1-D Stationary Wavelet Transform

This topic takes you through the features of 1-D discrete stationary wavelet analysis using the Wavelet Toolbox™ software. For more information see Nondecimated Discrete Stationary Wavelet Transforms (SWTs) in the Wavelet Toolbox User's Guide.

The toolbox provides these functions for 1-D discrete stationary wavelet analysis. For more information on the functions, see the reference pages.

Analysis-Decomposition Functions

Function Name

Purpose

swt

Decomposition

Synthesis-Reconstruction Functions

Function Name

Purpose

iswt

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 Wavelet Analyzer app, 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.

1-D Analysis Using the Command Line

This example involves a noisy Doppler test signal.

  1. Load a signal.

    From the MATLAB® prompt, type

    load noisdopp

  2. Set the variables. Type

    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 wextend function to extend it.

  3. 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
    NameSizeBytesClass
    noisdopp1x10248192double array
    s1x10248192double array
    swa1x10248192double array
    swd1x10248192double array

  4. 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')

  5. 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

  6. 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');

  7. 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
    NameSizeBytesClass
    noisdopp1x10248192double array
    s1x10248192double array
    swa3x102424576double array
    swd3x102424576double array

  8. 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

  9. 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');

  10. 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

  11. 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

  12. Remove noise by thresholding.

    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.

    Note

    All methods for choosing thresholds in the 1-D Discrete Wavelet Transform case are also valid for the 1-D Stationary Wavelet Transform, which are also those used by the Wavelet Analysis app. This is also true for the 2-D transforms.

[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 denoising, 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.

Interactive 1-D Stationary Wavelet Transform Denoising

Now we explore a strategy to denoise signals, based on the 1-D stationary wavelet analysis using the Wavelet Analyzer app. The basic idea is to average many slightly different discrete wavelet analyses.

  1. Start the Stationary Wavelet Transform Denoising 1-D Tool.

    From the MATLAB prompt, type waveletAnalyzer.

    The Wavelet Analyzer appears.

    Click the SWT Denoising 1-D menu item. The discrete stationary wavelet transform denoising tool for 1-D signals appears.

  2. Load data.

    At the MATLAB command prompt, type

    load noisbloc;
    In the SWT Denoising 1-D tool, select File > Import Signal from Workspace. When the Import from Workspace dialog box appears, select the noisbloc variable. Click OK to import the noisy blocks signal.

  3. 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).

  4. denoise the signal 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 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.

Importing and Exporting from the Wavelet Analysis App

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
NameSizeBytesClass
dnoibloc1x10248192double array
thrParams1x5580cell array
wname1x36char array

The denoised signal is given by dnoibloc. In addition, the parameters of the denoising 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 1-D 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.

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