinity in both directions.

However, transient signal components are non-zero only during a short interval.

In images, many important features like edges are highly localized in spatial

position. Such components do not resemble any of the Fourier basis functions and

they are not represented compactly in the transform coefficients. Thus the Fourier

Transform and other wave transforms are less optimal representations for

compressing and analyzing signals and images containing transient or localized

components.

To combat such a deficiency, mathematicians and engineers have explored

several approaches using transforms having basis functions of limited duration.

These basis functions vary in position as well as frequency. They are waves of

limited duration and are referred to as wavelets. Transforms based on them are

called Wavelet Transforms. Wavelets are a result of the time frequency analysis

of the signals in terms of a two-dimensional time-frequency space. According to

the time frequency analysis theory, each transient component of a signal maps to

21

position in the time frequency plane that corresponds to that component’s

predominant frequency and time of occurrence. For images the space is three-

dimensional and can be viewed as an image stack. The approach started with

Gabor’s windowed Fourier Transform, and led to short-time Fourier transforms

and then to the subband coding.

The serious drawback in transforming between the frequency domain and

the time information is that it leads to information loss. When looking at a Fourier

transform of a signal, it is impossible to say when a particular event took place.

Wavelet Transform is similar to the short time Fourier transform (STFT) to

overcome the resolution problem where the signal is multiplied with a function

[19]. It has high time resolution and low frequency resolution at high frequencies

together with high frequency resolution and low time resolution at low

frequencies. It is very suitable for short duration of higher frequency and longer

duration of lower frequency components.

The fundamental idea behind wavelets is to analyze according to scale.

Indeed, some researchers in the wavelet field feel that, by using wavelets, one

is adopting a whole new mindset or perspective in processing data. Wavelets

are functions that satisfy certain mathematical requirements and are used in

representing data or other functions. The idea is not new, since the early

1800’s approximation using superposition of functions has been existed,

when Joseph Fourier discovered that he could superpose sine and cosine to

represent other functions. However, in wavelet analysis, the scale that is used

to look at the data plays a special role. Wavelet algorithm processes the data at

different scales or resolutions. On looking at a signal with a large “window,”

One would notice coarse features. Similarly, looking at a signal with a small

“window,” One would notice fine features. The result in wavelet analysis is to

22

see both the forest and the trees. So in the following the basic concepts that

make wavelet analysis such a useful signal processing tool have been presented.

There are two types of wavelet transform, namely continuous wavelet transform

and Discrete Wavelet Transform.

1.9 CONTINUOUS WAVELET TRANSFORM

The Continuous wavelet transform of f (x) with respect to the wavelet

function Ψ(x) is then given as,

W (a, b) = <f, Ψa, b > (1.12)

where <> represents the inner product. For two-dimensional continuous wavelet

transform the basis function is of two dimensions, Ψ(x,y) and hence two

translation variables and one scaling variable is used. The important feature of the

wavelets is that they can be interpreted as filter bank. This implies that the

wavelet transform can be implemented as convolution of the input signal to the

filter. The filter coefficients depend on the mother wavelet that has been chosen

[59].

A continuous wavelet transform (CWT) is used to divide a continuous-

time function into wavelets. Unlike Fourier transform, the continuous wavelet

transform possesses the ability to construct a time-frequency representation of a

signal that offers very good time and frequency localization. The continuous

wavelet transform of a function )(tX at a scale (a>0), *+Raε and translational

value Rb ε is expressed by the following integral.

dta

bttxa

baX

−

= ∫∞

∞−

ψω )(1),(21 (1.13)

23

Where )(tψ is a continuous function in both the time domain and the frequency

domain called the mother wavelet and the over line represents operation of

complex conjugate. The main purpose of the mother wavelet is to provide a

source function to generate the daughter wavelets which are simply the translated

and scaled versions of the mother wavelet. To recover the original signal )(tx , the

first inverse continuous wavelet transform can be exploited.

(1.14)

ψ~ (t) is the dual function of )(Tψ and

(1.15)

Cψ is admissible constant, where hat means Fourier transform operator.

Sometimes, )()(~ tt ψψ = , then the admissible constant appears like

ωωωψ

ψ dC2)(~

∫∞∞−= (1.16)

Traditionally, this constant is called wavelet admissible constant. A wavelet

whose admissible constant satisfies ∞<< cψ0 is called an admissible

wavelet. An admissible wavelet implies that 0)0(~ =ψ , so that an admissible

wavelet must integrate to zero. To recover the original signal )(tx , the second

inverse continuous wavelet transform can be exploited.

dadba

btibaXa

tx

−

= ∫∫∞

∞−

∞

∞−

exp),(1)1(2

1)( 2 ωπψ (1.17)

ωω

ωψωψψ dt

C)(~))((

~̂∫∞

∞−

=

21 exp~1),()(

21 a

ddba

bta

baXCtx

−

= ∫∫∞

∞−

∞

∞−

− ψωψ

24

This inverse transform suggests that a wavelet should be defined as

(1.18)

where )(tw is the window. Such defined wavelet can be called as an analyzing

wavelet, because it admits to time-frequency analysis. An analyzing wavelet is

unnecessary to be admissible.

1.10 DISCRETE WAVELET TRANSFORM (DWT)

The Discrete wavelet transform used to calculate the wavelet coefficients

at every possible scale and it generates an awful lot of data. It turns out that if one

chooses scales and positions based on powers of two so called dyadic scales and

positions, then analysis will be much more efficient and accurate on capturing

both frequencies as well location information [59].

Figure 1.5 The Wavelet coefficients at II level Decomposition

1.10.1 Haar wavelets

The first DWT was invented by Hungarian mathematician Alfred

Haar. For an input represented by a list of 2n numbers, the Haar wavelet transform

may be considered to pair up input values, storing the difference and passing the

)exp()()( ittwt =ψ

25

sum. This process is repeated recursively, pairing up the sums to provide the next

scale, which leads to 2n -1 differences and a final sum.

1.10.2 Daubechies wavelets

The most commonly used set of discrete wavelet transforms was

formulated by the Belgian mathematician Ingrid Daubechies in 1988. This

formulation is based on the use of recurrence relations to generate progressively

finer discrete samplings of an implicit mother wavelet function; each resolution is

twice that of the previous scale. In her seminal paper, Daubechies derives a

family of wavelets, the first of which is the Haar wavelet. Interest in this field has

exploded since then, and many variations of Daubechies original wavelets have

been developed. Although the DCT-based image compression method used in the

JPEG standard, has been very successful in the several years, there is still some

scope for improvement [7].

Wavelet analysis is similar to the Fourier analysis in a sense that it breaks

a signal down into its constituent parts for analysis. The Fourier transforms break

the signal into a series of sine waves of different frequencies. Whereas the

wavelet transform breaks the signal into its "wavelets", scaled and shifted

versions of the "mother wavelet". There are some very distinct differences

between them as evident in Figure 1.6, which compares a sine wave to a typical

Debauches 5 wavelet. In comparison to the sine wave which is smooth and of

infinite length, the wavelet is irregular in shape and compactly supported. This

property makes the wavelets an ideal tool for analyzing signals of a non-

stationary nature. Their irregular shape lends them to analyzing signals with

discontinuity's or sharp changes, while their compact nature enables temporal

localisation of signals features. When analyzing signals of a non-stationary

nature, it is often beneficial to be able to acquire a correlation between the time

26

and frequency domains of a signal. The Fourier transform provides information

about the frequency domain, however time localised information is essentially

lost in the process.

Figure 1.6 Comparison of Sine wave and Daubechies 5 wavelet

The problem with this is the inability to associate features in the frequency

domain with their location in time, as an alteration in the frequency spectrum will

result in changes throughout the time domain. In contrast to the Fourier

transform, the wavelet transform allows exceptional localisation in both the time

domain via translations of the mother wavelet, and in the scale (frequency)

domain via dilations. The translation and dilation operations applied to the mother

wavelet are performed to calculate the wavelet coefficients, which represent the

correlation between the wavelet and a localised section of the signal. The wavelet

coefficients are calculated for each wavelet segment, giving a time-scale function

relating the wavelets correlation to the signal. This process of translation and

dilation of the mother wavelet is depicted in Figure 1.7. It should be noted that

the process examined here is the DWT, where the signal is broken into dyadic

blocks (shifting and scaling is based on a power of 2). The continuous wavelet

transform (CWT) still uses discretely sampled data, however the shifting process

is a smooth operation across the length of the sampled data, and the scaling can

be defined from the minimum (original signal scale) to a maximum chosen by the

27

user, thus giving much finer resolution. The trade off for this improved resolution

is an increased computational time and memory required to calculate the wavelet

coefficients. A comparison of the DWT and CWT representations of a noisy chirp

signal with a high frequency component is shown in figure 1.8 as example.

Figure 1.7 Scaling and shifting process of the DWT

Figure 1.8 Example of comparison between DWT and CWT [7]

28

In numerical analysis and functional analysis, a discrete wavelet transform

is any wavelet transform for which the wavelets are discretely sampled. As with

other wavelet transforms, it has temporal resolution, which is the key advantage

over Fourier transforms, it captures both frequency and location information

(location in time). An advantage of wavelet transforms is that the windows

vary. In order to isolate signal discontinuities, one would like to have some

very short basis functions. At the same time, in order to obtain detailed

frequency analysis, one would like to have some very long basis functions. A

way to achieve this is to have short high-frequency basis functions and

long low-frequency ones. One thing to remember is that wavelet transforms

do not have a single set of basis functions like the Fourier transform, which

utilizes just the sine and cosine functions. Instead, wavelet transforms have

an infinite set of possible basis functions. Thus wavelet analysis provides

immediate access to information which are difficult to analyze by other

time-frequency methods such as Fourier analysis.

1.10.3 DWT and Filter Banks

The Discrete Wavelet Transform is based on sub-band coding, it is

found to obtain a fast computation of Wavelet Transform. Discrete Wavelet

Transform is easy to implement and reduces the computation time. The technique

similar to sub-band coding is known as pyramidal coding, and is used in efficient

multi-resolution analysis schemes of image. In the case of DWT, time-scale

representation of the digital signal is obtained using digital filtering method.

These digital filters are mainly used to suppress either the high frequencies in the

image (smoothing the image), or the low frequencies, (enhancing or detecting

edges in the image).An image can be filtered either in the frequency or in the

spatial domain. So the signal to be analyzed is passed through filters with

29

different cutoff frequencies at different scales. The first involves transforming the

image into the frequency domain, multiplying it with the frequency filter function

and re-transforming the result into the spatial domain. The filter function is

shaped so as to attenuate some frequencies and enhance others.

The advantage of the DWT is that it performs multi-resolution analysis of

signals with localization both in time and frequency domain. Whereas DWT

decomposes a digital signal into different sub-bands so that the lower frequency

sub-bands have good frequency resolution and coarser time resolution as

compared to the higher frequency sub-bands. Discrete Wavelet Transform is

highly used in image compression due to the fact that the DWT supports features

like progressive image transmission by quality and resolution, and ease of image

compression coding and manipulation. Because of these characteristics, DWT is

the basis of the image compression standard. So, in the discrete wavelet

transform, the image signal can be analyzed by passing through an analysis filter

bank followed by decimation operation. This analysis filter banks consist of a

low-pass and high-pass filter at each decomposition stage of the process. When

the signal passes through these filters such as Low-pass and High pass, it split

through two bands. The low-pass filter of the filter bank, which corresponds to an

averaging operation of the image sample, extracts the coarse information of the

signal or image. The high-pass filter performed corresponds to a differencing

operation, and extracts the detail information of the signal or image. Then output

of the filtering operation is decimated by two. The two-dimensional

transformation is accomplished by performing two separate one-dimensional

transforms. The first, image is filtered along the row and decimated by two. Then

it is followed by filtering the subbands image along the column and decimated by

two. So this operation splits the image into four bands, such as, LL, LH, HL, and

HH respectively.

30

1.10.4 First level of Transform The DWT of a signal x is calculated by passing it through a series of

filters. First the samples are passed through a low pass filter with impulse

response g resulting in a convolution of the two:

[ ] [ ] [ ] [ ]∑∞

−∞=

−==k

kngkxngxng )*( (1.19)

The signal is also decomposed simultaneously using a high-pass

filter h . The outputs give the detail coefficients from the high-pass filter and the

approximate coefficients from the low-pass. It is important that the two filters are

related to each other and they are known as a quadrature mirror filter.

Figure 1.9 The filter analysis

The filter outputs are then sub sampled by 2. In the next two formulas, the

notation is the opposite: g- denotes high pass and h- low pass as is Mallat's and

the common notation:

[ ] [ ] [ ]∑∞

−∞=

−=k

low knhkxny 2 (1.20)

[ ] [ ] [ ]∑∞

−∞=

−=k

high kngkxny 2 (1.21)

This decomposition halved the time resolution since only half of each filter

output characterizes the signal. However, each output has half the frequency band

of the input. So, the frequency resolution has been doubled with the subsampling

operator↓2.

31

( )[ ] [ ]knynky =↓ (1.22)

The above summation can be written more concisely.

( ) 2* ↓= gxylow (1.23)

( ) 2* ↓= hxyhigh (1.24)

However, on computing a complete convolution gx * with subsequent

down sampling would waste computation time. The lifting scheme is an

optimization where these two computations are interleaved.

1.10.5 Cascading and Filter banks

In filter bank the decomposition is repeated further to increase the

frequency resolution and the approximation coefficients decomposed with high

and low pass filters and then down-sampled. This is represented as a binary tree

with nodes representing a sub-space with a different time-frequency localisation.

The tree is known as a filter bank. For the 2-D Discrete Wavelet transform

implementation is based on the pyramidal algorithm developed for

multiresolution analysis of the signals.

The Pyramidal Algorithm is based upon the Filter bank theory. The

wavelet function and the scaling function are chosen. These functions are then

used to form the dilation equation. The wavelet dilation equation represents the

high pass filter. The scaling dilation equation represents the low pass filter. These

filter coefficients are then used to construct the filters [28]. Let h(n) be the low

pass filter and g(n) be the high pass filter. Then for the perfect reconstruction, it

has to satisfy some properties, In frequency domain, such as,

H (w) 2 + H (w+Π) 2 =1 (1.25)

H (w) 2 + G (w) 2 =1 (1.26)

32

The Filter structure of the Pyramidal Algorithm for I level Discrete Wavelet

Transform (DWT) and Inverse Discrete Wavelet Transform (IDWT) is

shown in figure 1.10and 1.11.

