Documentation

Load an image.

From the MATLAB^® prompt, type

load noiswom 
whos

For the SWT, if a decomposition at level k is needed, 2^k must divide evenly into size(X,1) and size(X,2). If your original image is not of correct size, you can use the function wextend to extend it.

Perform a single-level Stationary Wavelet Decomposition.

Perform a single-level decomposition of the image using the db1 wavelet. Type

[swa,swh,swv,swd] = swt2(X,1,'db1');

This generates the coefficients matrices of the level-one approximation (swa) and horizontal, vertical and diagonal details (swh, swv, and swd, respectively). Both are of size-the-image size. Type

whos

Display the coefficients of approximation and details.

To display the coefficients of approximation and details at level 1, type

map = pink(size(map,1)); colormap(map) 
subplot(2,2,1), image(wcodemat(swa,192));
title('Approximation swa') 
subplot(2,2,2), image(wcodemat(swh,192));
title('Horiz. Detail swh') 
subplot(2,2,3), image(wcodemat(swv,192));
title('Vertical Detail swv') 
subplot(2,2,4), image(wcodemat(swd,192));
title('Diag. Detail swd');

Regenerate the image by Inverse Stationary Wavelet Transform.

To find the inverse transform, type

A0 = iswt2(swa,swh,swv,swd,'db1');

To check the perfect reconstruction, type

err = max(max(abs(X-A0)))

err =
     1.1369e-13

Construct and display approximation and details from the coefficients.

To construct the level 1 approximation and details (A1, H1, V1 and D1) from the coefficients swa, swh, swv and swd, type

nulcfs = zeros(size(swa));
A1 = iswt2(swa,nulcfs,nulcfs,nulcfs,'db1');  
H1 = iswt2(nulcfs,swh,nulcfs,nulcfs,'db1');
V1 = iswt2(nulcfs,nulcfs,swv,nulcfs,'db1'); 
D1 = iswt2(nulcfs,nulcfs,nulcfs,swd,'db1');

To display the approximation and details at level 1, type

colormap(map)
subplot(2,2,1), image(wcodemat(A1,192)); 
title('Approximation A1')
subplot(2,2,2), image(wcodemat(H1,192)); 
title('Horiz. Detail H1')
subplot(2,2,3), image(wcodemat(V1,192)); 
title('Vertical Detail V1')
subplot(2,2,4), image(wcodemat(D1,192)); 
title('Diag. Detail D1')

Perform a multilevel Stationary Wavelet Decomposition.

To perform a decomposition at level 3 of the image (again using the db1 wavelet), type

[swa,swh,swv,swd] = swt2(X,3,'db1');

This generates the coefficients of the approximations at levels 1, 2, and 3 (swa) and the coefficients of the details (swh, swv and swd). Observe that the matrices swa(:,:,i), swh(:,:,i), swv(:,:,i), and swd(:,:,i) for a given level i are of size-the-image size. Type

clear A0 A1 D1 H1 V1 err nulcfs 
whos 

Display the coefficients of approximations and details.

To display the coefficients of approximations and details, type

colormap(map)
kp = 0; 
for i = 1:3
    subplot(3,4,kp+1), image(wcodemat(swa(:,:,i),192));
    title(['Approx. cfs level ',num2str(i)])
    subplot(3,4,kp+2), image(wcodemat(swh(:,:,i),192));
    title(['Horiz. Det. cfs level ',num2str(i)]) 
    subplot(3,4,kp+3), image(wcodemat(swv(:,:,i),192));
    title(['Vert. Det. cfs level ',num2str(i)]) 
    subplot(3,4,kp+4), image(wcodemat(swd(:,:,i),192)); 
    title(['Diag. Det. cfs level ',num2str(i)])
    kp = kp + 4; 
end

Reconstruct approximation at Level 3 and details from coefficients.

