The fast Haar transform

Wavelet theory and its relatives (subband coding, filter banks and multiresolution analysis) have become hot this last decade. Like the sinusoids in Fourier analysis, wavelets form bases that can decompose (analyze) and reconstruct (synthesize) signals and images. As a by-product, we obtain the accompanying processing (filtering, compression, denoising, etc.). But unlike the sinusoids, there seems to be an inexhaustible variety of different wavelet types: orthogonal, biorthogonal, spline, smooth, rough, short, long and so forth. We construct the simplest wavelet algorithm possible, called the fast Haar transform (FHT). We then indicate how this may be generalized to other fast wavelet algorithms.