Practical applications of wavelets

Question

I know wavelets were all the rage a few years ago, but I missed that boat and am wondering if it is worth putting significant effort into learning about them. My impression is that they were a little bit of a fad, and now have more limited applications than first assumed (similar to compressed sensing).*

My (very limited) understanding is that the wavelet transform is similar to the Fourier transform, except the filter bank/basis function set can be many other things besides complex exponentials (as in the DFT). I suppose that would make the short time Fourier transform a subset of the wavelet transforms.

So I have two questions:

1. Is my understanding basically correct? If not, can anyone point me to any good tutorials?
2. What are the main practical applications of wavelet transforms? The main thing I'm aware of is image compression.

Thanks for any pointers!

*Lest any wavelet or compressed sensing aficionados get up in arms, I know both have useful applications. But unsurprisingly neither has turned out to be the silver bullet some initially thought they might be. There are truly very few seminal breakthroughs.

Edit:

To hopefully clarify the second question a bit, I'm interested in classes of DSP problems where wavelets are significantly superior to other methods in general. As an analogy, deep learning is obviously strong on image classification when you have a large and broad set of training data.

Answer 1

Wavelets remain relevant. There's lots to unpack, one by one

a fad, and now have more limited applications than first assumed

It's the other way around. Wavelet scattering beats neural nets on small datasets in classification and has uses for synthesis and generative modeling; it's a 2012 invention. What's true is there's often better alternatives, but that didn't make the spectrogram obsolete. Another major use is instantaneous frequency/amplitude extraction, with little competition I know of.

wavelet transform is similar to the Fourier transform, except the filter bank/basis function set can be many other things besides complex exponentials

Yes and no. Any CWT or STFT basis is a sum of Fourier bases, it's all LTI filtering.

short time Fourier transform a subset of the wavelet transforms

No, though they're related - refer to this post.

neither has turned out to be the silver bullet some initially thought they might be

The silver bullet you speak of might be artificial general intelligence, and wavelets certainly ain't that. Wavelets remain the best for some applications under some constraints, and that's all that can be said. We still bother because such constraints exist and aren't going away anytime soon.

I recommend this tutorial for getting started with CWT. The top GIF here may also help, also see under "Applications". My answer doesn't cover all relevant applications.