IMPLEMENTATION OF BCWT IN GUI WAVELET TOOLBOX. Spandana Kongara, B. Tech. A Thesis ELECTRICAL ENGINEERING

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1 IMPLEMENTATION OF BCWT IN GUI WAVELET TOOLBOX by Spandana Kongara, B. Tech A Thesis In ELECTRICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING Approved Dr. Tanja Karp Chair of Committee Dr.Sunanda Mitra Dr. Brian Nutter Ralph Ferguson Dean of the Graduate School December, 2010

2 Copyright 2010, Spandana Kongara

3 ACKNOWLEDGMENTS I am extremely thankful to Dr. Tanja Karp for her guidance and support in making the research work successful. I also express my sincere gratitude to Dr. Brian Nutter and Dr. Sunanda Mitra for their valuable insights on this research work. My special thanks to Enrique Corona, who helped me in finding solutions to some of the problems in the research work. I would like to thank my parents, Mr. Hanumantharao kongara and Mrs. Leelavathi Kongara and my brother, Venkat Suraj Kongara for their support and inspiration. I also thank all my friends for their encouragement. I specially thank Vivek and Kusum for their moral support during hard times. ii

4 TABLE OF CONTENTS ACKNOWLEDGMENTS... ii ABSTRACT...v LIST OF FIGURES... vi I. INTRODUCTION PROBLEM STATEMENT / MOTIVATION OUTLINE OF THESIS... 4 II. BACKGROUND SOURCE ENCODER SOURCE DECODER WAVELET TRANSFORM Discrete Wavelet Transform Two Dimensional Wavelet Transform COMPRESSION OF IMAGES THROUGH WAVELET TRANSFORM IMAGE COMPRESSION ALGORITHMS EZW (Embedded Zero Tree Wavelet) Algorithm Set Partitioning Of Hierarchical Trees (SPIHT) BACKWARD CODING OF WAVELET TREES BCWT Algorithm III. MATLAB WAVELET TOOLBOX EXISTING IMAGE COMPRESSION COMMAND LINE FUNCTIONS FUNCTION CALLS EXISTING GUI TOOLBOX iii

5 3.4 FUNCTION CALLS FOR GUI WAVELET TOOLBOX IV. COMMAND LINE FUNCTIONS FOR BCWT NEW WTC_BCWT FUNCTION V. GUI WAVELET TOOLBOX FOR BCWT...49 VI. RESULTS AND CONCLUSION RESULTS FOR COMMAND LINE FUNCTION WCOMPRESS COMPARISION OF BCWT WITH SPIHT RESULTS FOR GUI WAVELET TOOLBOX CONCLUSION REFERENCES...68 iv

6 ABSTRACT MATLAB has different tools available for Image processing applications such as Image Processing Toolbox, Wavelet Toolbox etc. The Wavelet Toolbox has different GUI interfaces in it for various wavelet applications which can be accessed by the command wavemenu. For Image Compression the Wavelet Toolbox has a GUI tool named True Compression 2D. It can also be accessed by the command wc2dtool. The user can also access a command line function instead of GUI toolbox using the command wcompress. The toolbox and the command line function have different compression algorithms available for compressing images such as EZW, SPIHT etc. The user can select the desired method for a particular application and compress the images. The BCWT Compression algorithm that is proposed by Jiangling Guo is more advantageous than some of the existing algorithms in the Wavelet toolbox such as EZW, SPIHT etc. BCWT algorithm is less complex, fast and uses less memory than EZW and SPIHT. The BCWT is added to the available compression algorithms in the toolbox and the command line function such that the user can access it similar to the other compression methods. BCWT is made simple for all users by integrating it into the GUI Wavelet Toolbox. v

7 LIST OF FIGURES 2.1 General Image Compression Model Source Encoder Source Decoder Encoder Decoder One Dimensional two band filter bank for Encoding and Decoding Spectrum Properties Wavelet Transform Wavelet tree, offsprings and leaves [1] Decreasing magnitudes of Wavelet Tree [1] Two levels Discrete wavelet transform [1] Original Lena image Wavelet Transformed Lena image Wavelet Transform Image Horizontal border information Vertical border information Diagonal border information Second level hierarchical subband decomposition Spatial Orientation of Wavelet Trees [7] Scanning Order of Coefficients [7] An example of an BCWT coding unit U(i,j) [1] An example output bit stream of an image [1] wavemenu Block diagram of function calls for wcompress Wavelet Transform using 'wc2dtool' Compression using 'wc2dtool' Block Diagram for Wavelet Decomposition Using 'wc2dtool' vi

8 3.6 Block Diagram of Compression using 'wc2dtool' Block Diagram of Compression using 'wcompress' Block Diagram of function 'wtc_bcwt' Block Diagram of Wavelet Decomposition for 'BCWT' Block Diagram of Compression for 'BCWT' Original Image of lena.bmp Difference Image (q_min=3) Difference Image (q_min=5) Difference Image (q_min=7) Difference Image (Level=5) Difference Image (Level=7) Decompressed Image(baboon.png) Decompressed Image (house_profile.png) Decompressed Image for SPIHT Results of GUI Wavelet Toolbox for BCWT Results of GUI Wavelet Toolbox for BCWT for q_min= Results for the GUI Wavelet Toolbox for BCWT for q_min= Results of GUI Wavelet Toolbox for SPIHT Results of GUI Wavelet Toolbox for BCWT vii

9 CHAPTER I INTRODUCTION Present day large amounts of images are stored, processed and transmitted and hence there is a great need for the compression of an image to save memory, transmission bandwidth etc. For many applications, simply reducing the file size or simple compression is not sufficient some additional scalable and embedded properties are also required. Broadly, compression techniques can be categorized into lossless and lossy compression methods. Lossless algorithms perform compression without any loss of information of the original signal, e.g. for medical records, whereas the lossy compression algorithms provide larger compression ratios but when the original information is reconstructed some information is lost. Most common widely used lossless image compression algorithms are Huffman encoding, LZW and arithmetic coding methods. Lossy compression algorithms are widely used in video conferencing, broadcast television, etc. where certain loss of information is tolerable for high rates of compression. Compression reduces the amount of data required to represent a particular amount of information. If certain data contain repeated information or information that is not required, that information is called redundant data. When two sets of data containing similar data are present at least one may have some repeated or irrelevant data, so that it contains data redundancy. The three basic types of redundancies in digital image compression are Coding redundancy, Interpixel redundancy and Psycho-visual redundancy. When one or more of these redundancies are reduced from the original data, then data compression is said to be achieved. The general image compression scheme contains an encoder that reduces the above redundancies and a decoder that reconstruct the original image from the encoded information. 1

10 In natural images some gray levels are more probable than others. Thus the number of bits used to represent more probable and less probable gray levels is the same, and hence coding redundancy is said to be present in the image. The number of bits used to represent the gray levels of the pixels in an image depends on the repetition of the gray values in that particular image. The process of assigning more bits to the less probable gray levels and fewer bits to the highly probable gray level values is known as variable length coding. By this coding, some amount of data compression is achieved and coding redundancy is said to be achieved. When the coding of the gray levels of the pixels of an image is performed such that more coding symbols are assigned than actually required for each gray level then the image outcome is said to have coding redundancy. The structural, geometric and interframe similarities in the objects in an image result in correlation between the pixels. As the information carried by each pixel is very little, any given pixel in an image can be predicted from the neighboring pixels. This happens to have spatial, geometrical and interframe redundancies. The combination of these three redundancies can be called the interpixel redundancy. That is, the image can be represented as the differences between the adjacent pixels in an image. Thus, interpixel redundancy can be achieved. As far as the human eye is considered some information has relatively less importance than others. That less important information is known as psycho-visual redundant data. In general, human perception of information in an image doesn t involve pixel to pixel observation but it searches for the edges and textural regions and combine to regional groupings and these groupings are correlated by the brain with some past knowledge and interprets an image. The visually unimportant data is removed and hence the psycho-visual redundant data is removed from the image. When this is done some of the information in the image is lost as it quantizes that result in loss of information. This is a lossy compression because the original image cannot be perfectly reconstructed from the compressed one. 1.1 PROBLEM STATEMENT / MOTIVATION Different compression algorithms evolved for the efficient compression of images considering various applications. Some of the well-known compression techniques are 2

