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.

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 .

The MRA is characterised by its scaling sequence

or, as Z-transform,

and its wavelet sequence

or

(some coefficients might be zero). Those allow to compute the wavelet coefficients, 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 .