Wavelets and their Applications

Chapter 1

A Guided Tour

1.1. Introduction

In this first chapter¹, we propose an overview with a short introduction to wavelets. We will focus on several applications with priority given to aspects related to statistics or signal and image processing. Wavelets are thus observed in action without preliminary knowledge. The chapter is organized as follows: apart from the introduction to wavelets, each section centers on a figure around which a comment is articulated.

First of all, the concept of wavelets and their capacity to describe the local behavior of signals at various time scales is presented. Discretizing time and scales, we then focus on orthonormal wavelet bases making it possible at the same time:

– to supplement the analysis of irregularities with those of local approximations;

– to organize wavelets by scale, from the finest to the coarsest;

– to define fast algorithms of linear complexity.

Next we treat concrete examples of real one-dimensional signals and then two-dimensional (images) to illustrate the three following topics:

– analysis or how to use the wavelet transform to scan the data and determine the pathways for a later stage of processing. Indeed, wavelets provide a framework for signal decomposition in the form of a sequence of signals known as approximation signals with decreasing resolution supplemented by a sequence of additional touches called details. A study of an electrical signal illustrates this point;

– denoising or estimation of functions. This ...