11 JPEG (Joint Photographic Experts Group), EZW (Embedded Zero tree wavelet coding), SPIHT (Set Partitioning of Hierarchical Trees) etc. One of the traditional methods, JPEG is a benchmark of the compression techniques that provides some of the features like scalability, editability, etc. The traditional JPEG uses DCT (Discrete Cosine Transform) and produces blocking artifacts in the images. JPEG2000 which is developed as extension of the JPEG uses the wavelet transform. But still ringing artifacts occur at the edges of the images. There are many advantages for wavelet based codecs rather than the non wavelet based ones. In 1993, J. M. Shapiro proposed the EZW algorithm which uses efficient principles such as building a zero tree and also provides rate embeddedness. But EZW has its own disadvantages due to its high memory usage and complexity. Hence, there are many disadvantages for the existing compression algorithms while considering compression time, memory usage, complexity and many other features. Jiangling Guo implemented an efficient image compression algorithm Backward Coding of Wavelet trees (BCWT), which adopts the coding principle of SPIHT and is more efficient than the other compression techniques mentioned above. BCWT builds a MQD (Maximum Quantization of Descendants) map that is one fourth the size of the original wavelet transform matrix. The number of iterations it undergoes is also less, and hence BCWT uses less memory and is also less complex while retaining the PSNR performance of more complex methods. The GUI (Graphical User Interface) wavelet toolbox is an excellent tool by which user can process data in 1-D or 2-D. A variety of techniques are embedded in it, such as wavelet transform, wavelet packet transform, denoising techniques, etc. The GUI wavelet toolbox in Matlab has a true compression icon that can be accessed from typing wavemenu in the command window of Matlab which directs the user to the compression toolbox window. The window can also be directly accessed by typing wc2dtool (wavelet compression tool) in the Command window of Matlab. Some of the effective compression algorithms that exist in the compression toolbox are EZW, SPIHT, LTW, etc. There is also a command line function wcompress for the compression of images with different compression methods available similar to the wavelet toolbox. The 3

12 detailed information of the GUI wavelet toolbox and the command line function wcompress is explained in chapter two. Undoubtedly, BCWT can be added to the compression techniques those already exists GUI Wavelet toolbox for its efficiency and other exceptional features. The aim of the thesis research work is to add the BCWT algorithm to the GUI toolbox and to the command line function wcompress in Matlab. 1.2 OUTLINE OF THESIS BCWT is implemented by Guo by using a CDF wavelet transform, but the efficient principle can be used with other wavelets. Hence, the algorithm is implemented for the user defined wavelets which are available in Matlab such as Haar, Coiflet, Daubechies, Symlet, etc. An attempt was made to implement BCWT in Graphical User Interface wavelet toolbox, which already contains some of the popular compression algorithms such as EZW, SPIHT, LTW, etc. Chapter four provides all the information of the GUI wavelet toolbox and how the efficient compression algorithm BCWT is added along with the other existing compression methods. The command line function in Matlab wcompress works with the same compression algorithms stated above. The BCWT compression algorithm is added to the function wcompress along with the other methods. The detailed information about the command line function and the implementation is explained in Chapter three. 4

13 CHAPTER II BACKGROUND Image compression is achieved by encoding and decoding of the pixels in an image. The general block diagram shown in figure 2.1 consists of the encoder, decoder and channel. If f(x,y) is an image that is given as input to the source encoder it is encoded and a set of symbols are produced that is transmitted through the channel, and the output of the channel is fed to the decoder, which decodes and produces the image fˆ(x,y). The decoded fˆ(x,y) may be identical to f(x,y) if the compression is lossless and transmission is error free, or fˆ(x,y) can be distorted from f(x,y). The channel encoder and channel decoder can be omitted from the design if the channel is noise free, and thus the encoder contains only source encoder and the decoder contains only source decoder. If the channel is not noise free, then channel, the encoder and the decoder are present in the design. The source encoder removes all the input redundancies, and the channel encoder introduces noise immunity in the source encoded information. At the receiver, the channel decoder is followed by the source decoder, which decodes the obtained symbol sequence and reconstructs the image. Figure 2.1 General Image Compression Model. 2.1 SOURCE ENCODER The source encoder eliminates the three redundancies present in the input image f(x,y). The mapper transforms the image into a set of data that is an array of coefficients such that the interpixel redundancy removal is possible. The quantizer block reduces psychovisual redundancies present in the image by reducing the accuracy of the mapper output by some pre-established criteria. The symbol encoder maps the quantizer output according to the code and produces a fixed length or variable length code as an encoded 5

14 output. The variable code represents the mapper and quantizer in many coding algorithms that reduce the coding redundancy. Figure 2.2 Source Encoder 2.2 SOURCE DECODER The source decoder performs the inverse operation of the source encoder. Because the quantization process is irreversible, that information is lost and hence there is no inverse Quantizer included in the decoder. Figure 2.3 shows the source decoder block diagram. Wavelet transform based image codecs: Figure 2.3 Source Decoder Wavelet transform coding is computationally efficient, and it reduces blocking artifacts in the decompressed image. Like other transform coding techniques, the wavelet transform also decorrelates the pixels of the image and packs most important visual information into a smaller number of coefficients. The remaining unimportant information is then quantized or set to zero. This may cause negligible distortion in the image. The wavelet transform decomposes large portions of the original image into horizontal, vertical and diagonal coefficients called subbands, and these subbands individually contain less information and are quantized or truncated to zero. The transformed coefficients are 6

15 coded to reduce coding and other redundancies. In the decoder, the inverse operations of the encoder are done except for the quantization. The block diagrams of the encoder and decoder of wavelet transform based image codec are shown in Figure 2.4 and Figure 2.5. Encoding:- Decoding:- Figure 2.4 Encoder Figure 2.5 Decoder Image area can typically be represented as the trends, the original image content, and anomalies, the edges or the object boundary information. The traditional transform coders such as JPEG use Discrete cosine transform (DCT) to decompose the images and represent the coefficients with a fixed spatial area and frequency bandwidth. The frequency and the bandwidth allocated for all coefficients are the same in the representation. The edge information normally disperses, and hence many non-zero coefficients are required to represent the borders. For the lower bit rates, traditional 7

16 coders represent the trends with many bits and anomalies with fewer bits, and hence blocking artifacts occur and hence they fail. The wavelet transform represents the trends and anomalies with different scales. The drawback of the wavelet transform is that the fine detail coefficients that represent possible anomalies occupy large number of coefficients, and hence much of the information is represented by position of the significant anomalies coefficients. The traditional methods of image compression such as JPEG 2000 degrade the image quality by producing ringing artifacts in the image. Some of the important features like high fidelity, progressive coding, and region of interest coding needs to be provided by the codec. The wavelet tree based codecs satisfy most of the demands of today s applications. 2.3 WAVELET TRANSFORM A signal is mathematically transformed to know the information from the signal that cannot be seen when it is original time domain signal. A signal that is in time-amplitude representation can be represented in frequency domain when transformed. The Fourier transform is one of the popular transforms that mathematicians use, and there are also many others like Discrete cosine transform (DCT), Hilbert transform, short-time Fourier transform, Wigner distributions, the Radon transform and the wavelet transform. Each of these transforms has its own advantages and disadvantages. The Fourier transform provides frequency information of the signal. However when an user needs to know the time information as well as the frequency information in the signal i.e., when the user needs to know the time where the frequency components of the signal occur then wavelet transform is a solution. Fourier and the wavelet transforms are reversible, the signal can be forward as well as reverse transformed. DCT converts the spatial domain to frequency domain, while the wavelet transform retains both the spatial and frequency information and therefore wavelet domain is sometimes referred as spatial-frequency domain. Also, wavelet transform has many advantages over DCT, for instance DWT has no blocking 8

