Wavelets and their Applications

Chapter 2

Mathematical Framework

2.1. Introduction

Several points of view are needed to realize the interest of wavelets as a mathematical tool of function analysis and representation. Two methods are proposed in this chapter.

The first method centers on the idea of building a tool for local analysis in time. We introduce the continuous wavelet transform by showing how it solves the problems that are from this point of view raised by the Fourier transform and the sliding window Fourier transform, also called Gabor transform. In fact, the first is a global transformation and second is local, but with fixed temporal resolution. On the other hand, the wavelet transform is not just a local analysis; its temporal resolution is variable. We underline its capacity to describe the local behavior of signals on various timescales.

The second method is the inversion of analysis and the search for economic representations. These are integral transforms: they are obtained by integrating the signal multiplied by the basic analyzing functions. As for any transformation, the question of reconstruction (or inversion) arises: if g = T (f) is the transform of f, is it possible “to recover”, to reconstruct f knowing g ? In the three cases studied the answer is: yes, under appropriate conditions. In the “very good cases”, it is even possible to reconstitute f from discrete values of the transform. This has major advantages from a numerical point of view. Thus, the discrete wavelet transform is introduced. ...