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## 2-D Stationary Wavelet Transform

This section takes you through the features of 2-D discrete stationary wavelet analysis using the Wavelet Toolbox™ software.

### Analysis-Decomposition Function

Function NamePurpose

swt2

Decomposition

### Synthesis-Reconstruction Function

Function NamePurpose

iswt2

Reconstruction

The stationary wavelet decomposition structure is more tractable than the wavelet one. So, the utilities useful for the wavelet case are not necessary for the Stationary Wavelet Transform (SWT).

In this section, you'll learn to

• Load an image

• Analyze an image

• Perform single-level and multilevel image decompositions and reconstructions

• Denoise an image

### 2-D Analysis

In this example, we'll show how you can use 2-D stationary wavelet analysis to denoise an image.

Note

Instead of using image(I) to visualize the image I, we use image(wcodemat(I)), which displays a rescaled version of I leading to a clearer presentation of the details and approximations (see the wcodemat reference page).

This example involves a image containing noise.

1. Load an image.

From the MATLAB® prompt, type

load noiswom
whos

NameSizeBytesClass
X96x9673728double array
map255x36120double array

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.

2. 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

NameSizeBytesClass
X96x9673728double array
map255x36120double array
swa96x9673728double array
swh96x9673728double array
swv96x9673728double array
swd96x9673728double array
3. 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');

4. 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

5. 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')

6. 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

NameSizeBytesClass
X96x9673728double array
map255x36120double array
swa96x96x3221184double array
swh96x96x3221184double array
swv96x96x3221184double array
swd96x96x3221184double array
7. 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

8. 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

9. 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

10. Remove noise by thresholding.

To denoise an image, use the ddencmp function to find the threshold value, use the wthresh function to perform the actual thresholding of the detail coefficients, and then use the iswt2 function 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.