17 rate, and so for lesser bit rate we can achieve better visual quality and also there is the possibility of achieving the progression of quality and resolution. The basic idea of wavelet analysis is to analyze the signal components according to the time scale. Instead of the Fourier analysis, which analyzes the signal with sines and cosines, wavelet functions are used. One feature of the wavelets is that all the different wavelet functions can be constructed using a single mother wavelet which is a small wave, by translating and dilating it. One distinguishes between continuous wavelets (i.e. continuous time functions) and discrete wavelets, where the wavelets are discretely sampled. The signal is transformed by passing through low-pass and high-pass filters and derived from the mother wavelet by down sampling it by factors of 2. The process continues for the higher levels of transform other than single level of decomposition. If the transform is to be applied to a 2D signal, then the single level transform is applied along rows and columns of the signal. The filter outputs at level n can be the approximation (a n ), horizontal (h n ), vertical (v n ) and diagonal (d n ) components of the 2D signal. There are different types of discrete wavelets: Haar, Coiflet (6, 12, 18, 24, 30), Daubechies, Daubechies biorthogonal or Cohen-Daubechies-Feauveau (CDF N/P), Symlet etc Discrete Wavelet Transform The filters used for the decomposition are known as analysis filters, and those for the reconstruction of the image are called the synthesis filters. H 0 (Z) and H 1 (Z) represent analysis filters, whereas G 0 (Z) and G 1 (Z) represent synthesis filter in Figure 2.6. The down arrow represents that the image is down sampled by 2, and the up arrow shows that the image is up sampled by 2. The block diagram of the image decomposition and reconstruction can be seen in the figure below. The image is first passed through the lowpass and high-pass filters H 0 (Z), H 1 (Z) and then down-sampled by two, and then in the reconstruction part the outcome of the analysis part is first up-sampled by two and then passes through the synthesis low-pass and high-pass filters, G 0 (Z) and G 1 (Z). 9

18 Figure 2.6 One Dimensional two band filter bank for Encoding and Decoding The low pass and high pass filters are chosen such that there is no loss of information in the signal during the decomposition and reconstruction. Figure 2.7 Spectrum Properties Two Dimensional Wavelet Transform A single level wavelet transform is done by performing one level of one dimensional transform on every row and column of the matrix of the image. 10

19 Figure 2.8 shows the resulting decomposition after one level, where h1, a1, v1, d1 are the subbands, a contains the approximation details, h shows the horizontal details, v shows the vertical details and d subband contains the diagonal details of the image. If the original image has M rows and N columns each subband is of the size M/2 and N/2. When higher levels of transform are applied to the image, then the a1 sub band is further divided into sub bands, i.e., a1 is divided or further transformed into sub bands a2, h2, v2 and d2 but the level one fluctuation sub bands remain unchanged. Similarly, for the higher levels the a band is further divided into a, h, v and d subbands, and the lower level fluctuation bands are unaltered. Figure 2.8 Wavelet Transform 11

20 The wavelet tree for an image can be represented as follows:- Figure 2.9 Wavelet tree, offsprings and leaves [1] The wavelet tree magnitudes decrease as shown in the figure. Figure 2.10 Decreasing magnitudes of Wavelet Tree [1] 2.4 COMPRESSION OF IMAGES THROUGH WAVELET TRANSFORM When a wavelet transform is applied to the signal, the image resembles a tree format. The higher value coefficients concentrate in an area that is in a1 sub band this is because of the reason that most of the real world components have more of the low frequency 12

21 components. Thus the most of the important content of the signal is concentrated in the low pass sub band. A threshold value is compared to the wavelet transformed matrix, and the values in the matrix that are less than the threshold value are set to zero and the significant values (i.e., non zero values) are transmitted. This significantly decreases the size of the data transmitted. At the inverse transformation, the zero values can be approximated. This inverse transformation step Here in the BCWT the wavelet transform is performed for an image X as follows, take an input image X, which is a two dimensional matrix having the gray values ranging from 0 to 255 for each corresponding pixel in the image. Each row of the 2D matrix is passed through the high-pass and low-pass filters and then down sampled by 2. Each column is passed through a different set of high and low pass filters and again down sampled by 2. The resulting, four matrices are four subbands of the image, as shown in the figure below. For further decomposition of another level of transform, the lower subband is passed through the high pass and low pass filters and down sampled by 2 in a similar manner as above, and for further levels of decomposition the lower subband is processed. The two level discrete wavelet transform is shown in the figure below. The inverse DWT is just the inverse filtering by up sampling by 2. The filters used for the forward transform are the analysis filters, while for the inverse are the synthesis filters. The filters are designed such that perfect reconstruction is obtained. 13

22 Figure 2.11 Two levels Discrete wavelet transform [1]. We can form a wavelet tree by keeping together all the wavelet coefficients of different subbands corresponding to the same spatial location. The coefficient at the top of the tree is the root or parent, the four children of the root are the offspring and the remaining descendents are the leaves. The offspring and the leaves together are called descendents. The coefficients in the tree generally follow the descending order of magnitude from top to bottom of the tree, but most of the image codecs exploit this trend in order achieve high coding performance and follow the zig-zag scanning order. 14

23 The wavelet transform applied to the Lena image Lena.png of size (512x512):- Figure 2.12 Original Lena image Figure 2.13 Wavelet Transformed Lena image Figure 2.14 Wavelet Transform Image Figure 2.15 Horizontal border information 15

24 Figure 2.16 Vertical border information Figure 2.17 Diagonal border information Border information can be seen in the subbands when wavelet transform is performed. 2.5 IMAGE COMPRESSION ALGORITHMS One of the popular wavelet tree based codecs is EZW (Embedded Zero tree Wavelet coding), which was proposed by J.M.Shapiro EZW become the pioneer for the tree based codecs, but it has many disadvantages like complexity, speed and other issues. Later, Pearlman proposed SPIHT (Set Partitioning of Hierarchical Trees) which is more efficient in building the tree by dividing into offspring and leaves and codes the coefficients according to priority. It efficiently encodes with better PSNR performance even without the arithmetic coding involved in its coding. Although SPIHT has many better features than previously proposed codecs, it also lacks some desirable features EZW (Embedded Zero Tree Wavelet) Algorithm J.M Shapiro proposed this simple yet remarkable image compression algorithm which is embedded in nature the bits in the bit stream are generated according to the order of importance. The embedded coding compression algorithm lets the encoder to cease the encoding bit stream at any time when the target bit rate is met. Also, the decoder can also 16

25 cease the decoding at any point and can still get back the same image as the image we get from decoding the entire bitstream. The important features of EZW compression algorithm are The discrete wavelet transform that gives the multiresolution representation of the image. The unique zero tree coding of the multiresolution representation which is wavelet transformed provides the significant maps by which we can understand the positions of the significant bits. The zero tree coding predicts the significant coefficients to represent the exponentially growing trees. The successive approximation in it allows the multi precision representation of the significant coefficients thus making the algorithm embedded in nature. The wavelet coefficients are prioritized, and ordering is done by considering the magnitude and the spatial location of the coefficients. The larger magnitude coefficients are given more importance than the smaller. The multilevel adaptive arithmetic coding provides the entropy coding of the strings and the sequential coding of the algorithm. It stops whenever the target bit rate is achieved. The discrete wavelet transform:- The discrete wavelet transform used in this coding is similar to hierarchical subband decomposition. The decomposition is done by dividing the image into four subbands. The low frequencies bandwidth range is 0< ω <, and the high frequencies bandwidth range representation is < ω <. The four subbands are obtained when horizontal and vertical filters are applied to the image. The subbands LH1, HH1 and HL1 represent the very finest details of the image. For the second level of decomposition LL1 band is further divided into subbands LL2, HL2, LH2 and HH2 as shown in figure 8. 17

