Appendix A

APPENDIX B

Appendix C

B.1. INTRODUCTION

In chapters 4, 5 and in the appendix C, the results of some tests directed at having a better comprehension of the most up to date and frequently used method of image and signal compression, the Wavelet Transform Method are illustrated. A complete explanation on wavelet transforms and especially on Discrete Wavelet Transform (DWT) and Inverse Discrete Wavelet Transform (IDWT) can be found in the first part of this project at the section 2.2.3.; in this introduction we want only to specify something regarding the development of the tests.

B.1.1. WAVELAB

All the tests explained in this appendix and in the chapters 4 and 5, are developed with Matlab 5  applications, precisely with the use of WaveLab, a toolbox developed by “Stanford University”  instead of the original Wavelet Toolbox developed by “The MathWorks” . WaveLab is actually not so complete as the Wavelet Toolbox in terms of functions and utilities, but it is freeware and it contains all the basic functions needed for the tests. There is the possibility with the functions FWT2_PO, FWT2_PBS, IWT2_PO and IWT2_PBS to make DWT and IDWT for orthogonal and biorthogonal wavelet and to build with MakeONFilter and MakeBSFilter the QMF  orthogonal and biorthogonal suitable for these transformations. These functions work only with greyscale images, u8int or double class arrays, and require images whose size is square and with dyadic length , usually 256 ´ 256 or 512 ´ 512, as the test images shown in Figure 3.2.

B.1.2. IMAGES USED

The results obtained from these tests will be extracted for each image and for the mean of the five images. The basic idea of these tests is to improve the knowledge about these subjects:

• the values of the wavelet coefficients obtained in each subband and then in every level of the decomposition as shown in Figure B.1,
• the distribution of the really interesting values, distinguishing them from the less important coefficients from an image quality point of view,
• the changing of the image quality because of the transforming of some groups of wavelet coefficients: the kind of modifications usually interesting in coding and compression techniques.

Figure B.1 : Division into Subbands and Levels within the wavelet coefficients matrix.

B.2. WAVELET COEFFICIENTS

Anytime a DWT is carried out on the 512 ´ 512 greyscale image a 512 ´ 512 matrix of double is obtained, in which every value represents the amplitude of a single WCF (wavelet coefficient). These coefficients, as explained in the section 2.2.3. are grouped together, for a 3 level DWT, in 10 groups, as in the left part of Figure B.1, and in 4 levels, as shown in the right part of the same figure:

• Groups H, I and L are 256 ´ 256 size matrices and they represent the first level of detail coefficients; each of them has 65536 coefficients for a total of 196608 coefficients in the first level group,
• Groups E,F and G are 128 ´ 128 size matrices and they represent the second level of detail coefficients; each of them has 16384 coefficients for a total of 49152 coefficients in the second level group,
• Groups B,C and D are 64 ´ 64 size matrices and they represent the third level of detail coefficients; each of them has 4096 coefficients for a total of 12288 coefficients in the third level group,
• For simplicity we talk about the Zero Level as the combination of these 3 levels, formed by 258046 coefficients, or to explain better as the big group that gathers all the detail coefficients,
• Group A is 64 ´ 64 sizes matrix and represent the third level of approximation coefficients and it is formed by 4096 coefficients.

B.3. QUADRATURE MIRROR FILTERS

B.3.1. QMF EXPLANATION

In every DWT for the subband decomposition, a particular kind of filters, as seen in section 2.2.3., is used: High and Low Pass filters; these filters are called Quadrature Mirror Filters (QMF) and are really important for this kind of transformation because through them it is possible to have a perfect reconstruction of the images after the process of the direct and inverse transformation. Wavelab uses two kind of functions to build the DWT, FWT2_PO and FWT2_PBS, and two functions for the IDWT, IWT2_PO and IWT2_PBS; actually WaveLab divides the QMF into 2 groups, orthogonal and biorthogonal filters, using these 2 functions, MakeONFilter and MakeBSFilter respectively, to build these filters; within the two groups it is possible to find further groups as explained later on. Before proceeding with the illustration of the results of the tests it is important to have a closer look at these groups of QMF so as to have a better knowledge of the filters used in the different tests.