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 (command line only)
denoise an image
2-D Analysis Using the Command Line
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 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 theiswt2
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
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.
Interactive 2-D Stationary Wavelet Transform Denoising
In this section, we explore a strategy for denoising images based on the 2-D stationary wavelet analysis using the Wavelet Analyzer app. The basic idea is to average many slightly different discrete wavelet analyses.
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
In the SWT Denoising 2-D tool, select File > Import Image from Workspace. When the Import from Workspace dialog box appears, select theload noiswom
X
variable. Click OK to import the image.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:
From the bottom for level 1 to the top for level 4
On the left horizontal coefficients, in the middle diagonal coefficients, and on the right vertical coefficients
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 imagewoman.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.
Importing and Exporting Information from the Wavelet Analyzer App
The tool lets you save the denoised image to disk. The toolbox creates a MAT-file in the current folder with a name you choose.
To save the denoised image from the present denoising process, use the menu
File > Save denoised Image. A dialog box
appears that lets you specify a folder and filename for storing the image. Type the
name dnoiswom
. After saving the image data to the file
dnoiswom.mat
, load the variables into your workspace:
load dnoiswom whos
Name | Size | Bytes | Class |
---|---|---|---|
X | 96x96 | 73728 | double array |
map | 255x3 | 6120 | double array |
valTHR | 3x4 | 96 | double array |
wname | 1x4 | 8 | char array |
The denoised image is X
and map
is the
colormap. In addition, the parameters of the denoising process are available. The
wavelet name is contained in wname
, and the level dependent
thresholds are encoded in valTHR
. The variable
valTHR
has four columns (the level of the decomposition) and
three rows (one for each detail orientation).