26 To obtain more levels of decomposition, the LL subband is further sent through the filters and further divided into subbands. Each coefficient after decomposition is said to occupy 2x2 area of the total original image. Figure 2.18 Second level hierarchical subband decomposition More details and explanation of the mechanisms of the wavelet transform are discussed in the third chapter. Shapiro s Zerotree wavelet structure:- In natural images, most of the significant information lies at low frequencies, and hence low frequency subbands carry more information than the high frequency subbands. The zerotree consists of hierarchically descending magnitudes of the coefficients. The highest magnitude coefficient is the node of the wavelet tree and it has descendents such as offsprings and leaves. The spatial orientation of the tree and the wavelet coefficients are shown in figure 9. The coding of the wavelet coefficient is done only if it is significant. The wavelet coefficient is compared to a threshold and found whether it is significant or not. If it is found significant, then the sign is kept at output and it checks the descendents (offspring 18

27 and leaves). If it has descendents, then the descendents are checked if they are significant and if significant then coding is performed. The scanning order of the coefficients is seen in figure 10. The nodes are scanned first and then the descendents. The scan moves from one subband to another if and only if all the positions in one subband are scanned. Figure 2.19 Spatial Orientation of Wavelet Trees [7] Figure 2.20 Scanning Order of Coefficients [7] 19

28 2.5.2 Set Partitioning Of Hierarchical Trees (SPIHT) SPIHT, proposed by Said and Pearlman, developed the existing principles in EZW and extended it to achieve better results. The progression of efficiency cannot be achieved by this embedded zero tree wavelet algorithm. SPIHT not only deals well with the complex techniques but is also very fast and produces an embedded bitstream i.e., the decoder can cease decoding at any point and get the decompressed image. If the full resolution of the image is desired then the full length bit stream is loaded and decoded. The principles of SPIHT are as follows: 1) Partial ordering of the wavelet transformed image coefficients by magnitude and transmission of order by subset partitioning algorithm which is duplicated at the decoder. 2) Ordered bit plane transmission. 3) Exploitation of self similarity of an image wavelet transform at different scales. The partial ordering is done by comparing each magnitude of the wavelet transformed coefficient with a threshold and determining whether it is significant or not by comparing with the threshold. SPIHT is a lossy coding. The wavelet transformed filters used for the wavelet transform have non-integer tap weights therefore non integer coefficients are generated out of the transform which are truncated to the nearest integers and hence the image cannot be perfectly reconstructed and thus there is little loss of information. If the integer multiresolution transform is used, then perfectly reversible decompression of the image can be achieved. 2.6 BACKWARD CODING OF WAVELET TREES While improving all the drawbacks of the EZW and SPIHT, BCWT retains the PSNR performance. The PSNR performance of BCWT is identical to SPIHT. It also capable of 20

29 providing many features like resolution scalability, low complexity, fast coding speed and low memory usage. BCWT overcomes the three bottlenecks of the wavelet based image codecs like tree scanning, bitplane coding and dynamic list management BCWT Algorithm Building MQD maps: The MQD map abbreviated as Maximum Quantization levels of Descendents map is built with variation of J.M. Shapiro s zerotree map and is represented in most efficient form. In Shapiro s zero tree map each node stores the maximum magnitude wavelet coefficients of the offspring and leaves of the corresponding wavelet trees, but in the MQD map maximum quantization level of the total wavelet tree is stored in the node. As the quantization levels are much smaller numbers than the coefficients, much of the memory usage is reduced. In the figure shown below, m i,j is denoted as the MQD node, which is the maximum quantization level of all the descendents of the wavelet coefficient of coordinate denoted as c i,j, where (i,j) is assumed to be in second level or higher level subbands. Thus, the MQD map is 1/4 th of the size of the wavelet coefficients matrix. Thus, the scanning procedure of the coefficients is reduced 1/4 th times, thus reducing the computation and the memory used to store all the nodes and descendents. 21

30 Figure 2.21 An example of an BCWT coding unit U(i,j) [1] The mechanism of Backward Coding of Wavelet Trees: The wavelet trees and their corresponding MQD map nodes are called BCWT coding units. BCWT coding unit spans only three levels with twenty wavelet coefficients and five MQD nodes. In the previous forward coding techniques, the wavelet tree must be recursively scanned and hence the whole tree should be kept in memory, but in BCWT backward coding only three levels of units are required and once the tree is scanned then much of the information is no longer needed in the memory and hence it can be removed. The basic outline of the operations of a BCWT coding unit is as follows: 1) q O(i,j), q L(i,j) and m i,j values are calculated. 22

31 2) The four offspring coefficients at O(2i,2j) are encoded. 3) The difference q L(i,j) -m 2i,2j is encoded. The steps 2 and 3 are repeated for O(2i, 2j+1), O(2i+1,2j), O(2i+1,2j+1) offspring coefficients. 4) The difference m i,j - q L(i,j) is encoded. The BCWT coding starts from the third level, and if all the coefficients in that level are completed then it moves to one level up until the top level is reached. The coefficients in the LL band are coded, and after all the coefficients are encoded the encoded output bit stream can be reversed, or read from the end of the bit stream. The decoding process is exactly reverse of the encoding process, and the MQD maps are rebuilt from the highest level to the lowest level. If we assume U(i,j) as one BCWT coding unit then B(x) is binary code of [x] and T(n) will be the binary code with only single one at a place (the right most bit n 0) eg: and as the binary code starting with n th and ending with m th rightmost bit eg:. When the coding unit U(i,j) is present in the third level subband, then there is an extra step of calculating the MQD nodes as the coding unit contains four MQD nodes which are calculated by its children. But when it comes to a situation of coding level 3 unit then the four MQD nodes are not needed and they are removed and so the MQD map size shrinks to 1/16 th of the wavelet coefficient matrix size and not 1/4 th size as it normally is. The coding of BCWT unit U(i,j): If (i,j) coordinates are in the level 3 subband, then the MQD nodes of level 2 are calculated, then the maximum quantization level of all the leaves is calculated, and if the leaves are found significant, then the coefficients in the leaves in four groups are encoded. If the coefficient is significant, then its sign is kept at the output. The coefficient 23

32 is encoded and kept at the output. The quantization difference between this group and leaves are encoded and kept at output. The maximum quantization level of the descendents are calculated, and if the descendents are found significant, then the quantization level difference between leaves and the descendents are encoded and kept at the output. Progressive coding of the algorithm: The figure below shows the output bit stream of the BCWT algorithm before bit reversing. It is resolution scalable because the bits in the bit stream are in the ascending order of coding units that is sorted from the lowest level coding units to the highest level coding units. Thus the progression of resolution is achieved. Figure 2.22 An example output bit stream of an image [1] After the encoding is done, the bit stream is reversed, and the user can stop the decoding at any higher level unit and reconstruct the smaller version of the original image if the decoding is resumed and decoded until the full resolution can be achieved. The encoder can be adjusted to provide any kind of progression types, such as progression of quality, progression of position etc. The bits in the bit stream are placed 24

33 such that different quantization levels are stored in different streams instead of placing the bits sequentially into single stream. All the different streams are chained together in the order of the quantization levels and the bit stream is reversed then the progression of quality is achieved. The progression of position is achieved such that user can arrange the order of the encoding of coding units corresponding to the spatial location i.e., the coding unit having same spatial location is encoded together and hence the progression of position can be achieved. As explained above the BCWT compression algorithm is more efficient than EZW and SPIHT. And so BCWT is added to the wavelet toolbox and to the command line function that are used for image compression in MATLAB. 25

