Optimisation in Signal and Image Processing

Chapter 3

Wavelets and Fractals for Signal and Image Analysis

3.1. Introduction

The determination of singularities and of scaling laws by multifractal analysis (which uses multiresolution techniques based on the concept of wavelets) can be found in more and more applications in the fields of natural science, engineering and economics.

In this chapter we highlight the different multifractal methods by analyzing 1D and 2D generic signals; the self-similarity of fractals using wavelet transform modulus maxima has allowed scientists to determine the distribution of the singularities of the very different complex signals which can be found in the domains of material physics, biology and medicine.

The different themes in the chapter have been chosen for their relevance in relation to the further improvements and advances in scientific know-how as well as for the improvement in the quality of results that have been obtained during scientific experiments. The aim of including these themes is to help the reader better understand and apply certain purely mathematical theories which are normally difficult to carry out.

Chapter written by Abdeldjalil OUAHABI and Djedjiga AIT AOUIT.

3.2. Some general points on fractals

They exist everywhere around us. These luminous, unusual, beautiful shapes are known as fractals.

3.2.1. Fractals and paradox

It is not easy to give a correct definition of fractals. Nevertheless, in terms of etymology the word fractal leads us to the idea of fractus meaning ...