It's misleading to say that they don't use Fourier. The theory around any family of functions useful for compression/feature detection (like wavelets) is going

It's misleading to say that they don't use Fourier. The theory around any family of functions useful for compression/feature detection (like wavelets) is going to have the property that they define a Hilbert basis. And the idea of how to conceive of such families of functions and their potential required them being generalized from the specific cases of the Fourier and Laplacian transformations. Moreover wavelets have properties/tradeoffs defined in terms of time and frequency which are couched in terms and based on theories that are derived from this early complex analysis.