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

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

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

`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)`

% 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')