Figure 1.10 I-Level DWT Filter Implementation

Figure 1.11 IDWT Filter Implementation

Hi (z) ↑2

Gi (z) ↑2

+

Hi (z) ↑2

Gi (z) ↑2

+

Hi (z) ↑2

Gi (z) ↑2

+

LL

LH

HL

HH

Image

Columns Rows

H(z) ↓2

G(z) ↓2

H(z) ↓2

G(z) ↓2

H(z) ↓2

G(z) ↓2

LL

LH

HL

HH Rows

Columns

Image

33

Where H(z) is the Low pass Analysis Filter and G(z) is High Pass Analysis Filter,

Hi(z) is the Low pass Synthesis Filter and Gi(z) is High pass Synthesis Filter.

The Interpretation of the 2-Dimensional DWT for an NxN Image is shown

in figure 1.12.

Figure 1.12 Subband Decomposition of Image

The Wavelets are particularly attractive, as they are capable of capturing

most image information in the highly sub sampled low frequency band (LL) also

called as the approximation signal. The additional localized edge information in

spatial clusters of coefficients will be in the high frequency bands (HL, LH, and

HH). Another attractive aspect of the coarse to fine nature of the wavelet

representation naturally facilitates a transmission feature that enables progressive

transmission as an embedded bit stream.

To be specific, the wavelet transform is a good fit for typical natural

images that have an exponentially decaying spectral density, with a mixture of

strong stationary low frequency components and perceptually important short

duration high frequency components. The fit is good because the wavelet

34

transform’s decomposition attributes have good frequency resolution at low

frequencies and good time resolution at high frequencies. There are, however,

important classes of images whose attributes go against those offered by the

wavelet decomposition, e.g., images having strong high frequency components.

These images are better matched with decomposition elements that have good

frequency localization at higher frequencies, which the wavelet decomposition

does not offer.

Although the task of finding an optimal decomposition for every

individual image in the world is an impossible task, the situation gets more

interesting if we consider a large but finite library of desirable transforms, and the

best transform in the library adaptive to an individual image. Here the problem of

maintaining the library and the search is going to be difficult and this met with

wavelet packets. The extra adaptivity of the wavelet packet is obtained at the

price of added computation in searching for the best wavelet packet basis. The

other alternative that can bypass this complexity of searching best basis is the

Multiwavelet transform for the images.

It is applied in fields that are making use of wavelets which include

astronomy, acoustics, nuclear engineering, sub-band coding, signal and

image processing, neurophysiology, music, magnetic resonance imaging,

speech discrimination, optics, fractals, turbulence, earthquake-prediction,

radar, human vision, and pure mathematics applications such as solving

partial differential equations.

1.11 FAST WAVELET TRANSFORM

The DWT matrix is not sparse in general, so faces the same

complexity issues that had previously faced by the discrete Fourier

transform. This is solved, similar to the FFT method, by factoring the DWT into

35

a product of a few sparse matrices using self-similarity properties. The result

of this algorithm is that requires only arrange ‘ n’ operations to transform an

n-sample vector.

1.12 2-D DISCRETE WAVELETS TRANSFORM

The concepts developed for the representation of one-dimensional signals

generalize easily to two-dimensional signals. The scaling functions of DWT

represent the theories of multiresolution analysis and wavelets can be generalized

to higher dimensions. In practice, the usual choice for a two-dimensional scaling

function or wavelet is a product of two one-dimensional functions [43]. For

example,

)()(),( yxyx ϕϕϕ = (1.27)

And the dilation equation assumes the form:

∑ −−=1,

)12,())1,((2),(k

ykkhyx ϕϕϕ (1.28)

1.13 INTRODUCTION TO INTEGER MULTIWAVELET

TRANSFORMS

Integer wavelet transforms implemented by selected wavelet transform

with truncation, have been successfully applied to lossless image coding. The

transformation reduces the pixel correlation and first order entropy of the original

image. The simplest integer wavelet transform is the S-transform, which is an

integer version of the Haar wavelet transform. However, Haar wavelet has only

one vanishing moment accounting for its reduced ability in minimizing highpass

coefficient energy. One approach to tackle this problem is to improve the S-

transform by introducing prediction to generate new set of highpass coefficient

36

based on the S-transform lowpass coefficients. Since the S-transform is a

block transform. The effect is to exploit the correlation among the

neighbor coefficient blocks. The vanishing moment of the resulting high

pass analysis filter is also increased.

Instead of improving the integer Haar wavelet system, construct an integer

version of this simplest multiwavelet system of multiplicity r=2 and replace the

integer Haar transform by the new integer multiwavelet transform for lossless

image coding. The main advantage of the integer multiwavelet system is its

higher order of approximation(with two vanishing moments in its nontruncated

system) implying higher energy compaction capability while maintaining the

symmetry and short support properties as compared with the Haar wavelet

system. The main difference between traditional wavelet system and multiwavelet

system is that multiwavelet transform is implemented by multifilter bank with

vector sequence as its input and output. Pre-filtering/pre-processing of the

original signal is required to extract vector input from the multifilter bank. Thus

the associated prefilter for the proposed IMWT has to be designed. When IMWT

is applied to lossless image compression, experimental results show that its

compression capability outperforms that of the S-transform and lossless JPEG

[66].The interior performance as compared with some best lossless coding

schemes such as CREW and S+P, is expected as the proposed IMWT has not

exploited the inter block correlation as the former schemes do on the S-transform.

However, IMWT is a better alternative to S-transform and by exploiting inter

block correlation, better performance is expected.

1.14 MULTIWAVELET TRANSFORMS

Multiwavelet transform is very similar to the wavelet transform. Wavelet

transform makes use of a single scaling function and wavelet function, hence also

37

called as scalar wavelet transform. Multiwavelet transform have more than one

scaling function and wavelet function. The scaling functions and wavelet

functions are grouped into vectors. The number of such functions that are

grouped forms the multiplicity of the transform. For notational convenience,

Multiwavelet transform with multiplicity ‘r’ can be written using a vector

notation φ(t) = [φ1(t), φ2(t)… φr(t)], the set of scaling functions and ψ(t) = [ψ1(t),

ψ2(t), ….., ψr(t)], wavelet functions. When r =1,it forms a scalar wavelet

transform. If r>=2, it becomes Multiwavelet Transform. Till date Multiwavelet

transforms of multiplicity r=2 have been studied. As with the scalar wavelet

transform, The Multiwavelet transform also has a set of dilation equation that

gives the filter coefficients for the low pass and high pass filters. Multiwavelet

transform with multiplicity two has two low pass filters and two high pass filters.

Examples include GHM, CL, and IMWT.

The two dilation equations of Multiwavelet resemble those of scalar

wavelets and are given as [43],

k)-φ(2tHφ(t)k

k∑= (1.29)

k)-ψ(2tGψ(t)k

k∑= (1.30)

where Hk, Gk are the low pass and high pass multifilter coefficients respectively.

With Multiwavelet, there are more degrees of freedom to design the system. For

instance, simultaneous possession of orthogonality, short support, symmetry and

high approximation order is possible in Multiwavelet system. Multiwavelet can

be used to reduce the restrictions on the filter properties. For example, it is well

known that a scalar wavelet cannot simultaneously have both orthogonality and

symmetric property. Symmetric filters are necessary for symmetric signal

extension, while orthogonality makes the transform easier to design and

38

implement. Also, the support length and vanishing moments are directly linked to

the filter length for scalar wavelets.

This means longer filter lengths are required to achieve higher order of

approximation at the expense of increasing the wavelet’s interval of support. A

higher order of approximation is desired for better coding gain, but shorter

support is generally preferred to achieve a better localized approximation of the

input function. In contrast to the limitations of scalar wavelets, Multiwavelet are

able to possess the best of all these properties simultaneously.

1.15 INTEGER MULTIWAVELET TRANSFORM FUNCTION

In a general multiresolution analysis (MRA) of multiplicity r, the

r scaling functions rϕϕ ..,,.1 and the corresponding Multiwavelet rψψ ..,,.1

are usually represented as vectors [ ]Trϕφφ ,....,1= and [ ]Trψψψ ,...,1= This will

satisfy the matrix dilation equation ∑ −=k k ktHt )2()( φφ and the matrix

wavelet equation ∑ −=k k ktGt )2()( ϕψ where kH and kG are the low pass

and high pass multifilter coefficients respectively. With multiple wavelets,

there is more degree of freedom to design the system. For instance,

simultaneous possession of orthogonality, short support, symmetry and

high approximation order is possible in Multiwavelet system. The Integer

Multiwavelet transform is based on the box and slope scaling functions. The

system is based on the multiscaling and Multiwavelet given by [30],

+

−

=

(2t)φ(2t)φ

2/12/101

(2t)φ(2t)φ

2/12/101

(t)φ(t)φ

2

1

2

1

2

1 (1.31)

+

=

)2(ψ)2(ψ

002/12/1

)2(ψ)2(ψ

002/12/1

)(ψ)(ψ

2

1

2

1

2

1

tt

tt

tt

(1.32)

39

The two scaling functions are box, )(1 tφ , and slope, )(2 tφ . Similar to Haar

transform, it has short support and symmetry, and is a block transform, not

overlapping the next pair of samples. The approximation order of this

system, however, is two as the combination of the )(1 tφ and )(2 tφ can

exactly reproduce linear functions [62]. So, by including one additional

scaling function )(2 tφ , the original Haar wavelet transform is generalized to a

Multiwavelet transform with higher approximation accuracy.

The two sequences { }c n)0(

,1 and { }c n)0(

,2 for the multifilter bank input are

the output of the pre-filter input sequence{ }nx, .As an approximation to the

non truncated system, equation 3.3 and 3.4, The forward integer Multiwavelet

transform can be expressed as a two step algorithm to compute the four

sequences as{ }{ }{ }{ } { }d1)(j

n2,and,d1)(j

n1,,c1)(j

n2,,c1)(j

n1,,c(0)

n1,−−−− .At scale level j-1 from the

two sequences { } { }c(j)

n2,andc(j)

n1, at scale level j:

Step 1 (Backward):

+−=

−+=

++=

++=

2

c(j)

12,2nc(j)2,2n

m(j)

n2,

c(j)1,2nc

(j)11,2nm

(j)n1,

c(j)

12,2nc(j)2,2ns

(j)n2,

2

c(j)

11,2nc(j)1,2n

s(j)

n1,

(1.33)

40

Step 2 (Forward):

−=−

−=−

+=−

=−

m(j)

n2,d1)(jn2,

m(j)

n1,s(j)

n2,d1)(j

n1,

2s(j)

n2,m(j)

n1,c

1)(jn2,

s(j)

n1,c1)(j

n1,

(1.34)

where corresponds to downward truncation.The block transformation of

the four elements c(j)

12,2n,c(j)2,2n,c

(j)11,2n,c

(j)1,2n ++ may be viewed as application

of integer Harr transform to the selected pair among the four elements,Thus

the transform is reversible and the inverse transform is simply the backward

running of the forward transform and is expressed as:

Step 1(Backward):

−=

−−=

+

−+−=

−=

d1)(jn2,m

(j)n2,

d1)(j

n1,s(j)

n2,m(j)

n1,

1)/2(d1)(j

n1,c1)(jn2,s

(j)n2,

c1)(j

1ns(j)

n1,

(1.35)

Step 2 (Backward):

−=−+

++=

−+=−

++=+

c(j)2,2nm

(j)n2,c

1)(j12,2n

/21s(j)

n2,m(j)

n2,c(j)2,2n

m(j)

n1,c(j)

11,2nc1)(j

2n

/21m(j)

n1,s(j)

n1,c(j)

11,2n

(1.36)

41

The Integer Multiwavelet Transform (IMWT) has short support,

symmetry, high approximation order of two. It is a block transform. It can be

efficiently implemented with bit shift and addition operations. Another advantage

of this transform is that, while it increases the approximation order, the dynamic

range of the coefficients will not be largely amplified, which an important

requirement for lossless coding.

1.16 MULTIWAVELET FILTER BANKS Multi wavelets of multiplicity ‘r’ require ‘r’ input streams to the multi

wavelet filter banks. A multi wavelet filter banks has taps that are rxr matrices.

Coefficients for the low pass filter bank Hk are given by four r x r matrices and

The same is true for High pass filters Gk. Here coefficients of high pass filter Gk

cannot be obtained by flipping low pass filter coefficients as in it is done in scalar

wavelets. It has to be designed separately. For r channel r x r matrix filter bank

operates on r input data streams and generates 2r output streams which are then

down sampled by 2. Every row of multifilters are a combination of r ordinary

filters each operating on different data stream. The multi wavelet theory is also

based on multi resolution analysis. If decompose an image using a scalar wavelet

to single level of decomposition, resultant data will correspond to four sub band

of low pass /high pass filter in both the dimensions[28]. Data in LH sub band is

the output from high pass filtering of rows first and then low pass filtering of

column. For multi wavelets with multiplicity r=2, will have two sets of scaling

coefficients. In case of multi wavelets subscript 1 and 2 along with L and H

corresponds to the channel 1 and 2 respectively.

1.17 MULTIWAVELET DECOMPOSITION

The Filter bank implementations of the Multiwavelet transform with

multiplicity two, need four filters. The pyramidal algorithm then needs four filters

42

followed by a downsampler of factor four. In such cases the loss of information is

more. Hence the down sampling process is split into two stages by using prefilter.

This is better in terms of loss of information and complexity of design.

Figure 1.13 2-D Multiwavelet decomposition of an image

The prefilter produces vector inputs that are needed for the filters. The

decomposition of the image by Multiwavelet transform uses prefilter as shown in

figure 1.13, the reconstruction uses the post filter to produce the images.

1.18 WAVELET AND MULTIWAVELET COMPARISON

The Multiwavelet idea originates from the generalization of scalar

wavelets. Instead of one scaling function and one wavelet, multiple scaling

functions and multiple wavelets are used. This leads to a more degree of freedom

in constructing wavelets. Therefore as opposed to scalar wavelets, properties such

as compact support, orthogonality, symmetry, vanishing moments and short

support can be obtained simultaneously in Multiwavelet, which are the

43

fundamental requirement in signal processing .The increase in degree of freedom

in Multiwavelet is obtained at the expense of replacing scalars with matrices,

scalar functions with vector functions and single matrices with block of matrices.

However, pre-filtering is an essential task which should be performed for any use

of Multiwavelet in signal processing. The comparisons between the scalar and the

Multiwavelet are listed in Table 1.2.

