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wbmpen

Penalized threshold for wavelet 1-D or 2-D denoising

Syntax

THR = wbmpen(C,L,SIGMA,ALPHA)
wbmpen(C,L,SIGMA,ALPHA,ARG)

Description

THR = wbmpen(C,L,SIGMA,ALPHA) returns global threshold THR for denoising. THR is obtained by a wavelet coefficients selection rule using a penalization method provided by Birgé-Massart.

[C,L] is the wavelet decomposition structure of the signal or image to be denoised.

SIGMA is the standard deviation of the zero mean Gaussian white noise in denoising model (see wnoisest for more information).

ALPHA is a tuning parameter for the penalty term. It must be a real number greater than 1. The sparsity of the wavelet representation of the denoised signal or image grows with ALPHA. Typically ALPHA = 2.

THR minimizes the penalized criterion given by the following:

Let t* be the minimizer of

crit(t) = -sum(c(k)^2,k≤t) + 2*SIGMA^2*t*(ALPHA + log(n/t)) 

where c(k) are the wavelet coefficients sorted in decreasing order of their absolute value and n is the number of coefficients; then THR=|c(t*)|.

wbmpen(C,L,SIGMA,ALPHA,ARG) computes the global threshold and, in addition, plots three curves:

  • 2*SIGMA^2*t*(ALPHA + log(n/t))

  • sum(c(k)^2,k¬≤t)

  • crit(t)

Examples

% Example 1: Signal denoising.
% Load noisy bumps signal.
load noisbump; x = noisbump;

% Perform a wavelet decomposition of the signal
% at level 5 using sym6.
wname = 'sym6'; lev = 5;
[c,l] = wavedec(x,lev,wname);
% Estimate the noise standard deviation from the
% detail coefficients at level 1, using wnoisest.
sigma = wnoisest(c,l,1);

% Use wbmpen for selecting global threshold  
% for signal denoising, using the tuning parameter.
alpha = 2;
thr = wbmpen(c,l,sigma,alpha)
thr =

    2.7681

% Use wdencmp for denoising the signal using the above
% threshold with soft thresholding and approximation kept.
keepapp = 1;
xd = wdencmp('gbl',c,l,wname,lev,thr,'s',keepapp);

% Plot original and denoised signals.
figure(1)
subplot(211), plot(x), title('Original signal')
subplot(212), plot(xd), title('De-noised signal') 

% Example 2: Image denoising.
% Load original image.
load noiswom; 
nbc = size(map,1);

% Perform a wavelet decomposition of the image
% at level 3 using coif2.
wname = 'coif2'; lev = 3;
[c,s] = wavedec2(X,lev,wname);

% Estimate the noise standard deviation from the
% detail coefficients at level 1.
det1 = detcoef2('compact',c,s,1);
sigma = median(abs(det1))/0.6745;

% Use wbmpen for selecting global threshold  
% for image denoising.
alpha = 1.2;
thr = wbmpen(c,l,sigma,alpha)

thr =

   36.0621

% Use wdencmp for denoising the image using the above
% thresholds with soft thresholding and approximation kept.
keepapp = 1;
xd = wdencmp('gbl',c,s,wname,lev,thr,'s',keepapp);

% Plot original and denoised images.
figure(2)
colormap(pink(nbc));
subplot(221), image(wcodemat(X,nbc))
title('Original image')
subplot(222), image(wcodemat(xd,nbc))
title('De-noised image')

Introduced before R2006a