IMPLEMENTATION OF MULTIWAVELET TRANSFORM CODING FOR IMAGE COMPRESSION

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

ABSTRACT iv 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.

TABLE OF CONTENTS vii CHAPTER TITLE PAGE ABSTRACT LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS AND SYMBOLS iv xi xii 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

1.19 MAGNITUDE SET-VARIABLE LENGTH INTEGER REPRESENTATION 45 1.20 ORGANIZATION OF THE THESIS 46 ix 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

TABLE NO. LIST OF TABLES TITLE xi 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

FIGURE NO. LIST OF FIGURES 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 1 st 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 xii

FIGURE NO. TITLE xiii PAGE 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 NO.

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

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)/ f c - 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 xv

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. 1

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 2

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 H 0 + H H RL = 1 (1.1) L0 + L1 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. 3

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. 4

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. 5

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. 6

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. 7

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 1 2 MSE( db ) = 10log 10 (Xi Yi) (1.2) NxN i= 0 Where X i is input uncompressed image, Y i 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. 2 255 PSNR 10 log10 db MSE = (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. 8

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. (2 * µ xµ y + c1 )(2 * σ xy + c 2 ) SSIM(x, y) = 2 2 2 2 ( µ + µ + c )( σ + σ + c ) x y 1 x y 2 (1.4) Where µ x - average of x; µ is the average of y; y σ is the variance of x; 2 σ y - variance of y; σ is the covariance of x and y; c xy 1 = (k 1 L) 2, c 2 = (k 2 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); k 1 = 0.01 and k 2 = 0.03 by default. 2 x 1.3.1.4 Structural Dissimilarity Structural dissimilarity (DSSIM) is a distance metric derived from SSIM (though the triangle inequality is not necessarily satisfied). 1 SSIM( x, y) DSSIM( x, y) = (1.5) 2 9

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. Uncompressed file size Compressio n Ratio (CR) = (1.6) Compressed file size 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). Bitsper pixel (bpp) Number of bits 8* Number of = Number of pixels NxN bytes (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 10

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 Original Image Pre- Processing Encoding Compressed Image Figure 1.1 The Block diagram of Compression Compressed Image Decoding Post- Processing Original Image 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 11

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 12

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 postprocessing 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. 13

1.5 OVERVIEW OF METHODOLOGY The methodology [24] for the compression process flow which takes an input image of (NxN) size is shown below. Original input Image Pre-filtration process Pre-analysis across row Pre-analysis across column Performing magnitude set & bit plane coding Performing run length encoding for decomposition Store the compressed Image 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 14

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 postanalysis is done across columns and rows as shown in figure1.4. Compressed Image Performing run length decoding Performing magnitude set & bit map plane coding Inverse for post-analysis across column Inverse post-filtration across row Inverse for post-filtration Reconstructed Image Figure 1.4 The Reconstruction on Reverse transform 15

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: 16

jωt F ( ω) = f ( t) e dt (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, it 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 n 2 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 x 0... x N-1 be complex numbers. The DFT is defined by the formula. x k N 1 n = 0 n n i 2πk N = x e k=0 N-1 (1.9) Evaluating this definition directly requires O (N 2 ) operations, there are N outputs X k, 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 17

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 {F n } that can be defined as: x i = N 1 1 N n= 0 F n e 2πnj ni N (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. 18

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, {Ψ a,b (x)},[59] can be generated by translating and scaling the basic wavelet as, Ψ a,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. 19

For adapted waveform analysis, researchers seek a basis in which the coefficients, when rearranged in decreasing order, decrease as rapidly as possible. To measure the rate of decrease, they use tools from classical harmonic analysis including calculation of information cost functions. This is defined at 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 20

position in the time frequency plane that corresponds to that component s predominant frequency and time of occurrence. For images the space is threedimensional 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 21

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, Ψ a, 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 continuoustime 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 X (t) at a scale (a>0), value b ε R is expressed by the following integral. +* aεr and translational 1 t b Xω ( a, b) = x( t)ψ dt (1.13) 1 2 a a 22

Where ψ (t) 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 x (t), the first inverse continuous wavelet transform can be exploited. x( t) 1 = Cψ Xω( a, b) 1 a 1 2 ~ t b ψ exp db a d a 2 (1.14) ~ ψ (t) is the dual function of ψ (T ) and Cψ = ~ˆ ψ ( t)( ω) ~ ψ ( ω) dω ω (1.15) Cψ is admissible constant, where hat means Fourier transform operator. Sometimes, ~ ψ ( t) = ψ ( t ), then the admissible constant appears like ~ 2 ψ ( ω) Cψ = dω (1.16) ω Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies 0 < < is called an admissible cψ wavelet. An admissible wavelet implies that ~ ψ (0) = 0, so that an admissible wavelet must integrate to zero. To recover the original signal x (t), the second inverse continuous wavelet transform can be exploited. 1 1 x t) = 2πψ (1) a X ( 2 ω t b ( a, b)exp i db a da (1.17) 23