Multiwavelet system can simultaneously provide perfect reconstruction

while preserving length due to orthogonality of filters, good performance at the

boundaries (via linear-phase symmetry) and a high order of approximation

(vanishing moments).Multiwavelet decomposition produce two low pass sub

bands and two high pass sub bands in each dimension. Figure 1.14 shows the

sub band structure after first level of Multiwavelet decomposition. Wavelet

decomposition yields four sub bands after first level of decomposition, where as

in Multiwavelet sixteen sub bands result after first level of decomposition. The

next step of the cascade will decompose the low-low-pass sub- matrices L1L1,

L2L1, L1L2 and L2L2 in a similar manner.

Table 1.2

Comparison of Scalar and Multiwavelet Transform

SCALAR WAVELETS MULTIWAVELETS

1. Both Scaling and Wavelet

functions are scalars

1. Both Scaling and Wavelet

functions are vectors

2. It has one Scaling and Wavelet

Function

2. Multiwavelet with multiplicity ‘r’

has r scaling and r wavelet function

3. The Solution of the Dilation

Equation results in one Low Pass

(L) and one High Pass (H) filter.

3. The Dilation Equation is of matrix

form. For Multiwavelet of

multiplicity 2, it results in four filters

44

with two Low pass L1, L2 and two

High pass H1, H2 filters.

4. The Input Image can be used as

it is for the 2D Pyramidal

Algorithm.

4. The input image has to be pre-

filtered to produce vectors that can

be applied to the 2D Algorithm.

5. It produces 4 subbands after I

level decomposition namely,

LL – Approximation.

LH – Horizontal Detail.

HL – Vertical Detail.

HH – Diagonal Detail.

5. Here, it produces Sixteen

Subbands namely L1L1, L1L2, L2L1,

L2L2, L1H1, L1H2, L2H1, L2H2, H1L1,

H1L2, H2L1, H2L2, H1H1, H1H2,

H2H1, H2H2.

L1

L2

H1

H2

L1L1

L1L2

L1H1

L1H2

L2L1

L2L2

L2H1

L2H2

H1L1

H1L1

H1H1

H1H2

H2L1

H2L2

H2H1

H2H2

a) Horizontal filtering b) Vertical direction after horizontal filtering

Figure 1.14 I - level of decomposition Subband structure of images

45

1.19 MAGNITUDE SET – VARIABLE LENGTH INTEGER

REPRESENTATION

The Integer Multiwavelet Transform produces coefficients of both positive

and negative magnitudes. These coefficients have to be coded efficiently so as to

achieve better compression ratios. The coefficients of the H1H1, H1H2, H2H1, and

H2H2 have only the edge information and are mostly zeros. This helps in having

some redundancy in the subband, which can be exploited for compression. Each

Transform coefficient has sign and magnitude part in it. Magnitude set coding is

used for the compression of the magnitude and Run length encoding is used for

coding the sign part of the coefficients. Two methods based on the Magnitude set

variable length integer (MS-VLI) and Run length encoding have been tested for

both lossy and lossless Compression of the Integer Multiwavelet transform

(IMWT) coefficients. The Transformed coefficients are grouped into different

magnitude sets in both the methods. Each coefficient has three parameters namely

(Set, Sign and Magnitude) in MS-VLI coding [76]. The set information is coded

using run length encoding, followed by a bit for sign then followed by magnitude

information in bits.

In MS-VLI the sign bit is eliminated from the parameter list. Separate

coded is done using RLE method. Each coefficient is coded with two parameters

(Set, Magnitude).The coefficients with zero magnitude have no sign information

for coding. The magnitude set is used. The decoding is simple. The magnitudes of

the coefficients are reproduced with the set and magnitude information. Then the

sign bits are applied to each. If a coefficient is zero in magnitude, no sign bit has

to be applied, search for the next non-zero coefficient. Searching the non-zero

coefficients according to the scan order, and applying the run length decoded sign

information remains the decoding algorithm. From the analysis of the Integer

Multiwavelet transform (IMWT), it has been found that the L1L1 sub band has

46

always-positive coefficients. Thus the sign information of that sub band is not

coded.

Table 1.3

Definition of Magnitude Set Variable Length Integer Representation

Magnitude Set Amplitude

Intervals Magnitude Bits

0 [0] 0

1 [1] 0

2 [2] 0

3 [3] 0

4 [4 -5] 1

5 [6 - 7] 1

6 [8 – 11] 2

7 [12 – 15] 2

….. ……. …..

Thus for a NXN image the sign information of a N/4 X N/4 is not required.

It is implied that the first subband values are positive. The magnitudes of the

coefficients are grouped into different Magnitude sets according to the table 1.3.

1.20 ORGANIZATION OF THE THESIS

This thesis is organized in five chapters. Chapter-1 This chapter of the thesis provides the Introduction of Data

compression and the Compression techniques also give the overview of the

Compression system and methodology used for the compression techniques. It

also provides the basic fundamental of wavelet and its transform. It also gives the

47

performance metrics which is evaluated in number of ways with the existing

compression formats and also discusses the functions of Integer Multiwavelet

transform and Multiwavelet transform that includes Multiwavelet filter banks on

decomposition. It also provides the different comparison between the wavelet and

Multiwavelet. It also highlights the representation of Magnitude set variable

length integer.

Chapter-2 This chapter of the thesis provides the study made on different

Literature survey on both Lossy and Lossless methods. It also provides the

reviews for various Enhanced compression Algorithms as well the Real time

applications that plays role on compression techniques. It also gives the

knowledge gap identified between these compression techniques.

Chapter-3 This Chapter of thesis briefs about the Implementation of IMWT that

made use of Integer prefilters during the Compression process and the

transformation for obtaining the low bit rate. It also provides Magnitude set

coding, run length coding as well as bit plane coding and its representation with

Integer Multiwavelet transform. It also provides the evaluation of SSIM and

DSSIM.

Chapter-4 This part of the thesis gives the brief discussion on algorithm and

results obtained using both (MS-VLI) Magnitudes set variable length integer

Representation performance with and without RLE algorithm for both Lossy and

Lossless compression methods and the Procedure used for obtaining the very low

bits.

Chapter-5 This chapter of the thesis concludes with a summary of the outcomes

of the research work, augmented with the future research directions that arise

from the investigations that have been carried out.

48

CHAPTER 2

LITERATURE SURVEY

2.1 INTRODUCTION

The purpose of this literature reviews is to provide the background

concepts of the compression techniques and their issues with that of other existing

are to be considered in this thesis and to highlight the relevance of the current

studies. The thesis enlightens the higher compression ratio that provides the

output with good quality obtained from compressed Input images (NxN) size.

In this literature review obtaining the low bits using lossy and lossless

compression technique, for some standard (NxN) size test images of Lena,

Baboon and Barbara, when compared with existing standard algorithm also

studied. The standard Lena, Barbara and Baboon images that were tested and the

quality of the output was calculated using PSNR (dB), SSIM and Bits per Pixels

(Bpp) studies was undergone. This thesis considers various aspects of lossy and

lossless compression techniques with special references on obtaining the high

quality of output images based on their mathematical measures. To study and

analyze compression techniques specifically that is applicable for IMWT. This

thesis also presents related efforts for enhancing performance of those techniques

to achieve minimal computational load that may consumes less power as possible

while maintaining acceptable visual quality.

2.2 ANALYSIS OF WAVELET AND PROCESSING

Raghuveer M.Rao [59] proposed in the wavelet analysis to generate a set

of basis functions by dilating and translating a single prototype function, Ψ(x),

which is the basic wavelet. This is some oscillatory function usually centered

49

upon the origin, and dies out quickly as x → ∞. A set of wavelet basis

functions, {Ψa,b(x)}, can be generated by translating and scaling the basic wavelet

as,

Ψa,b(x) = (1/√a) * Ψ((x-b)/a) (2.1)

where a and b are real numbers. The variable ‘a’ is a positive number that reflects

the scale (width of the basis wavelet) and the variable ‘b’ specifies its translated

position along the x-axis and Ψ(x) is also called as mother wavelet. There many

mother wavelets like Mexican Hat, Coifflet, Biorthogonal, etc. There are two

types of wavelet transform, namely continuous wavelet transform and Discrete

wavelet transform.

Raghuveer M.Rao [59] proposed the Pyramidal algorithm that is based

upon the filter bank theory. The wavelet function and the scaling function are

chosen. These functions are then used to form the dilation equation. The wavelet

dilation equation represents the high pass filter. The scaling dilation equation

represents the low pass filter. These filter coefficients are then used to construct

the filters. Let h(n) be the low pass filter and g(n) be the high pass filter. Then for

the perfect reconstruction, it has to satisfy some properties in frequency domain,

such as,

H (w) 2 + H (w+Π) 2 =1 (2.2)

H (w) 2 + G (w) 2 =1 (2.3)

Gang Lin [15] proposed notational convenience, Multiwavelet transform

with multiplicity ‘r’ can be written using a vector notation φ(t) = [φ1(t), φ2(t)…

φr(t)], the set of scaling functions and ψ(t) = [ψ1(t), ψ2(t), ….., ψr(t)], the set of

wavelet functions When r =1 then it forms the scalar wavelet transform. If r >=2

it becomes Multiwavelet transform. As with the scalar wavelet transform the

50

multiwavelet transform also has a set of dilation equation that gives the filter

coefficients for the low pass and high pass filters. Multiwavelet transform with

multiplicity two has two low pass filters and two high pass filters. examples

include GHM, CL, and IMWT.

Cotronei [43] proposed the Multiwavelet two dilation equations resemble

those of scalar wavelets and are given as.

k)-φ(2tHφ(t)k

k∑= (2.4)

k)-ψ(2tGψ(t)k

k∑= (2.5)

where Hk, Gk are the low pass and high pass multifilter coefficients

respectively.With Multiwavelet there are more degrees of freedom to design the

system. For instance, simultaneous possession of orthogonality, short support,

symmetry and high approximation order is possible in Multiwavelet system.

Tan [28] proposes a general paradigm for the analysis and application of

discrete multiwavelet transforms, particularly to image compression. Firstly,

establish the concept of an equivalent scalar (wavelet) filter bank system in which

present an equivalent and sufficient representation of a multiwavelet system of

multiplicity in terms of a set of equivalent scalar filter banks.This relationship

motivates a new measure called the good multifilter properties (GMP’s), which

define the desirable filter characteristics of the equivalent scalar filters.

Cheung [30] proposed the Integer Multiwavelet transform is based on the

box and slope scaling functions. The system is based on the multiscaling and

multiwavelts .The Integer Multiwavelet Transform (IMWT) has short support,

symmetry, high approximation order of two. It is a block transform. It can be

efficiently implemented with bit shift and addition operations. Added advantage

51

of this transform is that, while it increases the approximation order, the dynamic

range of the coefficients will not be largely amplified, an important property for

lossless coding.

Ngai-Fong Law [49] Proposed the computational complexity associated

with the over complete wavelet transform for the commonly used spline wavelet

family. by deriving general expressions for the computational complexity using

the conventional filtering implementation, Which show that the inverse transform

is significantly more costly in computation than the forward transform. To reduce

this computational complexity, It is been proposed a new spatial implementation

based on the exploitation of the correlation between the low pass and the band

pass outputs that are inherent in the over complete representation. Both

theoretical studies and experimental findings show that the proposed spatial

implementation can greatly simplify the computations associated with the inverse

transform. In particular, the complexity of the inverse transform using the

proposed implementation can be reduced to slightly less than that of the forward

transform using the conventional filtering implementation [14].

Triantafyllidis G.A. [76] proposed the transformed coefficients are

grouped into different magnitude Sets in both the methods. Each coefficient has

three parameters namely (Set, Sign, and Magnitude) in MS-VLI coding. The set

information is arithmetically coded, followed by a bit for Sign then followed by

magnitude information in bits. The magnitudes of the coefficients are grouped

into different magnitude sets according to the table 2.1.

52

Table 2.1

Definition of absolute magnitude Set variable length integer representation

Magnitude

Set

Amplitude

Intervals

Magnitude

Bits

0 [0] 0

1 [1] 0

2 [2] 0

3 [3] 0

4 [4 -5] 1

5 [6 - 7] 1

6 [8 – 11] 2

7 [12 – 15] 2

….. ……. …..

Xiaolin Wu [91] proposed low bit rate compression by a practical

approach of uniform down sampling in image space and yet making the sampling

adaptive by spatially varying, directional low-pass prefiltering. The resulting

down-sampled pre-filtered image remains a conventional square sample grid, and,

thus, it can be compressed and transmitted without any change to current image

coding standards and systems. The decoder first decompresses the low-resolution

image and then upconverts it to the original resolution in a constrained least

squares restoration process, using a 2-D piecewise autoregressive model and the

knowledge of directional low-pass prefiltering.

Suzuki [73] proposed the Image compression (coding) schemes can be

classified into two distinct categories, lossless and lossy. Lossless image coding is

used in high-end hardware for medical images, remote sensing, image archiving,

53

and satellite communications so on. Lossy image coding is used in low-end

hardware for digital camera and internet contents and so on. Although lossless

image coding provides the information integrity that maintained throughout the

entire encoding and decoding process.

Ghorbel [50] proposed the Discrete wavelet transform is a mathematical

transform that separates the data signal into fine-scale information known as

detail coefficients, and rough-scale information known as approximate

coefficients. Its major advantage is the multi-resolution representation and time-

frequency localization property for signals. DWT has the capability to encode the

finer resolution of the original time series with its hierarchical coefficients.

Esfandarani [20] proposes the low bit rate applications, such as cell phone

and wireless transmission of images, require compression schemes that could

keep acceptable levels of visual quality of the medium. In this work a multi layer

compression scheme is presented which is intended to preserve the texture details

of an image at low bit rates [78]. The first layer uses wavelet transform for

extraction of textures. Then in the second layer the strength of the contourlet

transform in preservation of textures is employed to compress the highlighted

textures of the image. The proposed method is compared with a number of low

bit rate methods and proved to be superior to these methods.

Zhang [90] proposed a novel scheme for lossy compression of an

encrypted image with flexible compression ratio. A pseudorandom permutation is

used to encrypt an original image, and the encrypted data are efficiently

compressed by discarding the excessively rough and fine information of

coefficients generated from orthogonal transform. After receiving the compressed

data, with the aid of spatial correlation in natural image, a receiver can

54

reconstruct the principal content of the original image by iteratively updating the

values of coefficients. This way, the higher the compression ratio and the

smoother the original image, the better the quality of the reconstructed image.

K Nagamani [48] proposed the wavelets offer an elegant technique for

representing the levels of details present in an image. When an image is

decomposed using wavelets, the high pass component carry less information, and

vice-versa. The possibility of elimination of the high pass components gives

higher compression ratio in the case of wavelet based image compression [11].

To achieve higher compression ratio, various coding schemes have been used.