To reconstruct the approximation at level 3, type

mzero = zeros(size(swd)); 
A = mzero; 
A(:,:,3) = iswt2(swa,mzero,mzero,mzero,'db1');

To reconstruct the details at levels 1, 2 and 3, type

H = mzero; V = mzero; 
D = mzero; 
for i = 1:3
    swcfs = mzero; swcfs(:,:,i) = swh(:,:,i); 
    H(:,:,i) = iswt2(mzero,swcfs,mzero,mzero,'db1');
    swcfs = mzero; swcfs(:,:,i) = swv(:,:,i); 
    V(:,:,i) = iswt2(mzero,mzero,swcfs,mzero,'db1');
    swcfs = mzero; swcfs(:,:,i) = swd(:,:,i); 
    D(:,:,i) = iswt2(mzero,mzero,mzero,swcfs,'db1');
end

Reconstruct and display approximations at Levels 1, 2 from approximation at Level 3 and details at Levels 1, 2, and 3.

To reconstruct the approximations at levels 2 and 3, type

A(:,:,2) = A(:,:,3) + H(:,:,3) + V(:,:,3) + D(:,:,3); 
A(:,:,1) = A(:,:,2) + H(:,:,2) + V(:,:,2) + D(:,:,2);

To display the approximations and details at levels 1, 2, and 3, type

colormap(map)
kp = 0; 
for i = 1:3 
    subplot(3,4,kp+1), image(wcodemat(A(:,:,i),192));
    title(['Approx. level ',num2str(i)]) 
    subplot(3,4,kp+2), image(wcodemat(H(:,:,i),192));
    title(['Horiz. Det. level ',num2str(i)]) 
    subplot(3,4,kp+3), image(wcodemat(V(:,:,i),192)); 
    title(['Vert. Det. level ',num2str(i)])
    subplot(3,4,kp+4), image(wcodemat(D(:,:,i),192)); 
    title(['Diag. Det. level ',num2str(i)]) 
    kp = kp + 4; 
end

Remove noise by thresholding.

To denoise an image, use the threshold value we find using the Wavelet Analyzer app tool (see the next section), use the wthresh command to perform the actual thresholding of the detail coefficients, and then use the iswt2 command to obtain the denoised image.

thr = 44.5; 
sorh = 's'; dswh = wthresh(swh,sorh,thr); 
dswv = wthresh(swv,sorh,thr);
dswd = wthresh(swd,sorh,thr); 
clean = iswt2(swa,dswh,dswv,dswd,'db1');

To display both the original and denoised images, type

colormap(map)
subplot(1,2,1), image(wcodemat(X,192)); 
title('Original image') 
subplot(1,2,2), image(wcodemat(clean,192)); 
title('denoised image')

A second syntax can be used for the swt2 and iswt2 functions, giving the same results:

lev= 4; 
swc = swt2(X,lev,'db1'); 
swcden = swc; 
swcden(:,:,1:end-1) =
wthresh(swcden(:,:,1:end-1),sorh,thr); 
clean = iswt2(swcden,'db1');

You obtain the same plot by using the plot commands in step 9 above.

Start the Stationary Wavelet Transform Denoising 2-D Tool.

From the MATLAB prompt, type waveletAnalyzer.

The Wavelet Analyzer appears:

Click the SWT Denoising 2-D menu item.

Load data.

At the MATLAB command prompt, type

load noiswom

Perform a Stationary Wavelet Decomposition.

Select the haar wavelet from the Wavelet menu, select 4 from the Level menu, and then click the Decompose Image button.

The tool displays the histograms of the stationary wavelet detail coefficients of the image on the left of the window. These histograms are organized as follows:

Denoise the image using the Stationary Wavelet Transform.

While a number of options are available for fine-tuning the denoising algorithm, we'll accept the defaults of fixed form soft thresholding and unscaled white noise. The sliders located to the right of the window control the level dependent thresholds indicated by the dashed lines running vertically through the histograms of the coefficients on the left of the window. Click the Denoise button.

The result seems to be oversmoothed and the selected thresholds too aggressive. Nevertheless, the histogram of the residuals is quite good since it is close to a Gaussian distribution, which is the noise introduced to produce the analyzed image noiswom.mat from a piece of the original image woman.mat.

Selecting a thresholding method.

From the Select thresholding method menu, choose the Penalize low item. The associated default for the thresholding mode is automatically set to hard; accept it. Use the Sparsity slider to adjust the threshold value close to 45.5, and then click the denoise button.

The result is quite satisfactory, although it is possible to improve it slightly.

Select the sym6 wavelet and click the Decompose Image button. Use the Sparsity slider to adjust the threshold value close to 40.44, and then click the denoise button.