34 CHAPTER III MATLAB WAVELET TOOLBOX MATLAB provides different tools and techniques for the processing of images such as the Image Processing Toolbox, the Wavelet Toolbox, etc. [1]. The Wavelet Toolbox provides different wavelets and wavelet packages that can be applied to user defined data. The built-in functions of the Wavelet Toolbox can be split into two different categories: command line functions, which can be accessed by typing the particular function name in the MATLAB Command Window or in an m-file, and Graphical Interactive tools consisting of a collection of GUI (Graphical User Interface) tools that can be accessed by typing wavemenu in the MATLAB Command Window. There are different m-files defined for the particular command line functions and the GUI tools and also for the functions they call. The Wavelet Toolbox has different GUI tools and those can be accessed for different wavelet applications. It is shown in Figure3.1. It provides different tools such as 1D to 3D wavelet transform, denoising, wavelet packet analysis, image fusion, regression, density estimations, signal extensions, compression etc. 26

35 Figure 3.1 wavemenu 3.1 EXISTING IMAGE COMPRESSION COMMAND LINE FUNCTIONS The command line function for image compression is wcompress. This command line function performs either compression or decompression of gray scale or true color images depending on the options chosen. 27

36 Syntax[5]: wcompress(option, varargin); wcompress('c', X, SAV_FILENAME, COMP_METHOD); wcompress('c', FILENAME,...); wcompress('c', I,...); wcompress(...,'parname1',parval1,'parname2',parval2,...); [COMPRAT,BPP] = wcompress('c',...); XC = wcompress('u', SAV_FILENAME); XC = wcompress('u', SAV_FILENAME, 'plot'); XC = wcompress('u', SAV_FILENAME, 'step'); There are two inputs for the function wcompress: option and varargin. The option input accepts either c for compression or u for decompression. The varargin input accepts a variable number of inputs such as input image, save_filename, compression method, level, wavelet name etc. The input image argument can be given as user defined input in single quotation marks, e.g. lena.png or lena.bmp. The dimensions of the image should strictly be a power of two as the DWT extension mode uses periodic extension. The save_filename is the name under which the user wants to save the compressed image data. The user can give any filename of interest with the extension.wtc such as ouput.wtc. Thus save_filename contains the compressed data which may be a 2D array that contains a grayscale image or a 3D array of uint8 containing a true color image. The compression methods are available in the Wavelet Toolbox, so that the user can select depending on the application. The compression methods that are readily available in the toolbox are Embedded Zerotree Wavelet (EZW), Set Partitioning of Hierarchial Trees (SPIHT), Spatial-orientation Tree Wavelet (STW), Wavelet Difference Reduction 28

37 (WDR), Adaptively Scanned Wavelet Difference Reduction (ASWDR), Set Partitioning in Hierarchical Trees 3D for true color images (SPIHT_3d), Subband thresholding of coefficients and Huffman encoding (LVL_MMC), Global thresholding of coefficients and fixed encoding (GBL_MMC_F) and Global thresholding of coefficients and Huffman encoding (GBL_MMC_H). The supported compression methods are broadly classified into three categories: EZW, SPIHT, STW, WDR, ASWDR and SPIHT_3d fall under PCSM (Progressive Coding Significant Methods); LVL_MMC belongs to the Coefficient Thresholding Method-1 (CTM-1), which is based on subband thresholding of the coefficients and Huffman encoding; GBL_MMC_F, GBL_MMC_H belong to the Coefficient Thresholding Method-2 (CTM-2). GBL_MMC_F uses global thresholding and then fixed encoding. GBL_MMC_H use the global thresholding and then Huffman encoding. The compression name is chosen in single quotation such as ezw, spiht, stw, etc. Additional parameters than can be given as input for the function, such as level, wname, it, cc. These parameters are set to default values if not provided by the user. The parameters and the values are given as input as: wcompress(...,'parname1',parval1,'parname2',parval2,...); The parameter level ranges from 1 to levmax depending on the level of compression the user desires. The levmax parameter depends on the size of the image. The wname parameter allows the user to choose different wavelets such as Haar, Daubechies, Biorthogonal, Symlets, Coieflets, Morlet, Mexican Hat, Meyer, or other real wavelets and complex wavelets. The wavelets are given as inputs in single quotation and the available wavelets are haar, db1 through db10, sym2 through sym8, coif1 through coif5, bior1.1 through bior6.8, rbio1.1 through rbio6.8 and dmey. The type of image transform is set by the parameter it or IT. The option n corresponds to no transform and the true color or the gray scale is automatically detected. 29

38 Option g selects the gray scale transformation of the image, option c is for the color transformation of the given image. The parameter cc or CC is the color conversion parameter. If its value is set to rgb or none, then there is no conversion involved which is default. If the value is set to yuv then it is YUV transform, klt correspond to Karhunen-Loeve transform, yiq correspond to YIQ color transform, and xyz correspond to CIEXYZ color space transform. For the PCSM methods, the parameter maxloop decides the maximum number of steps of the compression algorithm the default parameter value is 10, and the user can select any positive integer for this parameter value. Also the parameter plotpar sets the plotting parameter. If the value of it is plot, then the decompressed image is plotted, and if the value is step, then the step by step encoding process can be observed by the user. The parameters bpp and comprat are given only for CTM-1(LVL_MMC) and CTM-2 (GBL_MMC_F, GBL_MMC_H) thresholding methods. The parameter value for bpp ranges from 0 to 8 for gray scale and 0 to 24 for true color. The parameter value for comprat ranges from 0 to 100. The parameters threshold, nbcfs, percfs are only for CTM-2 methods. The parameter value for threshold should be a positive non-negative real number. The parameter value for nbcfs is between zero and total number of wavelet coefficients. The parameter value for percfs ranges from 0 to 100. The parameters can be given in any order, but care should be taken such that the particular parameter value should follow its corresponding parameter name. An example of image compression for a lena image using spiht method for level 3 and for a wavelet db2 can be implemented through the command line wcompress by the following syntax. 30

39 wcompress ( c, lena.png, output.wtc, spiht, wname, db2, level,3); The compression ratio (comprat) and the bits per pixel (bpp) values are returned by wcompress if used as follows: [COMPRAT, BPP] = wcompress ( c...); The compression ratio and bpp are calculated when the wcompress.m function calls wtcmngr.m for case save. The parameters which are returned by encoding are saved here for further use in the decoding in a file with a filename specified by the user. The wtcmngr.m function returns the filesize of the saved file. The compression ratio and bpp are calculated by the following equations: Compression ratio = (100 * filesize) / (number of elements in the image) Bpp = (8 * filesize * size(image,3)) / (number of elements in the image) For decompression of the compressed image, option u is used, and the input varargin is save_filename, which is given as input by the user during the compression implementation. For wcompress('u', SAV_FILENAME, 'plot') it plots the uncompressed image for PCSM methods when the parameter step is used, then the step by step decompression can be observed such as for wcompress('u', SAV_FILENAME, 'step'). For the same example, decompression is done by the command line as follows wcompress ( u, output.wtc ); wcompress ( u, output.wtc, plot ); wcompress ( u, output.wtc, step ); 31

40 More information on wcompress can be found in MATLAB Wavelet Toolbox documentation [1]. 3.2 FUNCTION CALLS The function wcompress.m calls wtcmngr.m and getcompresspar.m functions for different cases and wtcmngr.m calls the function wtc_xxxx.m (where xxxx can be any compression method which are available) in which four functions are defined to perform encoding, decoding, saving and loading. The getcompresspar function is called to load the approximate compression ratio whenever needed. Figure 3.2 shows the block diagram and the function call procedure, which is identical for all the compression methods that are available in wcompress command line function. 32

41 Figure 3.2 Block diagram of function calls for wcompress Compression can be directly accessed by typing the following command line in the Command Window of MATLAB or from an m-file. wcompress ( c, X, Save_filename, Comp_Method,.); As discussed in Section 3.1, the different additional parameters such as level, wavelet name, etc., can be given by the user as desired. If they are not specified by the user then default values are set to the variables that are necessary for compression in wcompress.m. wcompress.m calls wtcmngr.m for the case of encoding. wtcmngr.m itself has different cases such as 'meth_fam', 'meth_ident', 'meth_name', 'enc', 'write', 'dec', 'read', 'save', 'load' and a function getfunname. 'meth_fam' detects if the method of compression specified by the user is a PCSM 33