Some of the well known coding algorithms are EZW (Embedded zero-tree

wavelet), SPIHT (Set partitioning in hierarchical tree) and EBCOT (Embedded

block coding with optimal truncation).

Negahban [13] have discussed an important issue in image compression is

the volume of pixels which will be compressed. This work presents a novel

technique in image compression with different algorithms by using the transform

of wavelet accompanied by neural network as a predictor. The details subbands in

different low levels of image wavelet decomposition are used as training data for

neural network. In addition, It predicts high level details subbands using low level

details subbands. This work consists of four novel algorithms for image

compression as well as comparing them with each other and well- known jpeg

and jpeg2000 methods.

Cohen [23] have proposed a new face image compression scheme based on

the redundant tree-based wavelet transform (RTBWT).On learning the transform

from training set containing aligned face images, and use it as a redundant

dictionary when encoded images by applying sparse coding on them. Improved

55

quality Wang Z [85] results are obtained by using a filtering-based post-

processing scheme. It have demonstrated competitive performance compared to

other methods, and managed to obtain results of high visual quality for low bit-

rates.

2.3 ENHANCED COMPRESSION ALGORITHMS

The contribution on enhancing compression techniques review works have

been discoursed in this literature.

Phooi [4] proposed a review for image compression algorithms and

presented performance analysis between various techniques in terms of memory

requirements, computational load, system complexity, coding speed, and

compression quality. Authors found that Set Partitioning In Hierarchical Tree

(SPIHT) is the most suitable image compression algorithm in lossy Image

compression due to its high compression ratio and simplicity of computations,

since wireless transmission of bits requires low memory, speed processing, low

power consumption, high compression ratios, less complex system and low

computational load.

Bhardwaj [31] discussed a new approach that enhances compression

performance compared with JPEG [2] (Joint Photographic Experts Group)

techniques and they used MSE and PSNR as the quality measures. Their

approach was based on using singular value decomposition (SVD) and block

truncation coding (BTC) with Discrete Cosine Transform (DCT) in image

compression technique. They depended on decision making parameter (x) which

is based on observation of standard deviation (STD σ) for deciding what

compression technique can be used as follows:

If σ < x use DCT

Else if σ > x use SVD Else if 35 ⩽ σ ⩽ 45 BTC

56

Maaref [47] described a study investigation on efficient adaptive

compression scheme that ensures a significant computational and energy

reduction as well as communication with minimal degradation of the image

quality. Their scheme was based on wavelet image transform and distributed

image compression by sharing the processing tasks between clusters to extend the

overall lifetime of the network.

Ayedi W [17] described robust use of DCT and Discrete Wavelet

Transform (DWT) and their capabilities in WSN. They provided practical

performance comparison between those techniques for various image resolutions

and different transmission distances with 2 scenarios. The first scenario used two

nodes only as transmitter and receiver, while, the second scenario using

intermediate nodes between sender and receiver. The comparison was in terms of

packet loss, reconstructed image quality, transmission time, execution time, and

memory usage. They concluded that DWT is better than DCT as DWT had fewer

packet losses (for Lena 32 ∗ 32 it became clear from a distance 12 m and 7 m for

Lena 64 ∗ 64), higher image quality in terms of higher PSNR quality measure,

minimal transmission time, faster execution time but large memory usage than

DCT.

Abid M [18] extended their work on previous research Ghorbel O, Jabri I,

Ayedi W [17] and made compression performance analysis for DCT and DWT

with additional important parameter which is energy consumption. They

measured battery life time and concluded that DWT is better than DCT in terms

of image quality and energy consumption.

57

2.4 BEHAVIOURS OF JPEG AND JPEG 2000

Sharif H [42] surveyed multimedia compression techniques and

multimedia transmission techniques and provided analysis for energy efficiency

when applied to resource constrained platform [37]. For image compression they

discussed three important techniques JPEG (DCT), JPEG2000 (Embedded Block

Coding with Optimized Truncation EBCOT), and SPIHT. They analyzed their

work in terms of compression efficiency, memory requirement and computational

load. They concluded that SPIHT is the best choice for energy-efficient

compression algorithms due to its ability to provide higher compression ratio with

low complexity. JPEG2000 (EBCOT) achieved higher compression ratio which

mean better quality than SPHIT. However, complexity of EBCOT tier-1 and tier-

2 operations caused intensive complex coding, higher computational load and

more energy consumption for resource constrained systems.

Sivasankar A [16] proposed a low complexity compression method to

hyperspectral images using distributed source coding (DSC) [53]. DCT was

applied to the hyperspectral images. Set-partitioning-based approach was utilized

to reorganize DCT coefficients into wavelet like tree structure. Cellular automata

(CA) for bits and bytes error correcting codes (ECC) to high through put rate. The

CA-based scheme can easily be extended for correcting more than two byte

errors. Its performance is comparable to that of the DSC scheme based on

informed quantization at low bit rate.

Frayne [56] discussed many techniques that have been proposed to

accomplish this. One of these, the S-transform, provides simultaneous time and

frequency information similar to the wavelet transform, but uses sinusoidal basis

functions to produce frequency and globally referenced phase measurements. It

has shown promise in many medical imaging applications but has high

58

computational requirements [19]. This work presents a general transform that

describes Fourier-family transforms, including the Fourier, short-time Fourier,

and S-transforms.

Zheng [40] discoursed about low-power, high-speed architecture which

performs two-dimension forward and inverse discrete wavelet transform (DWT)

for the set of filters in JPEG2000 is proposed by using a line-based and lifting

scheme. It consists of one row processor and one column processor each of which

contains four sub-filters. And the row processor which is time-multiplexed

performs in parallel with the column processor. Optimized shift-add operations

are substituted for multiplications, and edge extension is implemented by

embedded circuit. The whole architecture which is optimized in the pipeline

design way to speed up and achieve higher hardware utilization has been

demonstrated in FPGA. Two pixels per clock cycle can be encoded at 100MHz.

The architecture can be used as a compact and independent IP core for JPEG2000

VLSI implementation and various real-time image/video applications.

Chakrabarti [32] proposed an architecture that performs the forward and

inverse discrete wavelet transform (DWT) using a lifting-based scheme for the set

of seven filters proposed in JPEG2000. The architecture consists of two row

processors, two column processors, and two memory modules. Each processor

contains two adders, one multiplier, and one shifter. The precision of the

multipliers and adders has been determined using extensive simulation. Each

memory module consists of four banks in order to support the high computational

bandwidth. The architecture has been designed to generate an output every cycle

for the JPEG2000 default filters. The schedules have been generated by hand and

the corresponding timings listed.

59

Wang [87] proposed a practical approach of uniform down sampling in

image space and yet making the sampling adaptive by spatially varying,

directional low-pass prefiltering. The resulting down-sampled prefiltered image

remains a conventional square sample grid, and, thus, it can be compressed and

transmitted without any change to current image coding standards and systems.

The decoder first decompresses the low-resolution image and then upconverts it

to the original resolution in a constrained least squares restoration process, using

a 2-D piecewise autoregressive model and the knowledge of directional low-pass

prefiltering.

Buccigrossi [57] approached with the probability model for natural

images, based on empirical observation of their statistics in the wavelet transform

domain. Pairs of wavelet coefficients, corresponding to basis functions at adjacent

spatial locations, orientations, and scales, are found to be non-Gaussian in both

their marginal and joint statistical properties. Specifically, their marginal are

heavy-tailed, and although they are typically decorrelated, their magnitudes are

highly correlated. The proposed Markov model that explains these dependencies

using a linear predictor for magnitude coupled with both multiplicative and

additive uncertainties, and show that it accounts for the statistics of a wide variety

of images including photographic images, graphical images, and medical images

[69].

Min Kyung [26] describes Increasing the image size of a video sequence

aggravates the memory bandwidth problem of a video coding system. Despite

many embedded compression (EC) algorithms proposed to overcome this

problem, no lossless EC algorithm able to handle high-definition (HD) size video

sequences has been proposed thus far [60]. In this a lossless EC algorithm for HD

video sequences and related hardware architecture is proposed. The proposed

60

algorithm consists of two steps. The first is a hierarchical prediction method

based on pixel averaging and copying. The second step involves significant bit

truncation (SBT) which encodes prediction errors in a group with the same

number of bits so that the multiple prediction errors are decoded in a clock cycle.

The theoretical lower bound of the compression ratio of the SBT coding was also

derived. Experimental results have shown a 60% reduction of memory bandwidth

on average [1]. Hardware implementation results have shown that a throughput of

14.2 pixels /cycle can be achieved with 36K gates, which is sufficient to handle

HD-size video sequences in real time.

2.5 REAL-TIME APPLICATIONS

Hao [38] approached with a compound image compression algorithm for

real-time applications of computer screen image transmission. It is called shape

primitive extraction and coding (SPEC). Real-time image transmission requires

that the compression algorithm should not only achieve high compression ratio,

but also have low complexity and provide excellent visual quality [21]. SPEC

first segments a compound image into text/graphics pixels and pictorial pixels,

and then compresses the text/graphics pixels with a new lossless coding algorithm

and the pictorial pixels with the standard lossy JPEG, respectively. The

segmentation first classifies image blocks into picture and text/graphics blocks by

thresholding the number of colors of each block, then extracts shape primitives of

text/graphics from picture blocks.

Dynamic color palette that tracks recent text/graphics colors is used to

separate small shape primitives of text/graphics from pictorial pixels. Shape

primitives are also extracted from text/graphics blocks. All shape primitives from

both block types are losslessly compressed by using a combined shape-based and

palette-based coding algorithm. Then, the losslessly coded bitstream is fed into a

61

LZW coder. Experimental results show that the SPEC has very low complexity

and provides visually lossless quality while keeping competitive compression

ratios.

Guillemot [6] proposed the new transform for image processing, based on

wavelets and the lifting paradigm. The lifting steps of a one-dimensional wavelet

are applied along a local orientation defined on a quincunx sampling grid. To

maximize energy compaction, the orientation minimizing the prediction error is

chosen adaptively. A fine-grained multiscale analysis is provided by iterating the

decomposition on the low-frequency band. In the context of image compression,

the multiresolution orientation map is coded using a quad tree.

The rate allocation between the orientation map and wavelet coefficients is

jointly optimized in a rate-distortion sense. For image denoising, a Markov model

is used to extract the orientations from the noisy image. As long as the map is

sufficiently homogeneous, interesting properties of the original wavelet are

preserved such as regularity and orthogonality. Perfect reconstruction is ensured

by the reversibility of the lifting scheme. The mutual information between the

wavelet coefficients is studied and compared to the one observed with a separable

wavelet transform. The rate-distortion performance of this new transform is

evaluated for image coding using state-of-the-art subband coders. Its performance

in a denoising application is also assessed against the performance obtained with

other transforms or denoising methods.

Cavenor [58] describe the Adaptive DPCM methods using linear

prediction are described for the lossless compression of Hyperspectral (224-band)

images recorded by the airborne visible infrared Imaging Spectrometer (AVIRIS).

The methods have two stages-predictive decorrelation (which produces residuals)

and residual encoding. Good predictors are described, whose performance closely

62

approaches limits imposed by sensor noise. It is imperative that these predictors

make use of the high spectral correlations between bands. The residuals are

encoded using variable-length coding (VLC) methods, and compression is

improved by using eight codebooks whose design depends on the sensor’s noise

characteristics.

2.6 PROPERTIES OF MULTIWAVELET IN FILTERS

M.M. Al-Akaidi [67] approach is to provide a like-with-like performance

comparison between the wavelet domain and the multiwavelet domain

watermarking, under a variety of attacks. The investigation is restricted to

balanced multiwavelets. Furthermore, for Multiwavelet domain watermarking,

both wavelet-style and multiwavelet-style embedding are investigated. It was

shown that none of the investigated techniques performs best across the board.

The waveletstyle multiwavelet technique is best suited for compression attacks,

whereas scalar wavelets are superior under cropping and scaling.

The multiwavelet-style multiwavelet is far superior under low-pass

filtering. On the basis of these results, it was concluded that for attacks which are

likely to affect mid-range frequencies, the wavelets are more suitable than

multiwavelets, whereas for attacks which are likely to affect low frequencies or

high frequencies, the multiwavelets are the best choice. Furthermore, the

multiwavelets generally offer better visual quality than scalar wavelets, for the

same peak signal-to-noise ratio (PSNR). This suggests that part of the available

channel capacity remains unused, and shows once more the potential of

multiwavelets for digital watermarking.

Yu-Hing Shum [74] approached with the Prefilters are generally applied to

the discrete Multiwavelet transform (DMWT) for processing scalar signals [89].

63

To fully utilize the benefit offered by DMWT, it is important to have the prefilter

designed appropriately so as to preserve the important properties of

multiwavelets.To this end, which had recently shown that it is possible to have

the prefilter designed to be maximally decimated, yet preserve the linear phase

and orthogonal properties as well as the approximation power of multiwavelets.It

can be very difficult to find a compatible filter bank structure and in some cases,

such filter structure simply does not exist, e.g.,for Multiwavelet of multiplicity 2.

Wilkes [33] approached with the Prefiltering a given discrete signal has

been shown to be an essential and necessary step in applications using unbalanced

multiwavelets. In this they have develop two methods to obtain optimal second-

order approximation preserving prefilters for a given orthogonal multiwavelet

basis. These procedures use the prefilter construction introduced in part-I.The

first prefilter optimization scheme exploits the Taylor series expansion of the

prefilter combined with the multiwavelet. The second one is achieved by

minimizing the energy compaction ratio (ECR) of the wavelet coefficients for an

experimentally determined average input spectrum.

Heil [80] approached on Multiwavelets are a new addition to the body of

wavelet theory. Realizable as matrix-valued filterbanks leading to wavelet bases,

multiwavelets offer simultaneous orthogonality, symmetry, and short support,

which is not possible with scalar two-channel wavelet systems. After reviewing

this recently developed theory, That examine the use of multiwavelets in a

filterbank setting for discrete-time signal and image processing. Multiwavelets

differ from scalar wavelet systems in requiring two or more input streams to the

multiwavelet filterbank.

64

Zhang [27] proposes a lossy compression scheme for Bayer images is

proposed. Recently, it was found that compression-first schemes outperform the

conventional demosaicking-first schemes in terms of output image quality.

Balanced multiwavelet packet transforms effectively remove CFA image

correlation between frequency bands. Wavelet coefficients shuffling exploring

subband correlation makes it suitable for zero tree coding. Improved SPIHT

algorithm further exploits data correlation in different direction under the same

resolution using one symbol to denote three zero trees while the SPIHT algorithm

using three symbols.

