10 - Wavelet transforms

Summary

Introduction

In Chapter 9 we dealt with filter banks, which are important in several applications. In this chapter, wavelet transforms are considered. They come from the area of functional analysis and generate great interest in the signal processing community, because of their ability to represent and analyze signals with varying time and frequency resolutions. Their digital implementation can be regarded as a special case of critically decimated filter banks. Multiresolution decompositions are then presented as an application of wavelet transforms. The concepts of regularity and number of vanishing moments of a wavelet transform are then explored. Two-dimensional wavelet transforms are introduced, with emphasis on image processing. Wavelet transforms of finite-length signals are also dealt with. We wrap up the chapter with a Do-it-yourself section followed by a brief description of functions from the Matlab Wavelet Toolbox which are useful for wavelets implementation.

Wavelet transforms

Wavelet transforms are a relatively recent development in functional analysis that have attracted a great deal of attention from the signal processing community (Daubechies, 1991). The wavelet transform of a function belonging to ℒ²{ℝ}, the space of the square integrable functions, is its decomposition in a base formed by expansions, compressions, and translations of a single mother function ψ(t), called a wavelet.

The applications of wavelet transforms range from quantum physics to signal coding. It can be shown that for digital signals the wavelet transform is a special case of critically decimated filter banks (Vetterli & Herley, 1992).