42 method or a CTM method. The case 'meth_ident' assigns a particular identity for all the compression methods available and hence when the user wants a particular compression method then its identity is identified and the user desired compression method is accessed. The case meth_name finds the compression method name from the particular identity. The function getfunname calls the cases 'meth_fam', 'meth_ident', 'meth_name' and does the method identification with a particular codeid (code identity which is an alphabet). When the function wtcmngr.m for case enc is called by wcompress.m, it takes the compression method specified by the user and calls the function getfunname. The function getfunname finds the actual compression method by matching the name given by the user with the already existing ones and assigns a code ID to the method. This procedure is done in meth_name and meth_ident and the function handles of the desired compression algorithm are called here. In the encode case, the function handles are called for enc (encoding) case. Thus, the encoding output of wtc_xxxx, which is a structure, is given as output to wcompress.m. Then wcompress.m function calls wtcmngr.m for the case save. It takes the filename and the compression method given by the user. Then again getfunname is called, which checks the compression method name and calls the function handles of wtc_xxxx. The structure which is the returned from encoding is saved by calling the function wtc_xxxx for save through function handles. Thus all the structure variables are stored in the file name given by the user and the file size is returned to wcompress.m. For decompression, wcompress.m function is accessed for case u by typing the following command line in the command window or from an m-file. wcompress ( u, Save_filename,..); The plot parameter and the step flag can be set by the user by giving additional input to the command line as discussed in Section 3.1.The function wcompress.m calls 34

43 wtcmngr.m for the case read. In the case read, the filename given by the user is taken, and it calls wtcmngr.m for case load, which returns the compression method and the structure which is stored in the previously saved file to the read case. Then it calls getfunname and matches the compression method and calls the particular function handle. Then for the dec case, the wtc_xxxx is called, and the decoded output which is an image, is returned to wcompress.m. 3.3 EXISTING GUI TOOLBOX The GUI tool used for image compression is accessed by hitting the true compression 2D tab in the main menu display of the wavemenu, see figure xxx. It can also be directly accessed by typing the wc2dtool command in the MATLAB Command Window or any m-file. First, the image is wavelet transformed according to the parameters given as input by the user, and then the compression is performed by selecting one of the compression methods. Supported image formats such as JPEG, BMP, PNG etc., can be loaded into the toolbox by following the path File Load Image Matlab Supported Formats. This directs to the current MATLAB directory, and from there an image can be loaded. Then the wavelet and the level parameters are chosen, and when the Decompose push button is hit, the wavelet transformed image appears as shown in figure

44 Figure 3.3 Wavelet Transform using 'wc2dtool' The normalized histogram of the original image and the histogram of the wavelet coefficients are shown in separate figures. The number of wavelet coefficients decomposed which are calculated is shown in Nb.Cfs box as shown in the figure3.3. The compression methods that are available for wcompress.m are also available in this wc2dtool toolbox. The desired compression method is chosen from the Compression Method tab and the Compress button is hit. Then the decompressed image and the compressed image appear. 36

45 For example, for SPIHT compression method, level=3, wavelet = db2, the results are shown in Figure 3.4. Figure 3.4 Compression using 'wc2dtool' The parameters M.S.E (Mean Squared Error), Max Error, L2 Norm Ratio, P.S.N.R (Peak Signal-to-Noise Ratio), B.P.P, Comp Ratio (Compression Ratio) measure the quality of the decompressed image and the achieved compression rate. The compressed image s normalized histogram is plotted in next to the original image histogram. 37

46 3.4 FUNCTION CALLS FOR GUI WAVELET TOOLBOX The toolbox parameters and the procedure is defined in the main function wc2dtool.m. All the push buttons, pop-up menus, sliders are programmed in this main function. There are four folders in the Matlab Wavelet Toolbox, compression, wavelet, wavedemo and multisig1d. The folder compression contains the main function wc2dtool.m and other functions that are defined for the different compression methods such as wtc_ezw.m for EZW compression, wtc_spiht.m for SPIHT compression etc. The main function wc2dtool.m calls various functions from the wavelet folder, which is outside the compression folder. Step-1: Wavelet Transform The image of Supported Matlab formats such as JPEG, PNG, BMP etc., is loaded into the Wavelet Toolbox. The Decompose button shown in Figure 3.3, then the toolbox calls the function pus_decompose_callback which is defined in the main function wc2dtool.m. The function pus_decompose_callback function calls the function cbanapar.m and from there it loads the parameters wname and level. The cbanapar.m function calls the function utanapar.m that detects the parameters that are given by the user using the popup menus named Wavelet and Level. Thus the parameters wname or wavelet name and level which are given by the user are loaded. The original image handles are loaded using the function get_original_image which is defined in the main function wc2dtool.m and then the function pus_decompose_callback gets the color type of the color of the image using the function get_color_type and then color conversion is done using the function wimgcolconv.m. The original wavelet tree is constructed for the original image X using the function wdectree.m for the parameters wname and 38

47 level ( DEPTH ) given by the user. DIMDATA is the dimension of the image. Syntax: T = wdectree(x, DIMDATA, DEPTH, WNAME); Then the function wd2uiorui2d.m is called, and the wavelet tree is converted from double to uint8, and the function imagesc scales the data and displays the image. The function wguiutils sets the axes title as Original Decomposition. Next, function show_hist is called, and the normalized histogram of the original image and the truncated histogram of the wavelet coefficients are plotted. Then the function Comp_Results_Callback is called which gets the compression parameters such as No. of loops, bpp, compression ratio using the function getcompresspar.m. The function getcompresspar.m loads the parameters from the.mat files exp_comprat_xxxx_bw.mat and exp_comprat_xxxx_col.mat., where xxxx can be any compression method name such as ezw, spiht, LVL_MMC available in toolbox e.g. exp_comprat_ezw_bw.mat, exp_comprat_ezw_col.mat etc. 39

48 Figure 3.5 shows all the function calls when Decompose push button is pushed. Figure 3.5 Block Diagram for Wavelet Decomposition Using 'wc2dtool' Step-2: Selecting the compression Method When the compression method is selected the function pop_method_callback is called and then the parameters, which are to be set for the particular compression method, selected to be visible and the remaining parameters, which are not important for the compression method, selected are set to invisible i.e. visibility is set to off for those parameters. Then the default parameters for the particular parameters are set. Step-3: Compression When the Compression button is hit, then the function wc2dtool.m is called, then the case for compression method chosen. Then the parameters wname and level are loaded using the function cbanapar.m, which uses utanapar.m to read the input given by the user has given earlier. Then the 40

49 original image handles are loaded using the function get_original_image. The parameters Maxloop, StepFlag and currentaxes are loaded for the PCSM methods, which were discussed in Section 3.2. It calls the function wtcmngr.m for the case enc, and from here the procedure for the command line and the GUI toolbox are the same. wtcmngr.m calls the function wtc_xxxx where xxxx is the compression method selected and this function, wtc_xxxx have the encode, save, load and decode functions defined in it. Here it calls the wtc_xxxx function for the encode function and the encoded bitstream is the output. The encode function of wtc_xxxx also returns the compressed image, which is displayed with a title Compressed Image. The PSNR and MSE parameters are calculated by calling the function show_psnr_and_mse. This function calls the function psnr_mse_maxerr.m, which actually calculates the psnr and mse using the decoded image and the original image handles. The decomposition of the decoded image is calculated using the function wdectree.m and the axis title is set as Decomposition of the Compressed Image. Then all the parameters are saved by calling the function wtcmngr for case save. The function call procedure and the mechanism is similar to the command line function calls the wtcmngr for the save case, which is described in Section 3.2. The compression ratio and bpp are calculated by calling the function wcomp_img_info. P.S.N.R, M.S.E, Max Error and L2-Norm Ratio are calculated using the function psnr_mse_maxerr. The Figure 3.6 shows the Block diagram of function calls when the Compression push button is hit. 41

