Signal denoising using wavelet-based methods  (Page 2/9)

This work presents a revision of the wavelets basic theory. Definitions of mother wavelets, wavelets decomposition and its advantage over Fourier analysis are discussed in Section 1. Section 2 presents a summary ofbasic methods developed for noise removal. Their main features and limitations are presented, and a comparison study taken into account computational efficiency is performed. Section 3 introduces an example application for denoisingmethods. A given function contaminated with Gaussian additive noise is used as testbed for the described methods. Conclusions about the performance of the denoising procedures and the utility of using wavelet decomposition for thistype of problem are presented.

Wavelet decomposition basics

Historical note

The term wavelet was first introduced by Jean Morlet while working on the analysis of signals for seismic analysis on oil-related projects. Before Morlet's work remarkable contributions were developed by Haar citep( ) and Zweig in 1975. After the work of Morlet and Grossmann on the definition of the continous wavelettransform (CWT), several developings have followed. The work of researchers as Stromberg, Duabechies, Mallat and Newland, among others, has pushed forward the theoretical frontiers of wavelets-based orthogonal decomposition and also augmentedthe scope of possible application fields.

Basic concepts

A review of basic concepts in the wavelets framework is presented in next lines. This review is based upon burrus1988.

The term wavelet is mostly used for denoting a particular wave whose energy is concentrated in a specific temporal location. A wavelet is therefore a known signal with some peculiar characteristics that allow it to beemployed for studying the properties of other signals simultaneously in the frequency and time domains. An typical plot of a wavelet is shown in

Based on a particular wavelet, it is possible to define a wavelet expansion . A wavelet expansion is the representation of a signal in terms of an orthogonal collection of real-valued functions generated by applyingsuitable transformations to the original given wavelet. These functions are called “daughter” wavelets while the original wavelet is dubbed “mother” wavelet, acknowledging its function as source of the orthogonal collection.If $f\left(t\right)$ is a given signal to be decomposed, then it is possible to represent the signal as:

In equation , ${\psi }_i\left(t\right)$ are the orthogonal basis functions, and the coefficients $a_i$ can be found through the inner product of $f\left(t\right)$ and the functions ${\psi }_i\left(t\right)$ (Equation ).

The previous equations represent the general formulation for orthogonal decomposition, and so they are the same equations discussed in the particular case of Fourier analysis. In the case of wavelets expansion, and consequently with theirdefinition, the wavelets basis functions have two integer indexes, and Equation must be rewritten as

Equation is the wavelet expansion of $f\left(t\right)$ while the collection of coefficients $a_{j,k}$ is the discrete wavelet transform (DWT) of $f\left(t\right)$ .