PDE Evolutions for M-Smoothers: From Common Myths to Robust Numerics

Abstract

Local M-smoothers constitute an interesting and important class of image processing techniques with many connections to other methods. In our paper we derive a family of partial differential equations (PDEs) that result as limiting processes from M-smoothers which are based on local order-p means within a disc the radius of which tends to zero. The order p may take any nonzero value \(>-1\). Thus, we also allow negative values which have never been considered before. In contrast to results from the literature, we show in the space-continuous case that mode filtering does not arise for \(p \rightarrow 0\), but for \(p \rightarrow -1\). Extending our filter class to p-values smaller than \(-1\) allows to include e.g. the classical image sharpening flow of Gabor. Since our PDE class is highly anisotropic and may contain backward parabolic operators, designing adequate numerical methods is difficult. We present an \(L^\infty \)-stable explicit finite difference scheme that satisfies a discrete maximum–minimum principle, is fairly efficient, and offers excellent rotation invariance. Although it solves parabolic PDEs, it makes consequent use of stabilisation concepts from the numerics of hyperbolic PDEs. Our experiments show that the PDEs for \(p<1\) are of specific interest: Their backward parabolic term creates favourable sharpening properties, while they appear to maintain the strong shape simplification properties of mean curvature motion.