Documentation |
This section takes you through the features of one-dimensional discrete stationary wavelet analysis using the Wavelet Toolbox™ software. For more information see Discrete Stationary Wavelet Transform (SWT) in the Wavelet Toolbox User's Guide.
The toolbox provides these functions for one-dimensional discrete stationary wavelet analysis. For more information on the functions, see the reference pages.
Function Name | Purpose |
---|---|
Decomposition |
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 a signal
Perform a stationary wavelet decomposition of a signal
Construct approximations and details from the coefficients
Display the approximation and detail at level 1
Regenerate a signal by using inverse stationary wavelet transform
Perform a multilevel stationary wavelet decomposition of a signal
Reconstruct the level 3 approximation
Reconstruct the level 1, 2, and 3 details
Reconstruct the level 1 and 2 approximations
Display the results of a decomposition
Reconstruct the original signal from the level 3 decomposition
Remove noise from a signal
Since you can perform analyses either from the command line or using the graphical interface tools, this section has subsections covering each method.
The final subsection discusses how to exchange signal and coefficient information between the disk and the graphical tools.
This example involves a noisy Doppler test signal.
From the MATLAB^{®} prompt, type
load noisdopp
s = noisdopp;
For the SWT, if a decomposition at level k is needed, 2^k must divide evenly into the length of the signal. If your original signal does not have the correct length, you can use the Signal Extension GUI tool or the wextend function to extend it.
Perform a single-level Stationary Wavelet Decomposition.
Perform a single-level decomposition of the signal using the db1 wavelet. Type
[swa,swd] = swt(s,1,'db1');
This generates the coefficients of the level 1 approximation (swa) and detail (swd). Both are of the same length as the signal. Type
whos
Name | Size | Bytes | Class |
---|---|---|---|
noisdopp | 1x1024 | 8192 | double array |
s | 1x1024 | 8192 | double array |
swa | 1x1024 | 8192 | double array |
swd | 1x1024 | 8192 | double array |
Display the coefficients of approximation and detail.
To display the coefficients of approximation and detail at level 1, type
subplot(1,2,1), plot(swa); title('Approximation cfs') subplot(1,2,2), plot(swd); title('Detail cfs')
Regenerate the signal by Inverse Stationary Wavelet Transform.
To find the inverse transform, type
A0 = iswt(swa,swd,'db1');
To check the perfect reconstruction, type
err = norm(s-A0) err = 2.1450e-14
Construct and display approximation and detail from the coefficients.
To construct the level 1 approximation and detail (A1 and D1) from the coefficients swa and swd, type
nulcfs = zeros(size(swa)); A1 = iswt(swa,nulcfs,'db1'); D1 = iswt(nulcfs,swd,'db1');
To display the approximation and detail at level 1, type
subplot(1,2,1), plot(A1); title('Approximation A1'); subplot(1,2,2), plot(D1); title('Detail D1');
Perform a multilevel Stationary Wavelet Decomposition.
To perform a decomposition at level 3 of the signal (again using the db1 wavelet), type
[swa,swd] = swt(s,3,'db1');
This generates the coefficients of the approximations at levels 1, 2, and 3 (swa) and the coefficients of the details (swd). Observe that the rows of swa and swd are the same length as the signal length. Type
clear A0 A1 D1 err nulcfs whos
Name | Size | Bytes | Class |
---|---|---|---|
noisdopp | 1x1024 | 8192 | double array |
s | 1x1024 | 8192 | double array |
swa | 3x1024 | 24576 | double array |
swd | 3x1024 | 24576 | double array |
Display the coefficients of approximations and details.
To display the coefficients of approximations and details, type
kp = 0; for i = 1:3 subplot(3,2,kp+1), plot(swa(i,:)); title(['Approx. cfs level ',num2str(i)]) subplot(3,2,kp+2), plot(swd(i,:)); title(['Detail cfs level ',num2str(i)]) kp = kp + 2; end
Reconstruct approximation at Level 3 From coefficients.
To reconstruct the approximation at level 3, type
mzero = zeros(size(swd)); A = mzero; A(3,:) = iswt(swa,mzero,'db1');
Reconstruct details from coefficients.
To reconstruct the details at levels 1, 2 and 3, type
D = mzero; for i = 1:3 swcfs = mzero; swcfs(i,:) = swd(i,:); D(i,:) = iswt(mzero,swcfs,'db1'); end
Reconstruct and display approximations at Levels 1 and 2 from approximation at Level 3 and details at Levels 2 and 3.
To reconstruct the approximations at levels 2 and 3, type
A(2,:) = A(3,:) + D(3,:); A(1,:) = A(2,:) + D(2,:);
To display the approximations and details at levels 1, 2 and 3, type
kp = 0; for i = 1:3 subplot(3,2,kp+1), plot(A(i,:)); title(['Approx. level ',num2str(i)]) subplot(3,2,kp+2), plot(D(i,:)); title(['Detail level ',num2str(i)]) kp = kp + 2; end
To denoise the signal, use the ddencmp command to calculate a default global threshold. Use the wthresh command to perform the actual thresholding of the detail coefficients, and then use the iswt command to obtain the denoised signal.
