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I have what it may be a ridiculous question (since I don't know much about wavelets), but here I go.

I am using different Discrete Wavelet families to extract texture features from images. I plan to use them to train a ML model to solve some classification problems. My question arises because I have decided to use 10 different families (from pywt - haar, db4, db8, coif1, coif4,'rbio1.3','rbio3.1','rbio3.3','rbio1.3.5','rbio4.') based in some previous literature, and I am not sure if there is redundancy in the information (texture) extracted from these families such that I will need to do some dimensionality reduction.

For instance, for the same problem, I am also using grey level co-occurrence matrix features (6 features in total, but with many sub options that make tens of different combinations) as textures. I plan to use PCA to reduce the dimensionality of this data, to obtain new features that are reduced to an optimal subset (as I see that others have done in the past).

So, I am not sure if the same approach (using PCA) can be used to get "the optimal" combination of wavelet families the get the best classification accuracy possible.

My gut tells me that each DWT family is a completely different thing, so it may not be appropriate to use such PCA approach (why nobody has done this in the past? at least I haven't seen this). Also, I understand that DWT are a dimensionality reduction technique by itself, so again, I am really confused about this PCA on DWT thing.

Cheers!

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The wavelet transform itself does not reduce dimension. If boundaries are taken in account in some sensible way, the wavelet transform is square and invertible.

The "named" transforms provide generally adapted basis functions for piecewise smooth functions. This means that on such input signals the detail bands are sparse, generally contains small coefficients except close to edges of the input. If the transform has sufficiently many levels, most of the transformed signal is detail coefficients. Extracting that sparsity pattern and approximating the signal with only these coefficients and basis functions can be seen as dimension reduction.

The wavelet families will differ more in their geometric properties, symmetry, orthogonality, than in the compression features. However, the compression rate will also depend on the smoothness of the wavelet functions. In my superstition symmetric, linear phase wavelets are better for image analysis than unsymmetric orthogonal wavelets. Note that if you increase the channel number in the wavelet from 2 to 3 or more then one can have smooth wavelets that are both symmetric and orthogonal.

Performing a PCA/SVD on top of a wavelet transform may produce results that are easier to interpret, as one can expect that the leading singular vectors are also sparse in their large coefficients.

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  • $\begingroup$ Thanks for your answer!! $\endgroup$
    – PPM
    Commented Jan 17, 2023 at 3:23

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