Wavelet Toolbox

The Discrete Wavelet Transform

Calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to make our calculations?

It turns out, rather remarkably, that if we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be much more efficient and just as accurate. We obtain such an analysis from the discrete wavelet transform (DWT). For more information on DWT, see Algorithms.

An efficient way to implement this scheme using filters was developed in 1988 by Mallat (see [Mal89] in References). The Mallat algorithm is in fact a classical scheme known in the signal processing community as a two-channel subband coder (see page 1 of the book Wavelets and Filter Banks, by Strang and Nguyen [StrN96]).

This very practical filtering algorithm yields a fast wavelet transform -- a box into which a signal passes, and out of which wavelet coefficients quickly emerge. Let's examine this in more depth.

What's Continuous About the Continuous Wavelet Transform?

One-Stage Filtering: Approximations and Details

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