This inverse transform suggests that a wavelet should be defined as ψ ( t ) = w( t)exp( it) (1.18) where w (t) 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 2 n numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the 24

sum. This process is repeated recursively, pairing up the sums to provide the next scale, which leads to 2 n -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 nonstationary 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 25

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 26

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] 27

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 28

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. 29

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: [ n] = x * g [ n] = x[ k] g[ n k] k = g ( ) (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: low [ n] = x[ k] h[ n k] k = y 2 (1.20) high [ n] = x[ k] g[ n k] k = y 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. 30

( y k )[ n] = y[ kn] (1.22) The above summation can be written more concisely. ( x * ) 2 y low = g (1.23) ( x * ) 2 y high = h (1.24) However, on computing a complete convolution x * g 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) 31

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. H(z) 2 H(z) 2 LL Image G(z) 2 LH H(z) 2 HL G(z) 2 G(z) 2 Rows Columns Figure 1.10 I-Level DWT Filter Implementation HH LL H i (z) 2 LH G i (z) 2 + H i (z) 2 HL H i (z) 2 + Image HH G i (z) 2 + G i (z) 2 Columns Rows Figure 1.11 IDWT Filter Implementation 32

Where H(z) is the Low pass Analysis Filter and G(z) is High Pass Analysis Filter, H i (z) is the Low pass Synthesis Filter and G i (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 33

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 34

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, ϕ ( x, y) = ϕ( x) ϕ( y) (1.27) And the dilation equation assumes the form: ϕ ( x, y) = 2 ( h( k,1) ϕ) ϕ( k,2 y 1) (1.28) k, 1 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 35

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 36

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) = [φ 1 (t), φ 2 (t) φ r (t)], the set of scaling functions and ψ(t) = [ψ 1 (t), ψ 2 (t),.., ψ r (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], φ(t) = Hk φ(2t - k) (1.29) k ψ(t) = Gk ψ(2t - k) (1.30) k where H k, G k 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 37

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 functionsϕ 1,...,ϕ r and the corresponding Multiwavelet ψ..,ψ r are usually represented as vectors φ = [ φ,..., ] T and ψ [ ψ,..., ] T 1 ϕ r 1 ψ r 1,. = This will satisfy the matrix dilation equation φ ( t ) = H k kφ(2t k) and the matrix wavelet equation ψ ( t ) = G k kϕ(2t k) where H k and Gk 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], φ 1(t) 1 = φ 2(t) 1/ 2 0 φ 1(2t) 1 + 1/ 2 φ 2(2t) 1/ 2 0 φ 1(2t) 1/ 2 φ 2(2t) (1.31) ψ 1( t) 1/ 2 = ψ2( t) 0 1/ 2 ψ 1(2t) 1/ 2 + 0 ψ2(2t) 0 1/ 2 ψ 1(2t) 0 ψ2(2t) (1.32) 38

The two scaling functions are box, φ ( ), and slope, φ ( ). 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 φ ( ) and φ ( ) can exactly reproduce linear functions [62]. So, by including one additional scaling function φ ( ), the original Haar wavelet transform is generalized to a 2 t 1 t Multiwavelet transform with higher approximation accuracy. c n (0) c, n (0) The two sequences { } and { } 1, for the multifilter bank input are 2 the output of the pre-filter input sequence{ }.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 x, n (j 1) (j 1) (j 1) (j 1) sequences as{ c (0) }{, c }{, c }{, d }, and { d } 1, n 1,n 2,n 1,n 2,n two sequences { } and { } c (j) c (j) 1, n 2, n at scale level j: 2 t 1 t 2 t.at scale level j-1 from the Step 1 (Backward): c (j) + c (j) 1,2n 1,2n+ 1 s (j) 1, n = 2 s (j) = c (j) + c (j) 2, n 2,2n 2,2n+ 1 = m (j) c (j) + c (j) 1, n 1,2n 1 1,2n c (j) c (j) 2,2n 2,2n+ 1 m (j) 2, n = 2 (1.33) 39

