Wavelets and Signal Processing ECE592

Denoising Fall 89

1. General

What is Denoising. Denoising refers to manipulation of wavelet coefficients for noise reduction in which coefficient values not exceeding a carefully selected threshold level are replaced by zero followed by an inverse transform of modified coefficients to recover denoised signal. Denoising by thresholding of wavelet coefficients is therefore a nonlinear (local) operation. Thresholding can be done globally in which a single threshold level is applied across all scales, or it can be scale-dependent where each scale is treated separately. It can also be zonal in which the given function is divided into several segments (zones) and at each segment different threshold level is applied.

Thresholding modifies empirical coefficients (coefficients belonging to the given signal) in an attempt to reconstruct a replica of the true signal. Reconstruction of the signal is aimed to achieve a best estimate of the true (noise-free) signal. Best estimate is defined in accordance with a particular criteria chosen for threshold selection. Several criteria are considered for thresholding, they are discussed in the latter sections of these notes.

Mathematically, thresholding of the coefficients can also be described by a transformation of the wavelet coefficients in which transform matrix is a diagonal matrix with elements 0 or 1. Zero elements forces the corresponding coefficient below or equal to a given threshold to be set to zero while others corresponding to one, retains the coefficients as original as shown below.

(=diag((1, (2,.. (N) with (i={0,1}, i=1,N.

For noise reduction it is necessary to estimate the true uncorrupted signal from the noisy data. Therefore denoising by wavelets is considered as a nonparametric estimation problem where for the true uncorrupted signal s(t), an estimate se (t) is obtained as described below.

Hard /Soft Thresholding. Two approaches are considered for denoising:

Hard Thresholding. In hard thresholding, only those wavelet coefficients with absolute values below or at the threshold level are affected, they are replaced by zero and others are kept unchanged.

Soft Thresholding. In soft thresholding, coefficients above threshold level are also modified where they are reduced by the amount of threshold.

Mathematically hard and soft thresholding can be described as below.

Hard threshold:

wm= w if |w|th,

wm= 0 if |w|