Documentation

# 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') ```  