Step 2 (Forward): (j 1) c = s (j) 1,n 1, n + m (j) (j 1) 1, n c2,n = 2 (j 1) = d s (j) 1,n 2, n (j 1) = d2,n m (j) 1, n m (j) 2, n s (j) 2, n (1.34) where corresponds to downward truncation.the block transformation of the four elements c (j) c (j) c (j), c (j) 1,2n, 1,2n+ 1, 2,2n 2,2n+ 1may 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): (j 1) s (j) = 1, n c1n (j 1) (j 1) s (j) = + + 1)/2 2, n c2,n (d 1,n (j 1) = m (j) s (j) 1, n 2, n d1,n (j 1) = m (j) 2, n d2,n (1.35) c (j) = + + + s (j) m (j) 1 /2 1,2n 1 1, n 1, n (j 1) Step 2 (Backward): c = c (j) + m (j) 2n 1,2n 1 1, n = + c (j) m (j) + 1 /2 s (j) 2,2n 2, n 2, n (j 1) = c + m (j) c (j) 2,2n 1 2, n 2,2n (1.36) 40

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 41

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 42

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 1. Both Scaling and Wavelet functions are scalars functions are vectors 2. It has one Scaling and Wavelet 2. Multiwavelet with multiplicity r Function has r scaling and r wavelet function 3. The Solution of the Dilation 3. The Dilation Equation is of matrix Equation results in one Low Pass form. For Multiwavelet of (L) and one High Pass (H) filter. multiplicity 2, it results in four filters 43

with two Low pass L 1, L 2 and two High pass H 1, H2 filters. 4. The Input Image can be used as it is for the 2D Pyramidal Algorithm. 5. It produces 4 subbands after I level decomposition namely, LL Approximation. LH Horizontal Detail. HL Vertical Detail. HH Diagonal Detail. 4. The input image has to be prefiltered to produce vectors that can be applied to the 2D Algorithm. 5. Here, it produces Sixteen Subbands namely L 1 L 1, L 1 L 2, L 2 L 1, L 2 L 2, L 1 H 1, L 1 H 2, L 2 H 1, L 2 H 2, H 1 L 1, H 1 L 2, H 2 L 1, H 2 L 2, H 1 H 1, H 1 H 2, H 2 H 1, H 2 H 2. L 1 L 2 H 1 H 2 L 1 L 1 L 1 L 2 L 1 H 1 L 1 H 2 L 2 L 1 L 2 L 2 L 2 H 1 L 2 H 2 H 1 L 1 H 1 L 1 H 1 H 1 H 1 H 2 H 2 L 1 H 2 L 2 H 2 H 1 H 2 H 2 a) Horizontal filtering b) Vertical direction after horizontal filtering Figure 1.14 I - level of decomposition Subband structure of images 44

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 H 1 H 1, H 1 H 2, H 2 H 1, and H 2 H 2 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 L 1 L 1 sub band has 45

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 46

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. 47

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 48

upon the origin, and dies out quickly as x. A set of wavelet basis functions, {Ψ a,b (x)}, can be generated by translating and scaling the basic wavelet as, Ψ a,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) = [φ 1 (t), φ 2 (t) φ r (t)], the set of scaling functions and ψ(t) = [ψ 1 (t), ψ 2 (t),.., ψ r (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 49

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. φ(t) = Hk φ(2t - k) (2.4) k ψ(t) = Gk ψ(2t - k) (2.5) k where H k, G k 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 50

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. 51

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, 52

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 timefrequency 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 53

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 54

quality Wang Z [85] results are obtained by using a filtering-based postprocessing scheme. It have demonstrated competitive performance compared to other methods, and managed to obtain results of high visual quality for low bitrates. 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 55

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. 56

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 57

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. 58

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 59

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 60

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 61

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]. 62

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 secondorder 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. 63

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 64

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 65

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 66

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. 67

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. 68

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. 69

3.2 OVERVIEW The principle of image compression and decompression using IMWT is explained in this chapter. Original Image (NxN) IMWT Preprocessing Magnitude set & run length encoding Compressed Image Figure 3.1The Compression Compressed Image Magnitude set & run length decoding IIMWT post processing Decompressed Image 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. 70

3.3 INTEGER PREFILTER In a multiwavelet system that uses matrix valued filters input sequence{ x n}, cannot be directly processed by the multifilters. It is necessary to (0) (0) obtain a vector input sequence with the two vector element c and 1, n c from the 2, n input sequence { xn}, through a pre-filter Q(z) as shown in figure 3.3.The equivalent non polyphase representation of the nontruncated multiwavelet system is shown in figure 3.4. X n 2 2 Q(z) c c (0) 1, n (0) 2, n H(z) G(z) Figure 3.3 Multifilter bank implementation of 1 st level Multiwavelet decomposition pre-filtering as polyphase representation 2 2 d c c d ( 1) 1, n ( 1) 2, n ( 1) 1 n ( 1) 2, n X n H ~ 1 (z) H ~ 2 (z) 2 ( 1) 2 2 c c 1, n ( 1) 2, n G ~ 1(z) G ~ 2 (z) 2 2 2 d d ( 1) 1, n ( 1) 2, n Figure 3.4 Multiwavelet decomposition pre-filtering as equivalent nonpolyphase representation 71