50 Compression Block diagram: Figure 3.6 Block Diagram of Compression using 'wc2dtool' Thus the image compression is performed for the existing compression algorithms in GUI wavelet toolbox. 42

51 CHAPTER IV COMMAND LINE FUNCTIONS FOR BCWT BCWT is added as an image compression algorithm option to the command line function wcompress.m and is accessed in the same way as the already existing compression techniques. The function wtc_bcwt.m is added to the existing set of functions in the wavelet compression toolbox. The basic block diagram of the compression and decompression using command wcompress.m is shown in Figure 3.2. The top-level m file calls the wcompress.m function for option c for compression. For decompression the top level m file calls the wcompress.m function for the option u. After the top level m file calls function wcompress.m, the same function call procedure repeats for the added function wtc_bcwt.m as for the other compression algorithms explained in Section 3.1. All the options, varargin inputs and the parameters are same for the BCWT as defined for the other compression algorithms in the Section 3.1, except that the variable q_min is added to the input parameters. Figure 4.1 shows the modified block diagram of function calls for added BCWT method. The red block in Figure 4.1 indicates the m- files or the function that had to be added to the existing ones in order to implement BCWT. The blocks with red borders but black names denote the function for which a BCWT option had to be added to the other compression methods or some required modifications were done to the function to make it work for BCWT. For BCWT the handles loaded are from wtc_bcwt.m. 43

52 Figure 4.1 Block Diagram of Compression using 'wcompress' 4.1 NEW WTC_BCWT FUNCTION wtc_bcwt.m function calls four functions defined internally such as encode, decode, save, and load. The function wtc_bcwt.m is called for these four functions separately whenever needed by wtcmngr.m. The function calls block diagram are shown in figure 4.2. Encoding: The encode function loads all the inputs given into the parameters such as image, wname, level, modedwt, Coltype, stepflag 44

53 and q_min. If the parameters are not given by the user, then the default values are loaded into the parameters. At first, the wavelet transform is applied to the image using wavedec2.m, and then the BCWT algorithm is performed. An index matrix is built to encode the coefficients. The function BCWT_Unit.m takes the wavelet coefficient matrix, index matrix coordinates, q_min and level as inputs and extracts the offsprings of the particular wavelet coefficient at index matrix coordinate and computes the quantization values of the offsprings and leaves and finds the maximum value and the MQD_node is returned. The maximum of all the offspring and leaves are compared with q_min, and if it is greater then it is encoded. Thus the MQD_node and the encoded bitstream of the coefficient are returned to main function. The function offspring calculates the offspring of the particular wavelet coefficient and the coordinates of the offspring. The function bin_section converts the decimal number to binary number to store the encoded coefficients as bitstreams. The sign_out function gives the sign bit as output. If the coefficient is greater than zero (i.e., positive) then the output bit is 1, and if the coefficient is less than zero (i.e., negative) then it gives 0 as output. The trun function truncates the output bitstream depending on the two mark inputs given as input. The Encode_4_group function encodes the offspring coefficients if they are found significant, and the bitstream is produced as output. Finally the get_mqd_node function calculates the maximum MQD node for the offspring. When the encoding is completed, the BCWT algorithm normally creates a header bit stream for all the variables. Here, however all 'the parameters needed further are loaded into structures, and the whole structure is returned as WTC_Struct to wtcmngr.m. 45

54 Saving: wtc_bcwt.m is called by wtcmngr.m. After encoding, the variables returned by the encoding function are saved using a Matlab built-in function fwrite. Here wtc_bcwt encoding function returns Row, Column, q_min, q_max, bit stream, level, wname (wavelet name). At first a temporary file is created for the storage of variables. Then the variables are written into the file using fwrite function in Matlab. fwrite (fileid, A, precision); This statement writes the variable A to the file giving a file identifier (fileid) with given precision. If the precision is not specified then it takes default precision as uint8. Thus all the variables needed for decoding are stored. Loading: The Load function at first loads the temporary file that is created in the save function using fopen: fileid = fopen (filename); This function opens the file with specified in filename and returns a file identifier fileid. It loads all the parameters into the variables in the same order they are written in the save function with the help of the fileid using function fread. A = fread (fileid, sizea, precision); A is the variable that is written into the file in the save function. The fread function loads the variable A by finding the fileid of that particular variable 46

55 and matches the size of that variable and the precision. The same precision that is used during the saving of the variable previously in save function should be used here for that particular variable. fileid = fopen (filename, permission, machineformat); Decoding: The Decode function loads all values of the parameters using the structures that are loaded in the load function. Then, BCWT decoding is done to the bitstream. Next the bitstream is subjected to inverse wavelet transform using waverec2.m. The decoded image is given as output of the function to wtcmngr.m, which gives the output to wcompress.m. As explained above the total red block indicate the functions in Figure 4.2 that are written separately and the borders of the blocks that are highlighted by red indicate that some required modifications are done in the already existing functions or.m files. The borders of the blocks that are highlighted in blue indicate the previous research work, which is taken from Jiangling Guo and Enrique Corona, Backward Coding of Wavelet Tree implementation [2]. As shown in the block diagram, the functions encode and decode are such that changes are made for the function to work for the BCWT option. The wtc_bcwt.m function is written by taking the other existing function wtc_spiht.m as reference. The encode and decode functions are organized similarly to the wtc_spiht.m function but the encoding and decoding procedure is modified for the BCWT option. The reason the save and load functions for BCWT are different from SPIHT is that the bit signs and the original coefficient bitstream are separately loaded into the parameters, and the bitstreams are later concatenated into a single bitstream in the save 47

56 function. In the BCWT option, the signs of the coefficients and the encoded coefficient bitstreams are concatenated in the encode function, and the total bitstream is loaded into WTC_Struct.bitstream parameter. Hence, it is sufficient to save the parameter WTC_Struct.bitstream instead of different bitstreams for signs and coefficients. Also, the parameters that are required for the BCWT are different from SPIHT such as maxloop, q_min etc. The block diagram of wtc_bcwt m file is shown below. Figure 4.2 Block Diagram of function 'wtc_bcwt' Thus the wcompress command line function and the built-in functions are modified such that it works for the BCWT Compression algorithm. 48

57 CHAPTER V GUI WAVELET TOOLBOX FOR BCWT The image compression procedure using GUI wavelet toolbox in detail is explained in Chapter II. The toolbox function wc2dtool.m is modified and BCWT is added to the existing compression algorithms in the toolbox. The wavelet toolbox for BCWT option: The function call procedure and the mechanism of the functions are similar to the existing procedure, which is defined in Section 3.4, but the BCWT compression method is added to the other compression methods. The extra parameter q_min, the minimum quantization level, is added to the existing parameters such as wname, level, etc. The parameters that are used by the other PCSM wavelet based compression algorithms are set as default for the BCWT option. For the BCWT option, the parameter, q_min is used as threshold for the wavelet coefficients and hence it has to be added to the inputs for compression. Step 1: Wavelet Decomposition The wavelet decomposition procedure is similar to the procedure defined for other compression algorithms. All the parameters required for the decomposition are loaded, and the wavelet decomposition is done in a similar manner as for the other compression methods. The function Comp_Results_Callback uses the.mat files exp_comprat_bcwt_bw.mat and exp_comprat_bcwt_col.mat to load the estimated compression ratios. The Figure 5.1 shows the block diagram for wavelet transform when Decompose button is hit in the toolbox. 49

