Fast algorithms for discrete and continuous wavelet transforms

Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm, the “a trous” algorithm and their generalizations by Shensa. The goal is 1) to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, 2) to compare the various algorithms obtained and give an idea of possible gains by providing operation counts. The computational structure of the algorithms rather than the mathematical relationship between transforms and algorithms, is focused upon. Most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform. Although this general method is already efficient, it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques (such as the FFT) in a suitable manner. The modified algorithms are termed “fast” because of their ability to reduce the computational complexity per computed coefficient from L to log L (within a small constant factor) for large filter lengths L. For short filters, we obtain smaller gains: “fast running FIR filtering” techniques allow one to achieve typically 30% save in computations. This is still of practical interest when heavy computation of wavelet transforms is required, and the resulting algorithms remain easy to implement.