From Box Filtering to Fast Explicit Diffusion

Abstract

There are two popular ways to implement anisotropic diffusion filters with a diffusion tensor: Explicit finite difference schemes are simple but become inefficient due to severe time step size restrictions, while semi-implicit schemes are more efficient but require to solve large linear systems of equations. In our paper we present a novel class of algorithms that combine the advantages of both worlds: They are based on simple explicit schemes, while being more efficient than semi-implicit approaches. These so-called fast explicit diffusion (FED) schemes perform cycles of explicit schemes with varying time step sizes that may violate the stability restriction in up to 50 percent of all cases. FED schemes can be motivated from a decomposition of box filters in terms of explicit schemes for linear diffusion problems. Experiments demonstrate the advantages of the FED approach for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. In the latter case FED schemes are speeded up substantially by embedding them in a cascadic coarse-to-fine approach.