Signal denoising using wavelet-based methods  (Page 4/9)

Once the noise representation is identified, the removal process starts. It has been proven that a suitable strategy for noise removal consists in making the coefficients associated to the noise frequency equal to zero. This statementrepresents a global perspective for noise removal, different methods for denoising differ in the way these coefficients are tracked and taken out from the representation. The conceptual details of several of these methods are presented in the nextsections. The main reference for the methods discussed here is antoniadis2001

Before attempting to describe the methods is convenient to discuss an alternative definition for wavelet representation used for noise removal. First, the description assumes that the representation is achieved using periodised wavelets bases on $[0,1]$ . Also, the basis functions are generated by dilation and translation of a compactely supported scaling function $\phi$ , also called father wavelet and a familiar mother wavelet function, $\psi$ . $\psi$ must be associated with an r-regular multiresolution analysis of $L^2\left(\mathbb{R}\right)$ . An advantage of this approach is that generated wavelets families allow integration of different kinds of smoothness and vanishing moments. This features lead to the fact that many signals in practicecan be represented sparsely (with few wavelets coefficients) and uniquely under wavelets decomposition. The decomposition expresion using a father and a mother wavelet is depicted in Equation .

where $j_0$ is a primary resolution level, and ${\alpha }_{j_{0}k}$ and ${\beta }_{jk}$ are calculated as the inner products shown in Equations and

When the discrete wavelet transform is used, the coefficients $c_{j_{0}k}$ , discrete scaling coefficients and $d_{jk}$ , discrete wavelet coefficients are used instead of the continous parameters ${\alpha }_{j_{0}k}$ and ${\beta }_{jk}$ . The discrete parameters can be approximately calculated by applying a $\sqrt{n}$ factor to the continous coefficients.

Finally, when the DWT is applied to Equation , these expressions are obtained (Equations and ):

Classical approach to wavelet thresholding

The original and simpler way to remove noise from a contaminated signal consists in modifying the wavelets coefficients in a smart way such that the “small” coefficients associated to the noise are basically neglected. The updatedcoefficients can thus be used to reconstruct the original underlying function free from the effects of noise. It is implicit in the strategy that only a few “large” wavelets coefficients $d_{jk}$ are associated with the original signal, and that their identification and elimination of any other coefficients will allow a perfect reconstruction of theunderlying signal $g$ . Several methods use this idea and implement it in different ways. When attempting to descrease the influence of noise wavelets coefficients it is possible to do it in particular ways, also the need of informationof the underlying signal leads to different statistical treatments of the available information.