Wavelet Toolbox

Two-Dimensional Analysis Using the Command Line

In this example, we'll show how you can use two-dimensional stationary wavelet analysis to de-noise 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.

1. From the MATLAB prompt, type

   • load noiswom
     whos 
     
     

     Name	Size	Bytes	Class
     X	96x96	73728	double array
     map	255x3	6120	double 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 Image Extension GUI tool or the function wextend to extend it.

2. Perform a single-level Stationary Wavelet Decomposition.

1. 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 
     
     

     Name	Size	Bytes	Class
     X	96x96	73728	double array
     map	255x3	6120	double array
     swa	96x96	73728	double array
     swh	96x96	73728	double array
     swv	96x96	73728	double array
     swd	96x96	73728	double array

3. Display the coefficients of approximation and details.

1. 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.

1. 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.

1. 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.

1. 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 
     
     

     Name	Size	Bytes	Class
     X	96x96	73728	double array
     map	255x3	6120	double array
     swa	96x96x3	221184	double array
     swh	96x96x3	221184	double array
     swv	96x96x3	221184	double array
     swd	96x96x3	221184	double array

7. Display the coefficients of approximations and details.

1. 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.

1. 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.

1. 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.

1. To de-noise an image, use the threshold value we find using the GUI 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 de-noised 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 de-noised images, type

   • colormap(map)
     subplot(1,2,1), image(wcodemat(X,192));
     title('Original image')
     subplot(1,2,2), image(wcodemat(clean,192));
     title('De-noised 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 than in Step 14 above,

Two-Dimensional Discrete Stationary Wavelet Analysis

Two-Dimensional Analysis for De-Noising Using the Graphical Interface

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