Chapter 3

CHAPTER 4

Chapter 5

4.1. DISTRIBUTION OF WCF  AMPLITUDES

A first interesting test consists of having a look at the values of the coefficients in a general way; for this reason we have thought about the kind of histogram, shown in Figure 4.1.

4.1.1. EXPLANATION HISTOGRAMS

For this test wavelet coefficients are grouped for level, for QMF families, for image components (Y, Cb and Cr). To improve the comprehension of the diagrams and histograms, line diagrams are used instead of bar and stem diagrams. After having divided the WCF into different groups, these coefficients are sorted and the coefficients whose amplitude falls in a determinate interval are counted; this division is made according to the maximum value of the WCF of the WFC matrix.
 

	X95,100	X90,95	X10,15 - X85,90	X5,10	X0,5
3 LEVEL - Daub	0.19 - 0.25	0.36 - 0.51	0.35 - 0.52	0.38 - 0.47	91.45 - 92.75
2 LEVEL - Daub	0.36 - 0.40	0.72 - 0.78	0.68 - 0.80	0.71 - 0.77	85.73 - 86.80
1 LEVEL - Daub	0.71 - 0.73	1.38 - 1.46	1.26 - 1.46	1.23 - 1.28	74.41 - 74.81
0 LEVEL - Daub	0.62 - 0.64	1.22 - 1.28	1.12 - 1.28	1.10 - 1.14	77.53 - 77.81
0 LEVEL - Coiflet	0.64 - 0.66	1.27 - 1.29	1.13 - 1.30	1.13 - 1.14	77.17 - 77.65
0 LEVEL - Symlet	0.64 - 0.65	1.26 - 1.27	1.14 - 1.29	1.11 - 1.13	77.35 - 77.37
0 LEVEL - CDF 1.x	1.53 - 3.58	0.60 - 0.74	0.86 - 2.96	0.53 - 0.79	80.58 - 82.38
0 LEVEL - CDF 2.x	1.15 - 1.24	0.80 - 0.96	0.73 - 2.45	0.73 - 0.88	75.14 - 75.62
0 LEVEL - CDF 3.x	0.60 - 0.66	1.69 - 1.78	0.64 - 1.77	1.27 - 1.35	73.97 - 74.87

Thus in abscissa there are 20 indices XN1,N2 (with N2 = N1+5) that represent 20 intervals of 5 percentage each, in which we have linearly divided the interval {0,100}, e.g. X45,50 for interval {45,50]; in ordinate the percentage of the WCF whose values belong to that determined interval, compared to the maximum value normalised to 100 are displayed. The result are depicted in different ways: comparing the two histograms shown in Figure 4.1, on which on the right there are the values ordered linearly and on the left the same values ordered logarithmically, it is possible to see that the logarithmic diagram is preferable to show the results most clearly.
 

4.1.2. RESULTS

A first important, and expected, result is shown more or less in all the tests carried out; this result is very interesting, thinking about the various compression algorithms that will be investigated in the a following chapter. All the results show this trend: a lot of WCF, from 67 up to more than 90 percent, have amplitude values belonging to the interval between 0 and 5 percent, compared with the maximum value of the WCF. All the other WCF are almost uniformly distributed among the other 19 intervals, depending on the kind of images and QMF used; among those 19 intervals, there are usually half as many WCF with values belonging to the interval {95,100} compared with the other 18 intervals. For that reason, the explanation especially will be concentrated on the results around the values of the {0,5} interval.

4.2. ANALYSIS OF WCF AMPLITUDE STATISTICS

Another interesting test that developed on the Wavelet matrices consists of having a look at the amplitude of the detailed WCF values with respect to their percentage position in the various groups and levels.

4.2.1. DESCRIPTION OF ANALYSIS

To perform this kind of test the same 3 groups of coefficients, belonging to first, second and third level of the 512 ´ 512 DWT coefficient matrix are used; the tests are differentiated on the Y, Cb and Cr components to have a comparison among them and the same 24 QMF are used to build the DWT and the IDWT. Taking every group of WCF and making a sorting operation on their coefficients some vector of coefficients are obtained; these vectors are important only to work out some statistics. Actually in these tests we are not interested in the position that every coefficient has in the original transform space, but only in its amplitude value compared with the maximum amplitude value within the WCF matrix. All these vectors have values that are sorted in descending order, as in Figure 4.7, and they could give some useful information about the scattering of the amplitude values of the coefficients for various QMF and levels.

Given that it is difficult to make a comparison between the diagrams, because all the lines are close together and crossing, whether for the linear scale or for the logarithmic scale, 10 amplitude values belonging to the index XN shown in Table 4.3 are extracted from each of these lines: in fact a percentage is associated to the index XN, where the percentage is calculated on the total size of the vector, that is the number of coefficients in that group. A more understandable meaning of these amplitude values is explained here: if, for example, the value of X0.5 is N, 0.5 percent of the WCF of that group or level has values bigger than N and 99.5 percent have values lower than N.
 
 

Index	X1	X2	X3	X4	X5	X6	X7	X8	X9	X10
Percentage	First Value	0.1	0.2	0.5	1	2	3	4	5	10

4.2.5. Y, CB AND CR COMPONENTS

The statistical results of the tests made variously on Y, Cb and Cr and the comparison between them show that in general the values of XN are quite different for the 3 components of the images, even if the trend regarding the different QMF remains almost the same for all the values of the 3 components; also for this comparison in Table 4.7 are depicted only the results obtained with Daubechies QMF.
 

	First value	0.1 %	0.2 %	0.5 %	1 %	2 %	3 %	4 %	5 %	10 %
Y	3.34 - 4.36	1.22 - 1.49	0.96 - 1.16	0.63 - 0.74	0.44 - 0.48	0.30 - 0.32	0.24 - 0.26	0.21 - 0.23	0.19 - 0.20	0.13 - 0.14
Cb	1.00 - 1.15	0.39 - 0.41	0.33 - 0.34	0.26 - 0.27	0.22 - 0.23	0.19	0.17	0.15 - 0.16	0.14	0.11
Cr	0.98 - 1.19	0.42 - 0.43	0.38 - 0.39	0.32 - 0.33	0.28 - 0.29	0.24 - 0.25	0.22	0.20 - 0.21	0.19	0.15

A normal feature for all the 5 images and for all the QMF used, is that the values of the XN are much lower for the components Cb and Cr than for the component Y, above all for the indices between X1 and X4. To have values about 0.5 and 0.33 for the component Y X5 and X6 (1 and 2 percent) are necessary, while for the component Cr it is enough X2 and X4, 0.1 and 0.5 percent respectively, and for the Cb, X2 and X3, 0.1 and 0.2 percent. This feature suggests a possibility from a compression point of view to use the WCF of the Y component, more than the Cb and Cr components. In fact even if the results found for the Cb and Cr are sometimes quite variable, depending on the main component of the colour, as in some of the 5 images, they are always lower than the results found for the Y component; this feature shows that it is possible to achieve a better level of compression compressing a lot these two colour components compared with the luminance component, or use more bits to code the Y component than the Cb and Cr components.