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Penalized threshold for wavelet packet de-noising

`THR = wpbmpen(T,SIGMA,ALPHA)`

wpbmpen(T,SIGMA,ALPHA,ARG)

`THR = wpbmpen(T,SIGMA,ALPHA)`

returns a
global threshold `THR`

for de-noising. `THR`

is
obtained by a wavelet packet coefficients selection rule using a penalization
method provided by Birge-Massart.

`T`

is a wavelet packet tree corresponding
to the wavelet packet decomposition of the signal or image to be de-noised.

`SIGMA`

is the standard deviation of the zero
mean Gaussian white noise in the 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 packet 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 packet coefficients
sorted in decreasing order of their absolute value and `n`

is
the number of coefficients, then THR=|c(t^{*})|.

`wpbmpen(T,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 chirp signal. load noischir; x = noischir; % Perform a wavelet packet decomposition of the signal % at level 5 using sym6. wname = 'sym6'; lev = 5; tree = wpdec(x,lev,wname); % Estimate the noise standard deviation from the % detail coefficients at level 1, % corresponding to the node index 2. det1 = wpcoef(tree,2); sigma = median(abs(det1))/0.6745; % Use wpbmpen for selecting global threshold % for signal de-noising, using the recommended parameter. alpha = 2; thr = wpbmpen(tree,sigma,alpha) thr = 4.5740 % Use wpdencmp for de-noising the signal using the above % threshold with soft thresholding and keeping the % approximation. keepapp = 1; xd = wpdencmp(tree,'s','nobest',thr,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 packet decomposition of the image % at level 3 using coif2. wname = 'coif2'; lev = 3; tree = wpdec2(X,lev,wname); % Estimate the noise standard deviation from the % detail coefficients at level 1. det1 = [wpcoef(tree,2) wpcoef(tree,3) wpcoef(tree,4)]; sigma = median(abs(det1(:)))/0.6745; % Use wpbmpen for selecting global threshold % for image de-noising. alpha = 1.1; thr = wpbmpen(tree,sigma,alpha) thr = 38.5125 % Use wpdencmp for de-noising the image using the above % thresholds with soft thresholding and keeping the % approximation. keepapp = 1; xd = wpdencmp(tree,'s','nobest',thr,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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