Overview

Wavelet packets are particular linear combinations or superpositions of wavelets. They have also been called arborescent wavelets [87]. They form bases which retain many of the orthogonality, smoothness, and localization properties of their parent wavelets. The coefficients in the linear combinations are computed by a factored or recursive algorithm, with the result that expansions in wavelet packet bases have low computational complexity.

A discrete wavelet packet analysis, or DWPA, is a transformation into coordinates with respect to a collection of wavelet packets, while a discrete wavelet packet transform or DWPT is just the coordinates with respect to a basis subset. Since there can be many wavelet packet bases in that collection, the DWPA is not fully specified unless we describe the chosen basis. This basis choice overhead is reduced to a manageable amount of information precisely because wavelet packets are efficiently coded combinations of wavelets, which are themselves a fixed basis requiring no description.

Going the other way, a discrete wavelet packet synthesis or DWPS takes a list of wavelet packet coefficient sequences and superposes the corresponding wavelet packets into the output sequence. If the input is a basis set of components, then DWPS reconstructs the signal perfectly.

A discrete wavelet transform is one special case of a DWPA, and an inverse discrete wavelet transform is a special case of DWPS. Thus DWPS and DWPA implementations may be used instead of the specific DWT and iDWT implementations in Chapter 6.