Fast wavelet transform

The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets.

The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain.

This algorithm was introduced in 1989 by Stéphane Mallat.

[1] It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA).

In the terms given there, one selects a sampling scale J with sampling rate of 2J per unit interval, and projects the given signal f onto the space

; in theory by computing the scalar products where

is the scaling function of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so is the orthogonal projection or at least some good approximation of the original signal in

, at least some range k=M,...,J-1, without having to approximate the integrals in the corresponding scalar products.

Instead, one can directly, with the help of convolution and decimation operators, compute those coefficients from the first approximation

For the discrete wavelet transform (DWT), one computes recursively, starting with the coefficient sequence

In the Z-transform notation: It follows that is the orthogonal projection of the original signal f or at least of the first approximation

In the Z-transform notation: G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms" Comm.