6 Conclusion . . . . . . Training Set 5 Choosing the Best Transform from Multiple Classes Custom Wavelet Packet Image Compression Design

Given a training set of images, a transform coding image compression algorithm may be rationally chosen from a class of fast splitting algorithms. The choice criterion is a cost function that, when low, yields high compression ratios for transform coding image compression. The method works for wavelet packet and local trigonometric transforms and thus produces well-conditioned compression and decompression methods of complexity O(P logP) for P-pixel images. Searching for the best choice itself costs O(N P log P), where N is the number of training images. Sample image 1 Sample image 2 Sample image N S u m o f s q u a r e s 1 2 N ∑ Figure 6: A joint best basis from a class of splitting algorithms is determined by a sample set of N images. Pixel splitting Subband splitting Costs of the joint best basis from the class class 1 class 2 class n s class 1 class 2 class n p Least cost determines the winning transform Figure 7: A meta-algorithm for deciding which splitting algorithm to use with a particular class of images. Karhunen{Lo eve (KL) or principal orthogonal basis 4], which is known to be the minimizer of the number of nonnegligible amplitudes. With the constraints, whose purpose is to speed things up, the chosen transform is just an approximation to KL. There is a meta-algorithm for relaxing the constraints a bit while preserving the speed. Namely, a custom transform can be chosen by checking many classes of splitting algorithms in order to further increase the expected number of negligible coeecients. This scheme was rst proposed by Yves Meyer, and is depicted in Figure 7. At the end of each path is a cost gure, the expected number of nonnegligible coeecients for the training set of images. The path that leads to the lowest cost determines which algorithm should be used to nd the custom transform for compressing the images represented by the training set. Examples of diierent classes are the diierent subband splitting schemes associated to diierent conjugate quadrature lters ((5], Chapter 5 and Appendix C), or the adapted local trigonometric bases determined by diierent windows ((5], Chapters 3 and 4). 5 Figure 4: Division of an image into orthogonal wavelet subbands to level 5, or into the WSQ subbands. Frequencies increase down and to the right. Figure 5: Splitting schemes produces quadtrees; custom bases are determined by the …