This doesn't use wavelets afaics (they are not mentioned in the article), it uses something called 'Laplacian Pyramid'. That can be used separate a signal by frequency but seems to do the whole image at once. Looks to be the good old add-laplacian (ie 'blur') and subtract-from-the-original method (to get remaining high frequency).

I think wavelets usually have a limited extent in space (depending which basis shape you choose I guess).

Maybe they could make use of FFT for this separation, but it's probably hard to beat blur-and subtract for simplicity of implementation.