Wavelet Decomposition and Reconstruction Using Arbitrary Kernels: A New Approach

The discrete wavelet decomposition and reconstruction remains one of the main issues of current signal and image compression. A lot of work was done with the purpose of nding wavelet kernels that would guarantee good compression results. The question of orthogonal wavelet transform was treated in detail in 5], 4], 2] and other articles, while the biorthogonal wavelets were considered in 6], 7] and so on. Both approaches have the same drawback. They put too many restrictions on the choice of the wavelet lters. The orthogonality condition makes the lters to be either very long or having bad compression properties (regularity and the number of vanishing moments). While the biorthogonal l-ters mitigate this drawback to a certain degree, one still should go through a careful process of choosing them. In this paper we present a method that allows to obtain reconstruction lter for virtually every decomposition lter. Also we consider several compression examples using the SPIHT compression algorithm (see 3]), some of which produce results that are better than those of Daubechies 7/9 lter (which is known to be one of the best among the existing lters). The lter bank, or wavelet processing, usually involves ltering a signal with low and high pass analysis or decomposition lters, discarding every other sample on the output of each of them. The obtained lter coeecients might be processed somehow, e.g. for the purpose of compression or noise reduction, and then the original signal is supposed to be reconstructed from the coeecients by passing low and high band 2-interpolated coeecients through the correspondingly low and high pass synthesis (reconstruction) lters and summing the results. For every discrete function f we denote by a subscript l its l-shift to the right: f l k] = fk ? l]. Also for any pair of discrete complex valued functions f 1