The Mallat Algorithm

Abstract

The theory of wavelets as presented in the previous chapters gives a harmonic analysis representation of an infinite-dimensional function space (like L²(R) for instance) in terms of an infinite orthonormal basis (or tight frame in the general case). For applications of this theory to real-world situations, it is necessary to deal with suitable finite-dimensional approximations of the functions being represented and the representing wavelet basis. If we assume a sampling of a given function on a fine scale on a compact region of some space, this will give us an approximate wavelet representation at that scale. By using the Mallat transformation on these data, we can obtain a multiresolution (multiscale) representation of the sampled function on a finite number of scales from the finest to the coarsest scale, which would be of the order of the diameter of the region in question. This approximate representation of the function in terms of the sampled data is similar in spirit to the representation of a function in terms of polynomials, splines, or a finite Fourier series.