Got to say the "low-pass/high-pass" description of how wavelets works is about 10 years out of date since the invention of Lifting Methods. Early descriptions used to concentrate on the FFT of the signal, but Lifting loses the reliance on the frequency domain and instead talks about dyadic descimation. IMO, everyone trying to understand wavelets needs to forget the FFT based proofs and start again using Lifting.

For example, there is a simpler starting point than "average and difference" - just drop every second sample. On a 1D signal the difference between the two techniques seems unimportant, but the dyadic descimation idea can be applied to many more situations than just 1D signals. You can dro pevery second vertex on a 2D patch, you can drop every second direction in a spherical projection, etc. Lifting methods has helped Wavelets be used in many more places than before, it's really the correct explanation for why they work.

Personally, I'd recommend removing this "easy" explanation from the public and replacing it with a Lifting based introduction. And find a way to lose the book "Ripples in Mathematics" down the back of the sofa for being out of date.

- Robin Green
PDI/Dreamworks