Wavelets and their Applications

Chapter 4

Wavelet Families

4.1. Introduction

Wavelet analysis, as opposed to Fourier analysis, provides additional freedom since the choice of atoms of the transform deduced from the analyzing wavelet is left to the user. Moreover, according to the objectives of wavelet processing, we may prefer the continuous transform to the discrete transform, if the redundancy is useful for analyzing the signal. We would make the opposite choice, if we were looking for signal compression. In the latter case we must restrict ourselves to wavelets with filters, whereas in the former case almost any zero integral function is appropriate.

Since the Haar base appeared at the beginning of the last century, since renamed the Haar wavelet, passing by Gaussian Morlet wavelets, Meyer wavelets [MEY 90] (obtained using ad hoc construction) and Daubechies wavelets [DAU 88] and [DAU 92] that are the most widely used, numerous wavelets regularly appear in books and are made available in specialized software applications. Construction of new wavelets was very intense in the first ten years of their young history, but recently it has become less regular and bears on increasingly specific goals, often associated with limited application contexts. In Chapter 5 we will find some wavelet construction methods.

This chapter proposes a mini-genealogy of well-known wavelet families. Their presentation is organized according to the numerous properties that make it possible to differentiate between them and make a selection ...