Fast algorithms for discrete and continuous wavelet transforms

Several algorithms are reviewed for computing various types of wavelet transforms: the Mallat algorithm (1989), the ´a trous´ algorithm, and their generalizations by Shensa. The goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently, and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts. 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 (fast Fourier transform), 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, smaller gains are obtained: ´fast running FIR (finite impulse response) filtering´ techniques allow one to achieve typically 30% savings in computations.<>