It shows the combined prefilter and multifilter metrics, H ( z) Q( z) and Hˆ l and G ( z) Q( z) the polyphase matrices of another two set of filters, ( z) ˆ ( z) l = 1,2. Using lowpass & bandpass criteria on these two equivalent sets of G l Filters, good prefilters should satisfy the conditions. H ˆ ( 1) = 0 and G ˆ ( ) 0, = l z = l 1, 2 (3.1) l Such that the first level decomposition separates the input { xn}, into low ( 1) ( 1) frequency approximation, and { c }, { c } 1, n 2,, n ( 1) ( 1) { } { } d, 1, n d 2,, n and the high frequency details, as the traditional wavelet decomposition does. By limiting the nonpolyphase equivalent filter length of Q ( z) to2, a reasonable choice for such a short supported multiwavelet system, the combined prefilter and multifilter polyphase matrixes are expressed as H 1 1+ z 1 ( 1+ z 2 0 1 (1 + z 2 ( z) Q( Z ) = 1 1 1 (1 z G( z) Q( Z ) = 2 0 1 ) ) 1 (1 + z 2 (1 z a X ) c 1 ) a X ) c 1 b d, (3.2) b d (3.3) The equivalent nonpolyphase matrixes can be expressed as, H l z) = ah l ( z) + ch l ( z) + bh l ( z) + dh ( z), l 1,2 (3.4) ( 1 2 1 l 2 = Gl z) = agl ( z) + cgl ( z) + bgl ( z) + dg ( z), l 1,2 (3.5) ( 1 2 1 l 2 = Where H andg lm forl = 1,2and m = 1, 2 are the elements of the matrixes. lm The filter bank implementations of the Multiwavelet transform with multiplicity two, need four filters. The pyramidal algorithm then needs four filters 72

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 prefiltered 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 73

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 L 1 L 1 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. 74

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 75

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 76

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 Bit Bit position-1 position-0 Magnitude bit required Total bits required ( 8-11 ) 8 0 0 2 9 0 1 2 10 1 0 2 2 11 1 1 2 77

Table 3.3 Amplitude Intervals example for (24 to 31) Amplitude No. of Bits at respective position Magnitude Intervals Bit position-2 Bit position-1 Bit position-0 bit required 24 0 0 0 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 Total bits required ( 24-31 ) 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 78

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 79

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. (0) P r1, i = [(P 2i + P 2i + 1) / 2] Pre-filter row : P (0) r2, i P2i 1 P2i (3.6) = +, (3.7) Step2: Perform the Integer Multiwavelet Transform for Pre-analysis along columns. (0) P c1, i = [(P 2i + P 2i + 1) / 2] Pre-filter column: P (0) c2, i P2i 1 P2i (3.8) = +, (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; 80

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; 81

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. 82

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. 83

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 Lossless Reconstruction Pixels MSE PSNR Lena 7.2734 40.8566 Boat 7.3104 40.8196 Baboon 17.8008 30.3292 Barbara 14.0855 34.0445 84

Table 4.2 The Bit Rate for Lossless Compression Image Lossless Reconstruction 512x 512 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 85

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

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: 87

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. 88

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 L 1 L 1 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). 89

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. 90

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 Image 256x256 Standard Lena Satellite urban Satellite rural Proposed IMWT based Lossy Reconstruction PSNR (db) SPIHT Compression Ratio(CR). PSNR (db) IMWT Compression Ratio(CR) 35.81 8 37.12 8 19.00 8 20.39 8 12.60 8 14.77 8 91

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 92

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 93

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 L 1 L 1, L 1 L 2, L 2 L 1, L 2 L 2 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 94

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. 95

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 96

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 Proposed Lossy Existing Lossy Compression pixels Compression (Bpp) 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 97

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 Existing Lossy Compression (PSNR) db Proposed Lossy Compression Pixels 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 98

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. 99

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. 100

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. 101

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 Lossless Lossy 512x512 Based Compression Based Compression Pixels Existing CREW LJPEG Modified IMWT 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 4.14 102

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]. 103

Image (512 x 512) Original Existing Lossless IMWT output Proposed Lossy IMWT output Lena Couple Man 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. 104

Table 4.9 PSNR and MSE values in db for Reconstructed Images Lossy Reconstruction Image With LL Subband alone 512x 512 Pixels 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 105

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 106

technique for bit reduction. The below figure 4.13 represents the proposed IMWT compression for obtaining the low bit reduction. Original Image (512x512) IMWT (78.80%) 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. 107

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