Recursive separable schemes for nonlinear diffusion filters

Abstract

Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Euler-forward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semi-implicit schemes is studied, so-called additive operator splitting (AOS) methods. They share the advantages of explicit and (semi-)implicit schemes by combining simplicity with absolute stability. They are reliable, since they satisfy recently established criteria for discrete nonlinear diffusion scale-spaces. Their efficiency is due to the fact that they can be separated into one-dimensional processes, for which a fast recursive algorithm with linear complexity is available. AOS schemes reveal good rotational invariance and they are symmetric with respect to all axes. Examples demonstrate that, under typical accuracy requirements, they are at least ten times more efficient than explicit schemes.