2.7 MEASUREMENTS AND QUALITY METRICS

Zriakhov [79] proposed that images can be subject to lossy compression in

such a way that introduced distortions are not visible. For this purpose, two

modern visual quality metrics, MSSIM and PSNR-HVS-M, can be used [93, 94].

Their values are to be provided not less than 0.99 and 40 dB, respectively, and the

corresponding lossy compression is to be carried out. Attained compression ratio

(CR) depends upon image properties and a coder used. The proposed

methodology of lossy compression can be successfully exploited in remote

sensing and medical imaging with producing CR by several times larger than the

best lossless image compression techniques.

Ruikar [75] Proposes the Satellite Images are major resource for various

earth scientists, geologist and metrologies for better perceptive of earth's

environment and conditions. The increasing availability of satellite images has

raised the need for compression of satellite image without significant loss of

perceptual image. The Discrete Wavelet Transform (DWT) offers the optimal

results for image compression. The purpose was made by selection of wavelet by

comparing various wavelet functions like Haar, Daubechies,Coiflets, Biorthogona

65

and Discrete Meyer wavelet for satellite image compression [46]. The fine pick of

wavelet function aids in improving the quality of image. The compressed image

performance is analyzed by using picture quality measures.

Ling Wu [92] approaches with the objective of 3D image quality

assessment play a key role for the development of compression standards and

various 3D multimedia applications. The quality assessment of 3D images faces

more new challenges, such as asymmetric stereo compression, depth perception,

and virtual view synthesis, than its 2D counterparts [53]. In addition, the widely

used 2D image quality metrics (e.g., PSNR and SSIM) cannot be directly applied

to deal with these newly introduced challenges. This statement can be verified by

the low correlation between the computed objective measures and the

subjectively measured mean opinion scores (MOSs), In order to meet these newly

introduced challenges besides traditional 2D image metrics.

Ekuakille [81] used with application specific information processing

(ASIP) unit in smart cameras,which requires sophisticated image processing

algorithms for image quality improvement and extraction of relevant features for

image understanding and machine vision. The improvement in performance as

well as robustness can be achieved by intelligent moderation of the parameters

both at algorithm (image resolution, contrast, compression, and so on) as well as

hardware levels (camera orientation, field of view, and so on).

Bandyopadhyay [68] proposed a histogram based image compression

technique is proposed based on multi-level image threshold. The gray scale of the

image is divided into crisp group of probabilistic partition. Shannon’s Entropy is

used to measure the randomness of the crisp grouping. The entropy function is

maximized using a popular metaheuristic named Differential Evolution to reduce

66

the computational time and standard deviation of optimized objective value.

Some images from popular image database of UC Berkeley and CMU are used as

benchmark images. Important image quality metrics- PSNR, WPSNR and storage

size of the compressed image file are used for comparison and testing.

2.8 THE KNOWLEDGE GAP IDENTIFIED IN THE EARLIER

INVESTIGATIONS

The literature survey presented here reveals the following knowledge gap

in field of Image processing. The Lossless compressions yield a low compression

with acceptable visual quality. In the Lossy compression, it results in a higher

compression with low visual quality. Even though there are many compression

techniques, such as RWT, JPEG, CREW and JPEG2000 etc., that utilize

Multiwavelet transform, none of them produce high quality output. The new

technique - Integer Multiwavelet transform, has unique capability in producing

reasonably high quality image and achieving higher compression ratios.

However, not much work has been carried out using this IMWT technique. This

was the gap identified with the existing techniques and thus this thesis work may

be further utilized for providing higher quality output.

The wavelet transform is found good fit for typical natural images that

have an exponentially decaying spectral density with a mixture of strong

stationary low frequency components. The transform coding is a form of block

coding done in the transform domain. This transform coding is achieved by

filtering and by eliminating some of the high frequency coefficients.

2.9 RESEARCH MOTIVATION

The key for successful compression scheme is retaining only the necessary

information to understand it. It must be differentiated between data and

67

information. In digital images, data refers to the pixel gray level values that

correspond to the brightness of a pixel at a point in space. The data are used to

convey information much like the way the alphabet is used to convey information

via words. Information is an interpretation of the data in a meaningful way, which

also can be application specific. The compression algorithms are developed by

taking advantage of the redundancy that is inherent in image data. There are four

primary types of redundancy that can be found in images like Coding, Interpixel,

Interband and Psychovisual redundancy. The coding redundancy occurs when the

data used to represent the image is not utilized in an optimal manner. The

Interpixel occurs because adjacent pixels tend to be highly correlated in most

images. The brightness levels do not change rapidly, but change gradually. The

Interband redundancy occurs in color images due to the correlation between

bands within an image if we extract the red, green and blue bands they look

similar. In Psychovisual redundancy some information is more important to the

human visual system than the other types of information.

The standard images used for testing by this lossy compression technique

provide a high quality of results on reconstruction [54]. The toughness was

storing the data in physical device leads to more problem, sending the data by

GPRS also leads to cost efficiency, sending through MMS belongs to upper

bound limit and shot to shot time latency provides customer satisfactory. The

PSNR for artificial images were identified high-quality by using proposed lossy

method. The performance of IMWT for images with high frequencies was

outstanding. The subjective quality of the proposed lossy reconstructed image by

retaining the LL subband information alone is equal to that of existing lossy

reconstruction. This proves the performance of Multiwavelet that allows more

design freedom.

68

The applications requiring high speed connection such as high definition

television, real-time teleconferencing and transmission of multiband high

resolution satellite images, make us to think that image compression is not only

desirable but necessary. This has motivated many researchers to work for a better

compression technique than the available ones. It has been identified that more

work is needed in getting better compression using IMWT, which is ideal for

processing the images. Combining IMWT and RLE is another area where more

work could be done on efficient compression technique. This idea leads to the

current research work on IMWT and RLE for image compression.

2.10 AIM

The aim of this research work is to reduce the image file size as much as

possible using lossy compression with higher compression ratio.

2.11 OBJECTIVE OF THE RESEARCH WORK

The objective of research work is to make use of memory space effectively

such that to store large amount of valuable data, so that all the advantages is of

small file size (Memory storage space, transmission time, transmission cost, etc.)

can be effectively utilized. On keeping the resolution and the visual quality of the

reconstructed image as close to the original image as possible. The steps followed

to obtain the maximum storage space as listed below:

• To perform pre-filter in the original input image and forward Integer

Multiwavelet transform for Pre-analysis along rows and columns.

• To apply magnitude set and run length encoding for decomposition across

transformed values. and the Inverse Integer Multiwavelet transform for

image reconstruction.

• To analyze the final output as resultant of compressed and reconstructed

(NxN) gray image.

69

CHAPTER 3

IMPLEMENTATION OF IMWT

3.1 INTRODUCTION

As mentioned earlier, the storage constraints and bandwidth limitations in

communication systems have necessitated the search for efficient image

compression techniques. For real-time video and multimedia applications, where a

sensible approximation to the original signal can be tolerated, lossy compression is

used. In the recent past, wavelet-based lossy image compression schemes have

gained wide acceptance. The inherent characteristics of the wavelet transform

provide compression results that outperform other techniques such as the discrete

cosine transform (DCT). Consequently, the JPEG2000 compression standard has

adopted a wavelet approach to image compression [8], [95]. The literature

provides some information about wavelets and Multiwavelet with different

properties. The inadequate information motivates the search for a set of desirable

properties suited to image compression with wavelets and Multiwavelet [69]. At

present, scalar wavelets are well understood in the context of image compression;

however more research is required in the area of Multiwavelet. The properties of a

new class of Multiwavelet called Integer Multiwavelet and their usefulness in

image compression have been investigated studied in this chapter. The literature

indicates that objective quality metrics like peak signal-to-noise ratio (PSNR) do

not correlate with perceived image quality at high compression ratios [23]. This

motivates the need for incorporating characteristics of the human visual system

(HVS) into compression schemes. This chapter analyses a recent HVS-based

transform technique where perceptually important frequencies are preserved in the

compressed image for enhanced subjective quality.

70

3.2 OVERVIEW

The principle of image compression and decompression using IMWT is

explained in this chapter.

Figure 3.1The Compression

Figure 3.2The Reconstruction

The compression consists of a forward IMWT preprocessing stage and

encoding stage shown in figure 3.1. Whereas, the decompression or

reconstruction consists of a decoding stage followed by an inverse IMWT post

processing stage as shown in figure 3.2. Before encoding, preprocessing is

performed to position the image for the encoding process and the processing

consists of number of operations that are application specific. Once the

compressed file has been decoded, post-processing can be performed to eradicate

some of the potentially undesirable artifacts obtained by the compression process.

Original Image (NxN)

IMWT Pre-

processing

Magnitude set & run

length encoding

Compressed Image

Compressed

Image

Magnitude set & run

length decoding

IIMWT post

processing

Decompressed

Image

71

3.3 INTEGER PREFILTER

In a multiwavelet system that uses matrix valued filters input

sequence{ },nx cannot be directly processed by the multifilters. It is necessary to

obtain a vector input sequence with the two vector element c n)0(

,1 and c n)0(

,2 from the

input sequence { },nx through a pre-filter )(zQ as shown in figure 3.3.The

equivalent non polyphase representation of the nontruncated multiwavelet system

is shown in figure 3.4.

Figure 3.3 Multifilter bank implementation of 1st level Multiwavelet decomposition pre-filtering as polyphase representation

Figure 3.4 Multiwavelet decomposition pre-filtering as equivalent nonpolyphase representation

Q(z)

2

2

2

)1(

,2

−

nc Xn

G(z)

H(z)

2

)1(

,1

−

nc

)1(

1

−

nd )1(

,2−nd

)0(

,1 nc

)0(

,2 nc

2

2

2

2

(z)H~ 2

(z)G~ 1

(z)G~ 2

(z)H~1

2

)1(

,2

−

nc Xn

2 )1(

,1

−

nc

)1(

,1

−

nd )1(

,2

−

nd

72

It shows the combined prefilter and multifilter metrics, )()( zQzH and

)()( zQzG the polyphase matrices of another two set of filters, )(ˆ zH l and

.2,1)(ˆ =lzGl Using lowpass & bandpass criteria on these two equivalent sets of Filters, good prefilters should satisfy the conditions.

0)1(ˆ =−lH and 2,1,0)(ˆ == lzGl (3.1)

Such that the first level decomposition separates the input { },nx into low

frequency approximation, and { }{ }cc nn)1(

,,2)1(

,1 , −− and the high frequency details,

{ }{ }dd nn)1(

,,2)1(

,1 , −−

as the traditional wavelet decomposition does. By limiting the

nonpolyphase equivalent filter length of ,2)( tozQ a reasonable choice for such a

short supported multiwavelet system, the combined prefilter and multifilter

polyphase matrixes are expressed as

++−

+= −−

−

dcba

Xzzz

ZQzH )1(21)1(

21

01)()( 11

1

, (3.2)

−

+−=−

−−

dcba

Xz

zzZQzG)1(0

)1(21)1(

21

)()(1

11

(3.3)

The equivalent nonpolyphase matrixes can be expressed as,

2,1),()()()()( 2121 =+++= lzdHzbHzcHzaHzH lllll (3.4)

2,1),()()()()( 2121 =+++= lzdGzbGzcGzaGzG lllll (3.5)

Where lmH and 2,12,1 == mandforlGlm are the elements of the matrixes.

The filter bank implementations of the Multiwavelet transform with

multiplicity two, need four filters. The pyramidal algorithm then needs four filters

73

followed by a downsampler of factor four. For this structure the loss of

information is high. Hence the downsampling process is split into two stages by

using prefilter. This is better in terms of loss of information and complexity of

design. The prefilter produces vector inputs that are needed for the filters. The

decomposition of the image by Multiwavelet transform uses pre-filter, the

reconstruction uses the post filter to produce the image. Initially, the image is pre-

filtered along the row direction, and then processed by the Multiwavelet filters in

the same direction. Then the same process is carried out in the column direction

for the resultant image. The final result produces the sixteen subbands.

The decomposition of a (NxN) image by Multiwavelet transform is

depicted in the figure 3.5. First the image is pre-filtered along the row direction,

and then processed by the Multiwavelet filters in the same direction. Then the

same processing is done in the column direction for the resultant image. The final

result produces sixteen subbands.

Figure 3.5 2-D Process Flow of Multiwavelet decomposition of an image

74

The following figure 3.6 shows the results of the IMWT decomposition for Lena,

Couple and Man.

Figure 3.6 I-level IMWT Decomposition of Lena, Couple and Man

The Integer Multiwavelet Transform was first implemented in Matlab. The

RLE algorithm was applied to various images and the MSE and PSNR values

were obtained. The sixteen subbands were also obtained with Matlab. The

reconstruction of the image from all the sixteen subbands corresponds to the

Lossy reconstruction. The IMWT was tested for various standard images. The I-

Level IMWT subband Decomposition for 512 x 512 images is expanded and

shown in figure 4.12. The First (I-level) Integer Multiwavelet Transform (IMWT)

decomposition of the images has sixteen subbands with the L1L1 subband in the

Top left corner.

3.4 TRANSFORMATION TO OBTAIN LOW BITS

The figure 3.7 represents an example of transformation to perform the

compression on the input image and obtain the compressed output with low bits

with the help of sign plane and magnitude set as the resultant of magnitude bit

map plane.

75

In this example, as 4-bytes of information is assumed to be transferred.

This contains signed and unsigned values, which undergo sign plane process in

order to eliminate the signed values. The resultant will be in the form of zeros and

ones (0, 1) as a single binary bit. Followed by this binary coding, magnitude set

has been performed in order to obtain the unsigned values. Finally, after the

Magnitude Bit map plane was done by referring the MS-VLI table 3.1. It is

shown that just 17 bits is sufficient to send 4-bytes of uncompressed information,

for which the transmit time will nearly be halved. So the transmission bandwidth

can be effectively utilized.

Figure 3.7 Low bit required for the Information to transfer

76

3.5 MS-VLI REPRESENTATION

The table 3.1 represents the Magnitude set variable length integer

representation with amplitude interval and magnitude bits. Since the gray scale

images has been considered as input image, it has the values between 0 and 255,

and this value act as the amplitude intervals.

Table 3.1

Magnitude Set Variable Length Integer Representation

Magnitude

Set

Amplitude

Interval

Magnitude

Bits

0 0 0

1 1 0

2 2 0

3 3 0

4 4-5 1

5 6-7 1

6 8-11 2

7 12-15 2

8 16-19 2

9 20-23 2

10 24-31 3

11 32-39 3

12 40-47 3

13 48-55 3

14 56-71 4

15 72-87 4

77

16 88-103 4

17 104-119 4

18 120-151 5

19 152-183 5

20 184-215 5

21 216-247 5

The table 3.2 represents amplitude intervals example for the number of bit at

respective position from (1and 0) for the values (8 to 11) and also table 3.3 for the

(24 to 31) as bit position from (2 – 1 – 0).