58 Block Diagram for Wavelet Decomposition: Figure 5.1 Block Diagram of Wavelet Decomposition for 'BCWT' Step 2: Selecting the compression method When the compression method BCWT is selected, then the pop-up menu No. of encoding loops is set to off i.e., it becomes invisible, and the parameter q_min gets visible. Step 3: Compression All the parameters are set as defined by the user such as wname, level using cbanapar.m and utanapar.m functions. The function Pop_qmin_Callback defines the q_min parameter. The q_min parameter ranges from 0 to 7 and when the user enters the desired minimum quantization level it reads into the options and given as input to the wtcmngr.m function. The function calls procedure and the mechanism of the functions is the same as for the command line function shown in figure

59 The Figure 5.2 shows the block diagram for BCWT option for compression. Block Diagram for Compression using option BCWT : Figure 5.2 Block Diagram of Compression for 'BCWT' The function wtc_bcwt.m used here is the same function that is used in the command line function. Thus the block diagram that is shown in figure xxx and the function call procedure are same, which is explained in detail in Chapter 4. 51

60 CHAPTER VI RESULTS AND CONCLUSION The results for the command line function and the GUI wavelet toolbox are shown. The result comparison for the compression algorithms SPIHT and BCWT is also performed. 6.1 RESULTS FOR COMMAND LINE FUNCTION WCOMPRESS The results for the command line function wcompress.m for the option BCWT are observed for different images by changing the parameters. The parameters for the BCWT are wname, level and q_min where wname is the wavelet name, level is the level of transform and coding the user desires to be done, and q_min is the minimum quantization level. For the simulations in this chapter, the input parameters take the following ranges and inputs: Image : The GUI toolbox and the command line function works only for images of size as powers of two, such as 512x512, 256x256, 128x128 etc. The different supported formats of images are JPEG /(Joint Photographic Experts Group), BMP /(Windows Bitmap), PNG /(Portable Network Graphics), HDF /(Hierarchical Data Format), TIFF /(Tagged Image File Format), GIF /(Graphical Interchange Format), PCX /(Windows Paintbrush), PGM /(portable Gray Map), PBM /(Portable Bitmap), PPM /(Portable Pixmap). wname : The parameter wname is the wavelet name which decides the wavelet that will be used for the wavelet transform of the image. BCWT compression algorithm, as proposed by Jiangling Guo works for the CDF 9/7 wavelet transform. Here, it 52

61 works for all the wavelets which are available in the wavelet toolbox, such as Haar, Daubechies, Biorthogonal, Symlets, Coieflets, Morlet, Mexican Hat, Meyer, or other real wavelets and complex wavelets, as explained in chapter 3.1. Level : Level parameter sets the number of levels for the wavelet transform when the coding is performed. The parameter Level can vary from 1 to 9. q_min : The minimum quantization level parameter, q_min, can range from 0 to 7. If the above parameters are not set by the user, then the default values are loaded. The default values are as follows: wname is bior4.4, Level =9, q_min =3. The following results are observed by gradually increasing the q_min, the minimum quantization level that is the threshold for the coefficients. The results plotted for riginal image lena.bmp image of size 512x512. Figure 6.1 Original Image of lena.bmp 53

62 wname = db2, Level = 3, q_min=3 Figure 6.2 Decompressed Image (q_min=3) Figure 6.3 Difference Image (q_min=3) The compression ratio is The bpp (bits per pixel) is For the parameters, wname= db2, Level=3, q_min=5 Figure 6.4 Decompressed Image (q_min=5) Figure 6.5 Difference Image (q_min=5) 54

63 The compression ratio is The bpp (bits per pixel) is For the parameters wname = db2, Level=3, q_min=7 Figure 6.6 Decompressed Image (q_min=7) Figure 6.7 Difference Image (q_min=7) The compression ratio is The bpp (bits per pixel) is Comparison and explanation of the results for different q_min: If X is the original image and XC is the approximated or the decompressed image, then the difference image is X D =X-XC. The performance of the compression algorithm can be known by observing the difference image as it shows how well the original image is reconstructed. The compression results are observed by increasing the minimum quantization level gradually, because as discussed above, the minimum quantization level acts like a threshold for the BCWT wavelet coefficients, and hence when the q_min increases, the threshold increases, and more compression is done with fewer coefficients. At first, compression is done for the image lena.bmp for the parameters wname = db2, level =3, q_min =3, which is shown in the figure xxx, and the difference image is 55

64 observed. We can see that the difference image is almost invisible, and hence the original image is reconstructed well. When the compression results are observed by increasing the q_min to 5, then more compression is done as the threshold increases. The decompressed image and the difference image for case q_min =5 can be seen in figure xxx and figure xxx respectively. The difference image is observed clearly when compared with the previous case. When the q_min is further increased to 7, then more compression is done, and thus the reconstructed image which is shown in figure xxx is much different from the original one. Here in this case the difference image can be observed clearly than for the previous cases of q_min in figure xxx. The compression ratio and bpp decrease as the q_min increase. Hence more compression is done, and thus the reconstructed image is poor. The compression results are observed by increasing the number of levels of decomposition. All the parameters are set as follows: wname= db2, Level=5, q_min=5 Figure 6.8 Decompressed Image (Level=5) Figure 6.9 Difference Image (Level=5) The compression ratio is The bpp (bits per pixel) is

65 For wname= db2, Level=7, q_min=5 Figure 6.10 Decompressed Image (Level=7) Figure 6.11 Difference Image (Level=7) The compression ratio is The bpp (bits per pixel) is Compression results for different images of different size: The results for BCWT for baboon.png of size 512x512 wname = db2, Level=3, q_min=3 Figure 6.12 Original Image(baboon.png) Figure 6.13 Decompressed Image(baboon.png) 57

66 For baboon.png, the Compression ratio is and the bpp is The results for BCWT for house_profile.png of size 256x256: wname = db2, Level=3, q_min=3 Figure 6.14 Original Image (house_profile.png) Figure 6.15 Decompressed Image (house_profile.png) The Compression ratio is The bpp is

67 6.1 COMPARISION OF BCWT WITH SPIHT The results for SPIHT for the following parameters are as shown for the parameters wname = bior2.4, Level=3 Figure 6.16 Original Image of lena.png Figure 6.17 Decompressed Image for SPIHT The compression ratio is The bpp (bits per pixel) is

68 The results for BCWT for the following parameters: wname = bior2.4, Level=3, q_min=1 Figure 6.18 Original Image of lena.png Figure 6.19 Decompressed Image for BCWT ( q_min=1) The compression ratio is The bpp (bits per pixel) is

69 The results for BCWT for the following parameters: wname = bior2.4, Level=3, q_min=2 Figure 6.20 Original Image of lena.png Figure 6.21 Decompressed Image for BCWT (q_min=2) The compression ratio is The bpp (bits per pixel) is

70 6.2 RESULTS FOR GUI WAVELET TOOLBOX The results for BCWT for the parameters q_min =3; level =3; wavelet = haar Figure 6.22 Results of GUI Wavelet Toolbox for BCWT 62

71 The results for BCWT for parameters q_min=3, Level=3, wavelet = haar Figure 6.23 Results of GUI Wavelet Toolbox for BCWT for q_min=3 Compression ratio: 15.81% BPP (Bits Per Pixel): P.S.N.R (Peak Signal To Noise Ratio) = L2- Norm Ratio: 99.98% Max Error (Maximum Error): 14 M.S.E (Mean Square error):

72 For parameters q_min =7, wname = haar, level =3 Figure 6.24 Results for the GUI Wavelet Toolbox for BCWT for q_min=7 Compression ratio: 1.52% BPP: P.S.N.R= 26.07, L2- Norm Ratio: 99.55%, Max Error: 138 M.S.E:

73 Comparison between SPIHT and BCWT : Results for SPIHT wname = haar, level =3, Nb. Encoding Loops =9 Figure 6.25 Results of GUI Wavelet Toolbox for SPIHT Compression ratio: 19.23% BPP: P.S.N.R= 36.85, L2- Norm Ratio: 96.58%, Max Error: 70 M.S.E:

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