Multivariate Wavelet Denoising Using the Wavelet Analyzer App

This section explores a denoising strategy for multivariate signals using the Wavelet Analyzer app.

Start the Multivariate Denoising Tool by first opening the Wavelet Analyzer app. Type waveletAnalyzer at the command line.
Click Multivariate Denoising to open the Multivariate Denoising portion of the app.
Load data.
At the MATLAB command prompt, type
```
load ex4mwden
```
In the Multivariate Denoising tool, select File > Import from Workspace. When the Import from Workspace dialog box appears, select the x variable. Click OK to import the noisy multivariate signal. The signal is a matrix containing four columns, where each column is a signal to be denoised.
These signals are noisy versions from simple combinations of the two original signals. The first one is “Blocks” which is irregular and the second is “HeavySine” which is regular except around time 750. The other two signals are the sum and the difference between the original signals. Multivariate Gaussian white noise exhibiting strong spatial correlation is added to the resulting four signals.
The following example illustrates the two different aspects of the proposed denoising method. First, perform a convenient change of basis to cope with spatial correlation and denoise in the new basis. Then, use PCA to take advantage of the relationships between the signals, leading to an additional denoising effect.
Perform a wavelet decomposition and diagonalize the noise covariance matrix.
Use the displayed default values for the Wavelet, the DWT Extension Mode, and the decomposition Level, and then click Decompose and Diagonalize. The tool displays the wavelet approximation and detail coefficients of the decomposition of each signal in the original basis.
Select Noise Adapted Basis to display the signals and their coefficients in the noise-adapted basis.
To see more information about this new basis, click More on Noise Adapted Basis. A new figure displays the robust noise covariance estimate matrix and the corresponding eigenvectors and eigenvalues.
Eigenvectors define the change of basis, and eigenvalues are the variances of uncorrelated noises in the new basis.
The multivariate denoising method proposed below is interesting if the noise covariance matrix is far from diagonal exhibiting spatial correlation, which, in this example, is the case.
denoise the multivariate signal.
A number of options are available for fine-tuning the denoising algorithm. However, we will use the defaults: fixed form soft thresholding, scaled white noise model, and the proposed numbers of retained principal components. In this case, the default values for PCA lead to retaining all the components.
Select Original Basis to return to the original basis and then click Denoise.
The results are satisfactory. Both of the two first signals are correctly recovered, but they can be improved by getting more information about the principal components. Click More on Principal Components.

A new figure displays information to select the numbers of components to keep for the PCA of approximations and for the final PCA after getting back to the original basis. You can see the percentages of variability explained by each principal component and the corresponding cumulative plot. Here, it is clear that only two principal components are of interest.

Close the More on Principal Components window. Select 2 as the Nb. of PC for APP. Select 2 as the Nb. of PC for final PCA, and then click denoise.

The results are better than those previously obtained. The first signal, which is irregular, is still correctly recovered. The second signal, which is more regular, is denoised better after this second stage of PCA. You can get more information by clicking Residuals.