Table 3.2

Amplitude Intervals example for (8 to 11)

Amplitude

Intervals

No. of Bits at respective

positions Magnitude

bit

required

Total bits

required

( 8- 11 ) Bit

position-1

Bit

position-0

8 0 0 2

2 9 0 1 2

10 1 0 2

11 1 1 2

78

Table 3.3

Amplitude Intervals example for (24 to 31)

Amplitude

Intervals

No. of Bits at respective position Magnitude

bit

required

Total bits

required

( 24- 31 )

Bit

position-2

Bit

position-1

Bit

position-0

24 0 0 0 3

3

25 0 0 1 3

26 0 1 0 3

27 0 1 1 3

28 1 0 0 3

29 1 0 1 3

30 1 1 0 3

31 1 1 1 3

The Visual Quality of the Standard images of Lena, Baboon and Barbara

are tabulated with the Quality factor known SSIM and DISSIM for window size

(8 x 8) as shown in Table 3.4 been calculated across all the rows and columns

with help of RLE algorithm as resultant of transformation and also for the

Window size (16 x 16) tabulated only using SSIM in table 3.5.

Table 3.4

SSIM and DSSIM Results

Windows (8 x 8) SSIM DSSIM

Lena 0.9871 0.0064

Baboon 0.9545 0.0227

Barbara 0.9997 0.00015

79

Table 3.5

SSIM on Different window size

Windows Size SSIM ( 8 x 8 ) SSIM ( 16 x 16 )

Lena 0.9871 0.9824

Baboon 0.9545 0.9588

Barbara 0.9997 0.9996

The window size (32 x 32) and (64x64) for Lena, baboon and

Barbara are tabulated only using SSIM in table 3.6.

Table 3.6

SSIM on Different window size

Windows Size SSIM ( 32x32 ) SSIM ( 64 x 64 )

Lena 0.9791 0.9961

Baboon 0.9374 0.9408

Barbara 0.9996 0.9996

The table 3.7 represents the SSIM and DSSIM values for the standard

images like Lena, Baboon and Barbara with the window size of (256x256).

Table 3.7

SSIM and DSSIM for (512x512) Image

Windows Size

(256x256) SSIM DSSIM

Lena 0.9976 0.0012

Baboon 0.9783 0.0109

Barbara 0.9977 0.0011

80

3.6 LOW BIT RATE USING IMWT COMPRESSION ALGORITHM

Step1: Assume original (NxN) gray Image as Input. Apply using pre-filter and

Forward Integer Multiwavelet Transform for Pre-analysis along rows.

Pre-filter row : [ ]2/) P (PP 12i2iir1,(0)

++= (3.6)

2i12iir2, PPP(0) −= + , (3.7)

Step2: Perform the Integer Multiwavelet Transform for Pre-analysis along

columns.

Pre-filter column: [ ]2/) P (PP 12i2iic1,(0)

++= (3.8)

2i12iic2, PPP(0) −= + , (3.9)

Step3: Apply Magnitude Set and Run length coding (Encoding) for

decomposition across transformed values.

Step4: Obtain the Encoded values and Store the resultant value and find the

compression ratio.

Step5: Obtain the Encoded value by decoding process and get the transformed

image.

Step6: On applying the Inverse Integer Multiwavelet transform for

reconstructing the image.

Step7: The final output is resultant of reconstructed (NxN) gray image.

3.7 PSEUDO CODE FOR SSIM AND DISSM

The Pseudo code represents the calibration of structural similarity (SSIM)

and dissimilarity (DSSIM) for the windowing technique. %The procedure to perform SSIM Calaculation

% On taking the consideration of 8 x8 window from both

reconstruction and image as input

windowsize =8;

81

sumx=0;

sumy=0;

for i= 1:windowsize

for j= 1:windowsize

sumx = sumx+imagein(i,j);

sumy = sumy+recon(i,j);

%The Average or mean of input and reconstruction to

obtained

%mux - input image average

%muy - reconstructed image average

mux = sumx/(windowsize^2);

muy = sumy/(windowsize^2);

sumsqx =0;

sumsqy =0;

for i= 1:windowsize

for j= 1:windowsize

sumsqx = sumsqx+((imagein(i,j) - mux)^2);

sumsqy = sumsqy+((recon(i,j) - muy)^2) ;

end;

end;

%The covariance between input and the reconstructed

image are obtained as

covariance = sumsqxy/(windowsize^2);

ssim = (((2*mux*muy)+const1)* ((2*covariance)

+const2))/((mux*mux)+(muy*muy)+const1)*(sigmax+sigmay+

const2))

%The structural dissimilarity can be obtained

dssim = (1-ssim)/2;

82

3.8 PERFORMANCE EVALUATION

This thesis presents the performance evaluation of orthogonal, Integer

Multiwavelet in image compression. Our analysis suggests those Multiwavelet

characteristics that are important to image compression. Our results are based on

a large database of standard test images. The following are the contributions of

this thesis.

A comprehensive analysis of the effect of the Multiwavelet filter bank

properties on image compression performance.

The Modification of the Integer Multiwavelet decomposition scheme to

obtain low bit rates.

The Subjective performance results of Integer Multiwavelet with the

existing compression techniques results were obtained.

3.9 SUMMARY

In this chapter a new Multiwavelet based integer block transform is on

simple Multiwavelet system. This transformation can be efficiently

implemented with bit shift and addition operations. Another advantage of

this transform is that, while it increases the approximation order, the

dynamic range of the coefficients will not be largely amplified. The

performance of the Integer Multiwavelet Transform for compression of images

was analyzed for various window sizes. It was found that the IMWT can be

used for compression transform techniques in wireless technology. As we see

the SSIM values are close to 1 which indicates structural similarity is good with

the Integer Multiwavelet Transform. So the investigations done based on the

related resultant, the mathematical calibrations were done like identifying the

PSNR and MSE values from some of the literature that is been identified for low

bit rates that plays a role with wavelet and Multiwavelet function.

83

CHAPTER 4

SIMULATION RESULTS AND ANALYSIS

4.1 LOSSLESS COMPRESSION USING IMWT

The performance of the IMWT for lossless compression of images with

Magnitude set coding has been obtained. The Transform coefficients are coded

with Magnitude set coding and run length Encoding techniques. The Simulation

has been done using Matlab on various images and the MSE and PSNR values

have been obtained.

4.1.1 Procedure to obtain Lossless Compression Using IMWT Algorithm

Step 1: Obtain the total number of Pixels for the Original Input (NxN) grey

Image.

Step 2: Identify the total number of bits required before Compression by

(NxN) x 8-bits.

Step 3: On using IMWT transform identify the number of Sign bits.

Step 4: To calculate the Number of bits in Sign Plane encoded in RLE which

represents the bytes.

Step 5: To Identify the Number of bits for magnitude alone.

Step 6: To Calculate Total bits Sum the equivalents of (Number of Sign Bits

obtained + Number of bits in Sign Plane encoded in RLE obtained +

obtained Number of bits for magnitude).

Step 7: Calculate the Compression Ratio by total bits divided by (NxN) size of

the image before compression.

84

4.1.2 Results of Reconstructed Images

The Matlab code takes input from the BMP (Bitmap) file and the

reconstructed image is stored in the bmp format. The sixteen subbands were also

obtained with Matlab. The reconstruction of the image from all the sixteen

subbands corresponds to the Lossless reconstruction [9]. Reconstruction with

only four of the LL subbands corresponds to Lossy reconstruction (LL subband

alone). The Reconstructed images are shown in figure 4.1 for few standard

images like Lena, Boat, Baboon, Barbara, pepper, couple and tank (512 x 512)

size etc. Table 4.1 gives the PSNR and MSE values in dB for reconstructed

selective test images. The Table 4.2 gives the Bit rate for lossless compression for

test images of size 512 x512. The table 4.3 compares the results of lossless

reconstruction with the results of the reconstruction using LL subband alone for

the standard 128x128 images.

Table 4.1

PSNR and MSE values in dB for Reconstructed Images

Image 512x 512

Pixels

Lossless Reconstruction

MSE PSNR

Lena 7.2734 40.8566

Boat 7.3104 40.8196

Baboon 17.8008 30.3292

Barbara 14.0855 34.0445

85

Table 4.2 The Bit Rate for Lossless Compression

Image

512x 512

Lossless Reconstruction

MS-VLI

MS-VLI with

Lena 6.3046 2.1008

Boat 6.3736 2.1593

Baboon 7.2361 3.1018

Barbara 6.5434 2.3512

Pepper 6.4398 2.2584

Aerial 6.7404 2.5739

Couple 6.5185 2.3457

Tank 6.5844 2.4086

Table 4.3

Lossless Reconstruction and Reconstruction on LL subband

Image

128 x 128 Pixels

Lossless

Reconstruction

Reconstruction with

LL subband alone

MSE PSNR MSE PSNR

Lena 4.105425 44.025379 17.205691 30.925112

Boat 4.348370 43.782433 19.031072 29.099731

Baboon 4.909476 43.221329 22.865621 25.264982

Aerial 5.39241 43.011562 14.039796 34.091007

Chart 1.710970 46.419834 13.167216 34.963589

Chemical 4.605789 43.525013 12.223023 35.907780

Couple 4.369938 43.760864 10.172509 37.958294

Moon 4.141120 43.989685 8.327264 39.803539

Tank 4.406320 43.724483 7.888510 40.242294

86

Figure 4.1 Reconstructed images after I-level IMWT for (512x512)

87

The reconstructed images were of good quality, where the lossless

compression techniques were used on standard images [9]. The MS-VLI without

RLE and With RLE for artificial images were good. The performance of IMWT

for images with high frequencies was good. The subjective quality of the

reconstructed image by retaining the LL subband information alone is equal to

that of Lossless reconstruction. This proves the performance of Multiwavelet that

allows more design freedom. The figure 4.2 represents the PSNR and MSE values

for the standard images of (512 x 512) size like Lena, Boat, Baboon and Barbara

using the lossless reconstruction techniques.

Figure 4.2 PSNR and MSE values on Lossless Reconstruction

4.1.3 Summary of Performance for Lossless compression

A sample calculation for the compressed output by the proposed scheme

(MS-VLI without RLE) for Lena (128 x128), (256*256) and (512 * 512) image is

given below:

88

MS-VLI without RLE for 128x128 image (Lena)

• Total Pixels before compression = (128 X 128) =16384. • The total number of bits = (16384 x 8) =131072. • The total Number of Sign Bits alone = 21816. • The total Number of bits in Sign Plane encoded in RLE alone =7479. • The total Number of bits for magnitude = (16384 x 5) = 81920. • The summed Total bits = (21816 + 7479 + 81920) = 111215. • Obtained bpp (bits per pixels) for the image = 111215 / 16384 = 6.7880. • The Compression ratio = 1.17.

MS-VLI without RLE for 256x256 image (Lena)

• Total Pixels = 65536. • Total Bits before compression = 524288. • Number of Sign Bits = 59828. • Number of bits in Sign Plane encoded in RLE = 28555. • Number of bits for magnitude = 327680. • Total bits = 416063. • Obtained bpp (bits per pixel) for the image = 6.3486bpp. • Compression ratio = 1.26.

MS-VLI without RLE for 512x512 (Lena)

• Total Pixels = 262144. • Total Bits before compression = 2097152. • Number of Sign Bits = 225667. • Number of bits in Sign Plane encoded in RLE = 116333. • Number of bits for magnitude = 1310720. • Total bits = 1652720. • Obtained bpp (bits per pixel) for the image = 6.3046bpp. • Compression ratio =1.26.

89

The reduction to fewer bits by this method is due to the omission of the sign bits

of L1L1 subband and the Run Length Encoding of the sign bits using bit planes.

This small reduction can prove useful for progressive transmission of images

where bandwidth is limited and satellite applications.

4.1.4 Analysis

The performance of the Integer Multiwavelet Transform for the Lossless

compression of images for (128 x 128),(256x256) and (512 x 512) has been

studied. It was found that the IMWT can be used for Lossless compression

techniques. The Subjective output quality of the image using Lossless

reconstructed was almost the same as that of the Input original image (N x N).

The reduction of 4.1 to 4.2 bits per pixels from the tested standard images is due

to the omission of the sign bits of L1L1 subband and the run length encoding of

the sign bits using bit planes. This small reduction can prove useful for

progressive transmission of images where bandwidth is limited such as satellite

applications.

4.2 LOSSY COMPRESSION USING IMWT The Integer Multiwavelet transform was first implemented in Matlab. The

algorithm was applied to various images and the MSE and PSNR values were

obtained. The Matlab takes input from the BMP (Bitmap) file and the

reconstructed image is stored in the bmp format. The sixteen subbands were also

obtained from Matlab. The reconstruction of the image for all the sixteen

subbands corresponds to the Lossy reconstruction. Reconstruction from four of

the LL subbands alone corresponds to lossy reconstruction. Due to the memory

limitations, the test images size were restricted to (128 x128), (256 x 256), (512 x

512) .

90

In this work, Integer Multiwavelet Transform (IMWT) algorithm for lossy

compression has been done for three different images - Standard Lena, Satellite

urban and Satellite rural. The IMWT shows high performance with reconstruction

of the images. The transform coefficients are coded using the Magnitude set

coding and run length encoding techniques. The sign information of the

coefficients is coded as bit plane with zero thresholds. The Peak Signal to Noise

Ratios (PSNR) and Mean Square Error (MSE) obtained for standard images using

the proposed IMWT lossy compression scheme. The effectiveness of the lossy

compression method has been evaluated by estimating PSNR and MSE for

various 256x256 Gray images. The results confirm that the proposed scheme is

better suited for Standard Lena, Satellite rural and urban images than the existing

SPIHT (Set Partitioning in Hierarchical Trees) lossy algorithm. The simulations

were done in Matlab.

4.2.1 Procedure to obtain Lossy Compression Using IMWT Algorithm

Step 1: Obtain the total number of Pixels for the Original Input (NxN) grey Image.

Step 2: Identify the total number of bits required before Compression by (NxN) x 8-bits.

Step 3: On using IMWT transform identify the number of Sign bits.

Step 4: To calculate the Number of bits in Sign Plane encoded in RLE which represents the bytes.

Step 5: To Identify the Number of bits for magnitude with RLE.

Step 6: To Calculate Total bits Sum the equivalents of (Number of Sign Bits obtained + Number of bits in Sign Plane encoded in RLE obtained + obtained Number of bits for magnitude with RLE).

