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Penalized threshold for wavelet 1-D or 2-D de-noising

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

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

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

returns global
threshold `THR`

for de-noising. `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 de-noised.

`SIGMA`

is the standard deviation of the zero
mean Gaussian white noise in de-noising 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 de-noised signal or image grows with `ALPHA`

.
Typically `ALPHA`

= 2.

`THR`

minimizes the penalized criterion given
by

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)`

% Example 1: Signal de-noising. % 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 de-noising, using the tuning parameter. alpha = 2; thr = wbmpen(c,l,sigma,alpha) thr = 2.7681 % Use wdencmp for de-noising 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 de-noised signals. figure(1) subplot(211), plot(x), title('Original signal') subplot(212), plot(xd), title('De-noised signal')

% Example 2: Image de-noising. % 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 de-noising. alpha = 1.2; thr = wbmpen(c,l,sigma,alpha) thr = 36.0621 % Use wdencmp for de-noising 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 de-noised 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')

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