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wavelet analysis & data compression 279L2LLL2DataInputHH H22LHFigure 11.8 A multifrequency dyadic (power-of-2) filter tree used for discrete wavelettransformation. The L boxes represent lowpass filters, while the H boxes represent highpassfilters, each of which performs a convolution (transform). The circles containing ↓2 filter outhalf of the signal that enters them, which is called subsampling or factor-of-2 decimation.integration point. Therefore, rather than tabulate explicit wavelet functions, a set offilter coefficients is all that is needed for discrete wavelet transforms.Because each filter in Figure 11.8 changes the relative strengths of the differentfrequency components, passing the signal through a series of filters is equivalent,in the wavelet sense, to analyzing the signal at different scales. This is the originof the name “multiresolution analysis.” Figure 11.8 shows how the pyramid algorithmpasses the signal through a series of highpass filters (H) and then through aseries of lowpass filters (L). Each filter changes the scale to that of the level below.Notice too, the circles containing ↓2 in Figure 11.8. This operation filters out halfof the signal and so is called subsampling or factor-of-2 decimation. It is the waywe keep the areas of each box in Figure 11.7 constant as we vary the scale andtranslation times. We consider subsampling further when we discuss the pyramidalgorithm.In summary, the DWT process decomposes the signal into smooth informationstored in the low-frequency components and detailed information stored in the highfrequencycomponents. Because high-resolution reproductions of signals requiremore information about details than about gross shape, the pyramid algorithmis an effective way to compress data while still maintaining high resolution (weimplement compression in Appendix G). In addition, because components of differentresolution are independent of each other, it is possible to lower the number ofdata stored by systematically eliminating higher-resolution components. The useof wavelet filters builds in progressive scaling, which is particularly appropriatefor fractal-like reproductions.11.5.1 Pyramid Scheme Implementation ⊙We now wish to implement the pyramid scheme outlined in Figure 11.8. The filtersL and H will be represented by matrices, which is an approximate way to perform−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 279