Step 7: Calculate the Compression Ratio by total bits divided by (NxN) size of the image before compression.

91

4.2.2 Results of Reconstructed Images

The table 4.4 shows the results of existing SPIHT algorithm based lossy

compression method for Standard Lena, Satellite urban and Satellite rural images

[15] in figure 4.4.The Proposed IMWT algorithm based lossy compression

performance is better than the existing SPIHT algorithm based lossy compression

method as shown in table 4.4. It must be pointed out that unlike the existing

SPIHT lossy method, the proposed IMWT lossy method is simpler and has does

not exploit the pixel correlation among the neighbor blocks. Thus Integer

Multiwavelet transform is a promising technique for the lossy compression.

Table 4.4

Comparison of PSNR and Compression ratio for Existing SPIHT and

Proposed IMWT based Lossy Reconstruction

.

Image

256x256

SPIHT IMWT

PSNR (dB)

Compression Ratio(CR)

PSNR (dB)

Compression Ratio(CR)

Standard Lena 35.81 8 37.12 8

Satellite urban 19.00 8 20.39 8

Satellite rural 12.60 8 14.77 8

92

Standard Lena Image (256x256)

Existing SPIHT algorithm based Lossy Reconstructed (PSNR is 35.81dB)

Proposed IMWT algorithm based Lossy Reconstructed (PSNR is 37.12dB)

Reconstructed with LL-Sub band alone

Lossy Distortion output

Figure 4.3 Comparison of Existing SPHIT and proposed IMWT Lossy

Reconstruction with Distortion of Standard Lena

93

Satellite Rural Image (256 x 256)

Existing SPIHT algorithm based Lossy Reconstructed (PSNR in dB is 12.60)

Proposed IMWT algorithm based Lossy Reconstructed (PSNR is 14.77dB)

Reconstructed with LL-Sub band alone

Lossy Distortion output

Figure 4.4 Comparison of Existing SPHIT and proposed IMWT Lossy

Reconstruction with Distortion of Satellite Rural

94

On considering the quality factor, the proposed lossy shows good quality

with low-distortion compared to existing lossy method. The difference between

the Standard Lena image and the proposed lossy reconstructed output has been

shown in figure 4.3. Similarly for the Satellite Rural figure 4.4.

The standard images used for testing by this lossy compression technique

provide a high quality of results on reconstruction. The PSNR for Satellite images

were of high-quality by using Lossy method. The performance of IMWT for

images with high frequencies was outstanding.. The subjective quality of the

reconstructed image by retaining the LL subband information alone (comprising

of L1L1, L1L2, L2L1, L2L2 subbands) is almost equal to that of Lossy

reconstruction. This proves the performance of Multiwavelet that allows more

intend choice. The figure 4.5 shows the existing SPIHT and Proposed IMWT

algorithm based Lossy methods for Standard Lena, Satellite Urban and Satellite

Rural Image and also the results calculated using PSNR.

Figure 4.5 Existing SPIHT and Proposed IMWT based Lossy method

95

It has been shown that the IMWT process alone reduces the file size by

16% for Lena. With the additional RLE process, the file size is further reduced to

83KB from the original size of 262KB. So it has been highlighted with the huge

reduction of 179KB as shown in the table 4.5.

Table 4.5

Reduced file size on Compression without RLE and with RLE

Image (512X512)

Uncompressed file (KB)

Compression without RLE

(KB)

Compression with RLE (KB)

Lena 262 220 83

Pepper 262 225 85

Tank 262 229 85

Aerial 262 234 86

Barbare 262 227 84

Baboon 262 249 88

Boat 262 223 83

Couple 262 227 85

The IMWT was tested for various standard images. The first (I-level)

IMWT subband decomposition for 512 x 512 images is expanded and shown in

below figure 4.6.The subjective quality of the reconstructed images with LL

subband alone for Lena, satellite rural and urban images are good. The quality of

the reconstructed images was good for other standard images used for testing the

compression techniques.

96

Figure 4.6 I level IMWT decomposition of Lena 512 x 512 Image

The MSE and PSNR for artificial images were good. The performance of

IMWT on images with high frequencies was good. The subjective quality of the

reconstructed image by retaining the LL subband information alone is equal to

that of lossless reconstruction. So the performance of Multiwavelet that allows

more intend choices as shown in figure 4.7 to obtain minimum distortion.

Original Image (512x512)

IMWT Lossy Reconstructed

97

IMWT Reconstructed with LL Band alone

IMWT Lossy Distortion Output

Figure 4.7 Original and reconstructed with LL band alone

Table 4.6 shows the required bits per pixels (bpp) for the proposed lossy

compression. Reduced bits were obtained for the proposed lossy method for

standard test images of Lena, Baboon and Barbara 512x512. Also from the table,

it can be identified that a maximum of 4 to 6 bit per pixels required for existing

lossy method. However, for the proposed IMWT lossy method, only a maximum

of 2 or 3-bpp is required. That is, the proposed lossy compression requires lower

(bpp) compared to existing lossy compression.

Table 4.6

Required bits per pixels for existing and proposed Lossy compression

Bits per

pixels

(Bpp)

Existing Lossy Compression Proposed Lossy

Compression

AIC JPEG JPEG2000 IMWT

Lena 4.5 4.7 4.3 2.0

Baboon 6.7 6.4 6.0 3.0

Barbara 5.0 5.1 4.6 2.0

98

The figure 4.8 shows comparison of the Bpp for the existing AIC, JPEG,

JPEG2000 [12], and the proposed IMWT algorithm based lossy compression for

Standard Lena, Barbara and Baboon images.

Figure 4.8 Bpp for the Existing and the Proposed Lossy compression

Table 4.7

PSNR values in existing and proposed reconstructed images

Image

512x 512

Pixels

Existing Lossy Compression

(PSNR) dB

Proposed Lossy

Compression

AIC JPEG JPEG2000 IMWT

Lena 46.81 54.08 61.83 40.85

Baboon 45.9 54.02 62.13 30.32

Barbara 46.72 54.09 61.76 34.04

99

Table 4.7 and figure 4.9 give the PSNR values of the reconstructed

standard test images for the existing and the proposed lossy techniques. The

proposed Lossy reconstruction is done only with LL-Subbands on the Lena,

Baboon and Barbara 512 x512 images, which provide minimum PSNR.

Figure 4.9 Existing and Proposed Lossy compression with PSNR

4.2.3 Summary of performance for Lossy compression

A sample calculation for the lossy compressed output by the proposed

scheme (MS-VLI with RLE) for Lena (128 x128), (256x256) and (512x512)

image is given below:

MS – VLI with RLE for 128x128 image (Lena)

• Total Pixels before compression = (128 X 128) =16384.

• The total number of bits = (16384 x 8) =131072.

• The total Number of Sign Bits alone = 21816.

• The total Number of bits in Sign Plane encoded in RLE alone =7479.

100

• The total Number of bits for magnitude using RLE alone =14213.

• The summed Total bits = (21816 + 7479 + 14213) = 43508.

• Obtained bpp (bits per pixels) for the image = 43508 / 16384 = 2.6555.

• The Compression ratio =3.01.

MS – VLI with RLE for 256x256 image (Lena)

• Total Pixels = 65536

• Total Bits before compression = 524288.

• Number of Sign Bits = 59828.

• Number of bits in Sign Plane encoded in RLE = 28555.

• Number of bits for magnitude using RLE = 52398.

• Total bits = 140781.

• Obtained bpp (Bits per pixel) for the image = 2.1481bpp.

• Compression ratio = 3.72.

MS – VLI with RLE for 512x512 image (Lena)

• Total Pixels = 262144.

• Total Bits before compression = 2097152.

• Number of Sign Bits = 225667.

• Number of bits in Sign Plane encoded in RLE = 116333.

• Number of bits for magnitude using RLE =208724.

• Total bits = 550724.

• Obtained bpp (bits per pixel) for the image = 2.1008bpp.

• Compression ratio = 3.80.

101

4.2.4 Analysis

The reduction of bits in the lossy method is due to the omission of the sign

bits of L1L1 subband and the Run Length Encoding of the sign bits using bit

planes. In information theory and coding theory with applications in computer

science and telecommunication, error detection and correction or error control

are techniques that enable reliable delivery of digital data over unreliable

communication channels. Many communication channels are subject to channel

noise, and thus errors may be introduced during transmission from the source to

a receiver. Error detection techniques allow detecting such errors, while error

correction enables reconstruction of the original data in many cases. This small

reduction can prove useful for progressive transmission of images where

bandwidth is limited in wireless technology in implementing robust error

detection and correction methodologies. The results are compared with JPEG

and JPEG2000 [7]. The JPEG and JPEG2000 have been the most widely

accepted compression engines with the advantage of having able to offer higher

compression ratios for lossy compression [6]. Hence we have taken those as

benchmarks and compared the same with the IMWT as compared to the DCT

which is used by JPEG standard [16] and [4]. The table 4.6 shows the results of

reduced bit per pixels for existing and proposed lossy IMWT compression

images compared with AIC, JPEG and JPEG2000 [20] and [52]. The IMWT

produces good results even with artificial images and images with more high

frequency content like satellite urban and rural images etc.

102

4.3 COMPARISION OF EXISTING LOSSLESS WITH PROPOSED

LOSSY COMPRESSION TECHNIQUES

The reason behind the comparing the results with Lossless and Lossy is to

show that the proposed lossy is almost equal to the existing lossless compression

techniques. Simulation results were obtained for existing lossless to propose lossy

with some of the standard test images like Lena, Barbara and Baboon etc. The

table 4.8 shows the results of lossless compression [3] of three 8-bit 512 x 512

images. The lossy compression performance of IMWT is close to that of the

lossless IMWT and JPEG (LJPEG). As expected by higher desertion moment as

compared with lossless method. The increasing energy optimization competence

of lossy IMWT and thus resultant in better compression performance than that of

lossless IMWT .The table 4.8 also include the compression results for lossless

based CREW, LJPEG and IMWT [3] which are among the best lossless image

coding schemes. It must be pointed out that unlike lossless LJPEG and IMWT

schemes the proposed lossy based IMWT being the simplest Integer Multiwavelet

transformation and has not exploits the pixel correlation among the neighbor

blocks,

Table 4.8

Proposed Lossy and Existing Lossless based Compression

Image

512x512

Pixels

Lossless

Based Compression

Lossy

Based Compression

CREW LJPEG Existing

IMWT Modified IMWT

Lena 4.35 4.65 4.42 4.20

Couple 4.91 5.19 4.94 4.17

Man 4.51 4.88 4.82 4.14

103

Figure 4.10 Bits per pixels of proposed Lossy and existing lossless

compression

Thus Integer Multiwavelet transform is promising direction for lossy

coding. The figure 4.10 shows the graphical representation of number of Bits per

pixels required for proposed lossy and existing lossless compression. The existing

lossless and proposed lossy IMWT are shown in figure 4.11. on considering the

quality factor, the proposed Lossy shows good quality with less distortion

compared to existing Lossy method [61].

104

Figure 4.11 Existing Lossless and proposed Lossy IMWT

The table 4.9 gives the PSNR and MSE values for reconstructed standard

test images using Lossy method. The Lena, Couple and Man 512 x512 images on

Lossy reconstruction with LL-Subband alone provide minimum MSE and

Maximum PSNR.

Image (512 x 512)

Original

Existing Lossless IMWT output

Proposed Lossy IMWT output

Lena

Couple

Man

105

Table 4.9

PSNR and MSE values in dB for Reconstructed Images

Image

512x 512

Pixels

Lossy Reconstruction

With LL Subband alone

MSE ( dB)

PSNR (dB)

Lena 7.2734 40.8566

Couple 8.8752 39.2548

Man 12.3610 35.7690

The figure 4.12 shows the graphical representation of PSNR and MSE for

reconstructed images.

Figure 4.12 PSNR and MSE in dB for reconstructed images

The standard images used for testing in this lossy compression technique

provide a high quality reconstruction. The MSE and PSNR for artificial images

106

were identified high-quality by using Lossy method. The performance of IMWT

for images with high frequencies was outstanding. The subjective quality of the

reconstructed image by retaining the LL subband information alone is equal to

that of Lossy reconstruction.

4.3.1 Analysis

The performance of the Integer Multiwavelet Transform for the Lossy

compression of images for (512 x 512) size was analyzed. It was found that the

IMWT can be used for Lossy compression techniques. The Subjective quality of

the Lossy reconstructed images was almost the same as that obtained using

lossless reconstruction. The IMWT produces good results even for artificial

images and for images with more high frequency content like satellite images,

forest scenes, etc. The bit rate obtained using the MS-VLI with RLE scheme is

about 4.1 bpp to 4.2 bpp, which less than that is obtained using MS-VLI without

RLE scheme.

4.4 COMPARISION OF REAL AND BINARY WAVELET WITH

INTEGER MULTIWAVELET TRANSFORM

The IMWT compression scheme constantly gives output high bit

reduction. When compared with the existing RWT and BWT techniques.

Increase in the energy optimization capability of IMWT results in high bit

reduction using IMWT and thus resultant in better compression performance than

existing wavelets. In Table 4.10 are the compression results for 512x512 test

images using RWT and BWT and IMWT which are among the best low bit

reduction coding schemes are shown. It must be pointed out that unlike existing

RWT and BWT schemes the proposed low bit IMWT is the simplest Integer

Multiwavelet transformation and has not exploited the pixel correlation among

the neighbor blocks. Thus the Integer Multiwavelet transform is a very promising

107

technique for bit reduction. The below figure 4.13 represents the proposed IMWT

compression for obtaining the low bit reduction.

Existing RWT(85.92%) Low bit reduction

Existing BWT(84.30%) Low bit reduction

Figure 4.13 Comparison to obtain Bit reduction in

Percentage using IMWT

4.4.1 Analysis

The table 4.10 shows the compression ratio of IMWT is compared with the

existing techniques based on Real wavelet transform (RWT) and Binary wavelet

transform (BWT). The 8-bit standard images (512x512) have been used for this

experiment. The average reduction of 80.638% for IMWT compressed images is

due to the omission of the sign bits in LL bands and the Run Length Encoding of

the sign bits using bit planes.

Original Image (512x512)

IMWT (78.80%) Low bit reduction

108

Standard images

(512x512)

RWT (%)

BWT (%)

IMWT (%)

Lena

85.927 84.302 78. 807 Barbara 84.256 78.393 81. 481

Gold hill 85.718 82.442 81.793 Pepper 85.564 83.812 81.582 Boats 82.951 81.115 80.498

Couple 82.729 80.958 79.671 Average

reduction

84.524

81.837

80.638

Table 4.10

Existing RWT and BWT with proposed IMWT for bit Reduction

The figure 4.14 shows the comparison chart for the bit reduction for the

existing RWT, BWT techniques and the proposed IMWT techniques.

