Diffusion Filters and Wavelets: What Can They Learn from Each Other?

Nonlinear diffusion filtering and wavelet shrinkage are two methods that serve the same purpose, namely discontinuity-preserving denoising. In this chapter we give a survey on relations between both paradigms when space-discrete or fully discrete versions of nonlinear diffusion filters are considered. For the case of space-discrete diffusion, we show equivalence between soft Haar wavelet shrinkage and total variation (TV) diffusion for 2-pixel signals. For the general case of N -pixel signals, this leads us to a numerical scheme for TV diffusion with many favourable properties. Both considerations are then extended to 2-D images, where an analytical solution for 2 × 2 pixel images serves as building block for a wavelet-inspired numerical scheme for TV diffusion. When replacing spacediscrete diffusion by fully discrete one with an explicit time discretisation, we obtain a general relation between the shrinkage function of a shiftinvariant Haar wavelet shrinkage on a single scale and the diffusivity of a nonlinear diffusion filter. This allows to study novel, diffusion-inspired shrinkage functions with competitive performance, to suggest new shrinkage rules for 2-D images with better rotation invariance, and to propose coupled shrinkage rules for colour images where a desynchronisation of the colour channels is avoided. Finally we present a new result which shows that one is not restricted to shrinkage with Haar wavelets: By using wavelets with a higher number of vanishing moments, equivalences to higher-order diffusion-like PDEs are discovered.