implementation of multiwavelet transform coding...
TRANSCRIPT
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;
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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);
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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.