Wavelets are basically a generalisation of Fourier with an arbitrary basis function which can be selected to highlight/match specific features in the data.

They've been used in image filtering for a long time. I remember a wavelet-based grain removal and denoising plugin for Photoshop from around 15 years ago.

In this work I suspect that trying to implement wavelets in an audio editor - which generates some useful but slightly quirky representations and editing options - cued the realisation they could be applied to stochastic PDEs.