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 Name | Purpose |
---|---|
Decomposition |
Synthesis-Reconstruction Function
Function Name | Purpose |
---|---|
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.
Load an image.
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 intosize(X,1)
andsize(X,2)
. If your original image is not of correct size, you can use the functionwextend
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
, andswd
, respectively). Both are of size-the-image size. Typewhos
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
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
andD1
) from the coefficientsswa
,swh
,swv
andswd
, typenulcfs = 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
andswd
). Observe that the matricesswa(:,:,i)
,swh(:,:,i)
,swv(:,:,i)
, andswd(:,:,i)
for a given leveli
are of size-the-image size. Typeclear 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
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
To denoise an image, use the
ddencmp
function to find the threshold value, use thewthresh
function to perform the actual thresholding of the detail coefficients, and then use theiswt2
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
andiswt2
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.