[thr,sorh] = ddencmp('den','wv',s); dswd = wthresh(swd,sorh,thr); clean = iswt(swa,dswd,'db1');
To display both the original and denoised signals, type
subplot(2,1,1), plot(s); title('Original signal') subplot(2,1,2), plot(clean); title('denoised signal')
The obtained signal remains a little bit noisy. The result can be improved by considering the decomposition of s at level 5 instead of level 3, and repeating steps 14 and 15. To improve the previous de-noising, type
[swa,swd] = swt(s,5,'db1'); [thr,sorh] = ddencmp('den','wv',s); dswd = wthresh(swd,sorh,thr); clean = iswt(swa,dswd,'db1'); subplot(2,1,1), plot(s); title('Original signal') subplot(2,1,2), plot(clean); title('denoised signal')
A second syntax can be used for the swt and iswt functions, giving the same results:
lev = 5; swc = swt(s,lev,'db1'); swcden = swc; swcden(1:end-1,:) = wthresh(swcden(1:end-1,:),sorh,thr); clean = iswt(swcden,'db1');
You can obtain the same plot by using the same plot commands as in step 16 above.
Now we explore a strategy to denoise signals, based on the one-dimensional stationary wavelet analysis using the graphical interface tools. The basic idea is to average many slightly different discrete wavelet analyses.
Start the Stationary Wavelet Transform De-Noising 1-D Tool.
From the MATLAB prompt, type
wavemenu
The Wavelet Toolbox Main Menu appears.
Click the SWT De-noising 1-D menu item. The discrete stationary wavelet transform de-noising tool for one-dimensional signals appears.
From the File menu, choose the Load Signal option.
When the Load Signal dialog box appears, select the MAT-file noisbloc.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo.
Click the OK button. The noisy blocks signal is loaded into the SWT De-noising 1-D tool.
Perform a Stationary Wavelet Decomposition.
Select the db1 wavelet from the Wavelet menu and select 5 from the Level menu, and then click the Decompose Signal button. After a pause for computation, the tool displays the stationary wavelet approximation and detail coefficients of the decomposition. These are also called nondecimated coefficients since they are obtained using the same scheme as for the DWT, but omitting the decimation step (see Fast Wavelet Transform (FWT) Algorithm in the Wavelet Toolbox User's Guide).
denoise the signal using the Stationary Wavelet Transform.
While a number of options are available for fine-tuning the de-noising algorithm, we'll accept the defaults of fixed form soft thresholding and unscaled white noise. The sliders located on the right part of the window control the level-dependent thresholds, indicated by yellow dotted lines running horizontally through the graphs of the detail coefficients to the left of the window. The yellow dotted lines can also be dragged directly using the left mouse button over the graphs.
Note that the approximation coefficients are not thresholded.
Click the denoise button.
The result is quite satisfactory, but seems to be oversmoothed around the discontinuities of the signal. This can be seen by looking at the residuals, and zooming on a breakdown point, for example around position 800.
Selecting a Thresholding Method. Select hard for the thresholding mode instead of soft, and then click the denoise button.
The result is of good quality and the residuals look like a white noise sample. To investigate this last point, you can get more information on residuals by clicking the Residuals button.
The tool lets you save the denoised signal to disk. The toolbox creates a MAT-file in the current folder with a name of your choice.
To save the above denoised signal, use the menu option File > Save denoised Signal. A dialog box appears that lets you specify a folder and filename for storing the signal. Type the name dnoibloc. After saving the signal data to the file dnoibloc.mat, load the variables into your workspace:
load dnoibloc whos
Name | Size | Bytes | Class |
---|---|---|---|
dnoibloc | 1x1024 | 8192 | double array |
thrParams | 1x5 | 580 | cell array |
wname | 1x3 | 6 | char array |
The denoised signal is given by dnoibloc. In addition, the parameters of the de-noising process are available. The wavelet name is contained in wname:
wname wname = db1
and the level dependent thresholds are encoded in thrParams, which is a cell array of length 5 (the level of the decomposition). For i from 1 to 5, thrParams{i} contains the lower and upper bounds of the interval of thresholding and the threshold value (since interval dependent thresholds are allowed). For more information, see One-Dimensional Adaptive Thresholding of Wavelet Coefficients.
For example, for level 1,
thrParams{1} ans = 1.0e+03 * 0.0010 1.0240 0.0041
Here the lower bound is 1, the upper bound is 1024, and the threshold value is 4.1. The total time-interval is not segmented and the procedure does not use the interval dependent thresholds.