Wavelets and their Applications

Chapter 7

Signal Denoising and Compression

7.1. Introduction

Two of the great successes of wavelets are signal and image denoising and compression, which are often regarded as particularly difficult problems. This chapter tries to explain the reasons for this, by focusing on one-dimensional signals.

Denoising and the estimation of functions based on wavelet representations lead to simple and powerful algorithms that are often easier to fine-tune than the traditional methods of functional estimation. Signal compression constitutes a field where wavelet methods also appear very competitive for reasons fundamentally close to those that make wavelet-band denoising work. Indeed, the signals that we are interested in have in many cases very sparse wavelet representations and are very well represented using few coefficients. Here we deal with denoising followed by signal compression, exposing the first topic in greater detail.

First of all, we tackle the principle of denoising by wavelets, then a statistical introduction to the thresholding methods and finally some examples. We initially focus on a model of form Y_t = f(t) + ε_t. This model is very simple and the assumptions about noise are very strong. Fortunately, the “attraction basin” of denoising methods is much broader. Thus, we examine two extensions of this model through examples: in the case of noise with multiple rupture charges of the variance and in that of a real signal where the noise structure is unknown.

We then go further ...