Figure 4.14 Comparison of Bit reduction between Existing RWT, BWT and

Proposed IMWT compression

109

The table 4.11 shows the total number of bits required after the

compression in the proposed IMWT and its percentage for standard images.

Table 4.11

The Bit Reduction using IMWT Image Compression

Standard Image

(512 x512) Pixels No of Bits (Bpp) Percentage (%)

Lena 1652720 78. 807

Barbara 1715334 81.481

Gold hill 1710909 81.793

Pepper 1688179 81.582

Boats 1670824 80.498

Couple 1708793 81. 481

The total number of Bits required for a standard 512x512 images is equal to

(512x512x8=2097152-Bits), After the compressing for the low bit reduction

using proposed IMWT, the total number of bits obtained for Lena is 1652720-

Bits.The amount of bit reduction using proposed IMWT is identified as 78.807.

4.5 SUMMARY

In this work, we have compared the performances of lossy and lossless

compression results with the other existing compression techniques like JPEG,

JPEG 2000, CREW, LJPEG and SPIHT. It has been found that the proposed

IMWT provides better output results with higher quality of images. Different

comparative results have been tabulated for the PSNR and MSE. The advantages

of this proposed IMWT results with higher compression ratio, with invisible loss

of data to the original images during compression and decompression. The

IMWT produces excellent results even with artificial images and images with

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more high frequency content like satellite images, forensic images etc. The bit

rate obtained using the IMWT scheme results in an average reduction of

80.638%, which is less than that of the existing RWT (84.524%) and BWT

(81.837%). The performance of IMWT for images with high frequencies is

outstanding.

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CHAPTER 5

CONCLUSION AND FUTURE WORKS

Applications requiring high speed connections, such as high definition

television, real-time teleconferencing and transmission of multiband high

resolution satellite images, lead us to the finding that image compression is not

only desirable but necessary. In this work, the performance of the Integer

Multiwavelet transform for lossless and lossy compression of images has been

analyzed. The IMWT based lossless and lossy compression on the standard

images show high quality results on reconstruction. A high PSNR is obtained for

artificial images by using proposed lossy method. The SSIM values are close to 1,

which indicate the structural similarity is good with the IMWT. The Subjective

quality of the reconstructed images using proposed IMWT is almost the same as

that obtained using existing lossy reconstruction. The IMWT produces excellent

results even with artificial images and images with more high frequency content

like satellite images, forensic images etc. This proves the performance of

Multiwavelet that allows more design freedom.

The IMWT image compression results in with a very low bit rate, which

results in a smaller file size. This indicates that the IMWT can be used for

wireless technology with the benefits of very low storage space, low probability

of transmission error, high security and low transmission cost.

Applications that require image compression are many and varied such as:

Internet, Businesses, Multimedia, Satellite imaging, Medical imaging and

forensic etc. The reduction in file size is necessary to meet the bandwidth

112

requirements for many transmission systems and for the storage requirements in

computer databases.

On the interest to service of the society, this thesis work can be used for

the big data base collection of images of all the public for the AATHAAR card,

the PAN card, the voter ID, the Ration card, the driving license and etc. Storing

other information such as the biometric finger print and the live video of the

traffic signals in the major cities, and the voice / VoIP recording for the entire

network service providers will also be possible. Generally, it is suitable for all

those applications that require memory for storing large amount of data.

5.1 Contribution of the Thesis

The wavelet transform is found to be a good fit for typical natural images

that have an exponentially decaying spectral density with a mixture of strong

stationary low frequency components. The newer techniques such as IMWT can

achieve reasonably good image quality with higher compression ratios. The

Integer Multiwavelet transform (IMWT) has short support, symmetry, high

approximation order of two. The key concept of the thesis in image compression

algorithm is the development to determine the minimal data required to retain the

necessary information.

The run-length coding works by counting adjacent pixels with the same

gray level value called the run-length, which is then encoded and stored. The

transform coding is a form of block coding done in the transform domain. This

transform coding is achieved by filtering and by eliminating some of the high

frequency coefficients.

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5.2 Limitation and Future Works

This research has mainly focused on the Lossy image compression

techniques to improve the high visual quality of image resultant with the

decompression process. The future work can be extended to Lossless

image compression techniques to enhance the high visual quality of

images and to obtain higher compression ratio.

This research provides the results with the only standard test gray

scale images using lossy compression. As the modern world is towards the

regular use of color images the enhancement can be done on RGB using

lossy compression.

This work is focused on input images with the size of (NxN) alone.

As the part of enhancement, the further work may be carried with different

dimensions of images size of (NxM) of standard test images.

114

APPENDIX – A Program for MS-VLI Representation %To perform the Magnitude set variable length integer (MS-VLI) coding. magnitudemap = zeros (1, sz*sz); signbytes = uint32(0); % Magnitude Set| Amplitude Interval | Magnitude Bits % 0 0 0 % 1 1 0 % 2 2 0 % 3 3 0 % 4 4-5 1 % 5 6-7 1 % 6 8-11 2 % 7 12-15 2 % 8 16-19 2 % 9 20-23 2 % 10 24-31 3 % 11 32-39 3 % 12 40-47 3 % 13 48-55 3 % 14 56-71 4 % 15 72-87 4 % 16 88-103 4 % 17 104-119 4 % 18 120-151 5 % 19 152-183 5 % 20 184-215 5 % 21 216-247 5 % Any values that found above 247 is rounded to Magnitude Set as 21. for i = 1:sz for j = 1:sz if(magnitude(i,j) == 0) magnitudemap(i*sz+j) = 0; elseif(magnitude(i,j) == 1) magnitudemap(i*sz+j) = 1; elseif(magnitude(i,j) == 2) magnitudemap(i*sz+j) = 2; elseif(magnitude(i,j) == 3) magnitudemap(i*sz+j) = 3; elseif (magnitude(i,j) == 4 || magnitude(i,j) == 5) magnitudemap(i*sz+j) = 4; signbytes = signbytes +1; elseif (magnitude(i,j) == 6 || magnitude(i,j) == 7)

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magnitudemap(i*sz+j) = 5; signbytes = signbytes + 1; elseif (magnitude(i,j) >=8 && magnitude(i,j) <= 11) magnitudemap(i*sz+j) = 6; signbytes = signbytes + 2; elseif (magnitude(i,j) >=12 && magnitude(i,j) <= 15) magnitudemap(i*sz+j) = 7; signbytes = signbytes + 2; elseif (magnitude(i,j) >=16 && magnitude(i,j) <= 19) magnitudemap(i*sz+j) = 8; signbytes = signbytes + 2; elseif (magnitude(i,j) >=20 && magnitude(i,j) <= 23) magnitudemap(i*sz+j) = 9; signbytes = signbytes + 2; elseif (magnitude(i,j) >=24 && magnitude(i,j) <= 31) signbytes = signbytes + 3; magnitudemap(i*sz+j) = 10; elseif (magnitude(i,j) >=32 && magnitude(i,j) <= 39) signbytes = signbytes + 3; magnitudemap(i*sz+j) = 11; elseif (magnitude(i,j) >=40 && magnitude(i,j) <= 47) signbytes = signbytes + 3; magnitudemap(i*sz+j) = 12; elseif (magnitude(i,j) >=48 && magnitude(i,j) <= 55) signbytes = signbytes + 3; magnitudemap(i*sz+j) = 13; elseif (magnitude(i,j) >=56 && magnitude(i,j) <= 71) signbytes = signbytes + 4; magnitudemap(i*sz+j) = 14; elseif (magnitude(i,j) >=72 && magnitude(i,j) <= 87) signbytes = signbytes + 4; magnitudemap(i*sz+j) = 15; elseif (magnitude(i,j) >=88 && magnitude(i,j) <= 103) signbytes = signbytes + 4; magnitudemap(i*sz+j) = 16; elseif (magnitude(i,j) >=104 && magnitude(i,j) <= 119) signbytes = signbytes + 4; magnitudemap(i*sz+j) = 17; elseif (magnitude(i,j) >=120 && magnitude(i,j) <= 151) signbytes = signbytes + 5; magnitudemap(i*sz+j) = 18; elseif (magnitude(i,j) >=152 && magnitude(i,j) <= 183) signbytes = signbytes + 5; magnitudemap(i*sz+j) = 19; elseif (magnitude(i,j) >=184 && magnitude(i,j) <= 215) signbytes = signbytes + 5; magnitudemap(i*sz+j) = 20; elseif (magnitude(i,j) >=216 && magnitude(i,j) <= 255)

116

signbytes = signbytes + 5; magnitudemap(i*sz+j) = 21; end % Minimum of 5bits per Magnitude set required for dictionary. %The runlenbytes represents the - sign plane encoded as runlength bits % The signbytes represents- \magnitude bits according to the set. fixedtotalbits = signbytes + runlenbytes + (sz*sz*5); bppfixedmap = fixedtotalbits/(sz*sz); cmpratiofixed = double(8.0/double(bppfixedmap)); fwrite(fid,zeros(1,signbytes),'ubit1'); fwrite(fid, magnitudemap, 'ubit5'); compressedmag = rle(uint8(magnitudemap)); compmagsize = size(compressedmag{1,1}); rlemagbits = compmagsize(2); %for signbytes - no: of bytes for the sign information based on the set %for rlemagbits - run length coded magnuitude map. %for runlenbytes - run length coded sign map before MS. rletotalbits = signbytes+ runlenbytes+ rlemagbits; %To perform the write operation we follow this command % fwrite(fid1,zeros(1,signbytes),'ubit1'); fwrite(fid1,signplane,'ubit1'); fwrite(fid1,compressedmag{1,1},'ubit2'); bpprlemag = rletotalbits/(sz*sz); cmpratiorle = double((8.0)/double(bpprlemag)); fclose(fid); fclose(fid1); % Procedure to obtain Lossy Reconstruction coding for i = 1:sz for j = 1:sz if i>= sz/2 && j>= sz/2 iout1(i,j) = 0; end %The first I- level reconstruction start for i = 1:sz % The first I- level column Inverse IMWT can be obtained for j = 1:sz/4 c1n(1,j) = iout1(j,i); c2n(1,j) = iout1(j+sz/4,i);

117

d1n(1,j) = iout1(j+sz/2,i); d2n(1,j) = iout1(j+((3*sz)/4),i); s1n(1,j) = c1n(1,j); s2n(1,j) = c2n(1,j) + floor((d1n(1,j)+1)/2); m1n(1,j) = s2n(1,j) - d1n(1,j); m2n(1,j) = -d2n(1,j); end for j = 1:sz/4 c1n(1,2*j) = s1n(1,j) + floor((m1n(1,j)+1)/2); c1n(1,2*j-1) = c1n(1,2*j) - m1n(1,j); c2n(1,2*j-1) = m2n(1,j) + floor((s2n(1,j)+1)/2); c2n(1,2*j) = s2n(1,j) - c2n(1,2*j-1); end for j = 1:sz/2 iout(2*j,i) = c1n(1,j) + floor((c2n(1,j)+1)/2); iout(2*j-1,i) = iout(2*j,i) - c2n(1,j); end %The I- level row reconstruction for i = 1:sz %for the I- level row Inverse IMWT can be obtained for j = 1:sz/4 c1n(1,j) = iout(i,j); c2n(1,j) = iout(i,sz/4+j); d1n(1,j) = iout(i,sz/2+j); d2n(1,j) = iout(i,(3*sz/4)+j); s1n(1,j) = c1n(1,j); s2n(1,j) = c2n(j) + floor((d1n(1,j)+1)/2); m1n(1,j) = s2n(1,j) - d1n(1,j); m2n(1,j) = - d2n(1,j); end for j = 1:sz/4 c1n(1,2*j) = s1n(1,j) + floor((m1n(1,j)+1)/2); c1n(1,2*j-1) = c1n(1,2*j) - m1n(1,j); c2n(1,2*j-1) = m2n(1,j) + floor((s2n(1,j)+1)/2); c2n(1,2*j) = s2n(1,j) - c2n(1,2*j-1); end for j = 1:sz/2 recon(i,2*j) = c1n(1,j) + floor((c2n(1,j)+1)/2); recon(i,2*j-1) = recon(i,2*j) - c2n(1,j); end

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129

LIST OF PUBLICATIONS

International Journals

1. K.Rajakumar and T.Arivoli. ‘‘Image Compression for Low bit

reduction using Integer Multiwavelet Transform” an

International Journal of Applied Engineering Research, ISSN 0973-

4562, Vol.10, No.1 (2015) pp.122-128.

2. K.Rajakumar and T.Arivoli. ‘‘Lossy Image Compression using

Multiwavelet Transform for Wireless Transmission” a Springer an

International Journal of Wireless Personal Communication,

Vol.83,No.2, May(ll) 2015. ISSN 0929-6212,DOI: 10.1007/s11277-

015-2637-2,

3. K.Rajakumar and T.Arivoli. ‘‘IMWT coding using Lossy Image

Compression Techniques for Satellite Images” an ARPN Journal of

Engineering and Applied Science, ISSN 1819-6608, Vol.10,

No.9.May. 2015, pp.4234-4242.

130

International Conference

1. K.Rajakumar and T.Arivoli. ‘‘Implementation of Multiwavelet

Transform coding for Lossless Image Compression’’, an IEEE

International Conference on Information Communication and

Embedded systems, ISBN: 978-1-4673-5786-9. Feb. 2013.

2. K.Rajakumar and T.Arivoli. ‘‘Lossy Image Compression using

Multiwavelet Transform Coding’’, an IEEE International

Conference on Information Communication and Embedded systems,

ISBN: 978-1-4799-3834-6 / 14. Feb. 2014.

131

CURRICULAM VITAE

The author Rajakumar.K, born on 20-02-1981, has graduated in

Instrumentation and Control Engineering from Madurai Kamaraj University in

the year 2002. He did his post graduation study in Embedded Systems from Anna

University, Chennai in the year 2006. He has totally 13 years of professional

experience which includes both industry and teaching. He started his career as

lecturer in Sri Ram Engineering College at Chennai from 2002 till 2006 and

worked as senior lecturer from 2006 till 2008. He served as development

Engineer in the Research and Development section of Techknowsys, Chennai for

a couple of month. Then he joined in Kalasalingam University, Krishnankoil as a

Faculty in the Department of Information Technology. His research area includes

Image Processing, Wireless Communication and Embedded System.