This example discusses the problem of signal recovery from noisy data. The general de-noising procedure involves three steps. The basic version of the procedure follows the steps described below:
Decompose: Choose a wavelet, choose a level N. Compute the wavelet decomposition of the signal at level N.
Threshold detail coefficients: For each level from 1 to N, select a threshold and apply soft thresholding to the detail coefficients.
Reconstruct: Compute wavelet reconstruction using the original approximation coefficients of level N and the modified detail coefficients of levels from 1 to N.
Two points must be addressed in particular:
how to choose the threshold,
and how to perform the thresholding.
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Thresholding can be done using the function WTHRESH which returns soft or hard thresholding of the input signal. Hard thresholding is the simplest method but soft thresholding has nice mathematical properties. Let thr denote the threshold.
thr = 0.4;
Hard thresholding can be described as the usual process of setting to zero the elements whose absolute values are lower than the threshold. The hard threshold signal is x if x>thr, and is 0 if x<=thr.
y = linspace(-1,1,100); ythard = wthresh(y,'h',thr);
Soft thresholding is an extension of hard thresholding, first setting to zero the elements whose absolute values are lower than the threshold, and then shrinking the nonzero coefficients towards 0. The soft threshold signal is sign(x)(x-thr) if x>thr and is 0 if x<=thr.
ytsoft = wthresh(y,'s',thr); subplot(1,3,1) plot(y) title('Original') subplot(1,3,2) plot(ythard) title('Hard Thresholding') subplot(1,3,3) plot(ytsoft) title('Soft Thresholding')
As can be seen in the figure above, the hard procedure creates discontinuities at x = t, while the soft procedure does not.
Recalling step 2 of the de-noise procedure, the function THSELECT performs a threshold selection, and then each level is thresholded. This second step can be done using WTHCOEF, directly handling the wavelet decomposition structure of the original signal. Four threshold selection rules are implemented in the function THSELECT. Typically it is interesting to show them in action when the input signal is a Gaussian white noise.
rng default; y = randn(1,1000);
Rule 1: Selection using principle of Stein's Unbiased Risk Estimate (SURE)
thr = thselect(y,'rigrsure')
thr = 2.0518
Rule 2: Fixed form threshold equal to sqrt(2*log(length(y)))
thr = thselect(y,'sqtwolog')
thr = 3.7169
Rule 3: Selection using a mixture of the first two options
thr = thselect(y,'heursure')
thr = 3.7169
Rule 4: Selection using minimax principle
thr = thselect(y,'minimaxi')
thr = 2.2163
Minimax and SURE threshold selection rules are more conservative and would be more convenient when small details of the signal lie near the noise range. The two other rules remove the noise more efficiently.
Let us use the "blocks" test data credited to Donoho and Johnstone as first example. Generate original signal xref and a noisy version x adding a standard Gaussian white noise.
sqrt_snr = 4; % Set signal to noise ratio init = 2055615866; % Set rand seed [xref,x] = wnoise(1,11,sqrt_snr,init);
De-noise noisy signal using soft heuristic SURE thresholding on detail coefficients obtained from the decomposition of x, at level 3 by sym8 wavelet.
scal = 'one'; % Use model assuming standard Gaussian white noise. xd = wden(x,'heursure','s',scal,3,'sym8'); Nx = length(x); subplot(3,1,1),plot(xref), xlim([1 Nx]) title('Original Signal') subplot(3,1,2),plot(x), xlim([1 Nx]) title('Noisy Signal') subplot(3,1,3),plot(xd), xlim([1 Nx]) title('De-noised Signal - Signal to noise ratio = 4')
Since only a small number of large coefficients characterize the original signal, the method performs very well.
When you suspect a nonwhite noise, thresholds must be rescaled by a level-dependent estimation of the level noise. As a second example, let us try the method on the highly perturbed part of an electrical signal. Let us use db3 wavelet and decompose at level 3. To deal with the composite noise nature, let us try a level-dependent noise size estimation.
load leleccum; indx = 2000:3450; x = leleccum(indx); % Load electrical signal and select part of it.
Find first value in order to avoid edge effects.
deb = x(1);
De-noise signal using soft fixed form thresholding and unknown noise option.
scal = 'mln'; % Use a level-dependent estimation of the level noise xd = wden(x-deb,'sqtwolog','s',scal,3,'db3')+deb; Nx = length(x); subplot(2,1,1),plot(x), xlim([1 Nx]) title('Original Signal') subplot(2,1,2),plot(xd), xlim([1 Nx]) title('De-noised Signal')
The result is quite good in spite of the time heterogeneity of the nature of the noise after and before the beginning of the sensor failure around time 2450.
The de-noising method described for the one-dimensional case applies also to images and applies well to geometrical images. The two-dimensional de-noising procedure has the same three steps and uses two-dimensional wavelet tools instead of one-dimensional ones. For the threshold selection, prod(size(y)) is used instead of length(y) if the fixed form threshold is used.
Generate a noisy image.
load woman init = 2055615866; rng(init); x = X + 15*randn(size(X));
In this case fixed form threshold is used with estimation of level noise, thresholding mode is soft and the approximation coefficients are kept.
[thr,sorh,keepapp] = ddencmp('den','wv',x); thr
thr = 107.9838
De-noise image using global thresholding option.
xd = wdencmp('gbl',x,'sym4',2,thr,sorh,keepapp); figure('Color','white') colormap(pink(255)), sm = size(map,1); image(wcodemat(X,sm)), title('Original Image') figure('Color','white') colormap(pink(255)) image(wcodemat(x,sm)), title('Noisy Image')
image(wcodemat(xd,sm)), title('De-Noised Image')
De-noising methods based on wavelet decomposition is one of the most significant applications of wavelets. More details can be found in the section of the documentation entitled "More About the Thresholding Strategies."