L2-stable nonstandard finite differences for anisotropic diffusion

Abstract

Anisotropic diffusion filters with a diffusion tensor are successfully used in many image processing and computer vision applications, ranging from image denoising over compression to optic flow computation. However, finding adequate numerical schemes is difficult: Implementations may suffer from dissipative artifacts, poor approximation of rotation invariance, and they may lack provable stability guarantees. In our paper we propose a general framework for finite difference discretisations of anisotropic diffusion filters on a 3 x 3 stencil. It is based on a gradient descent of a discrete quadratic energy where the occurring derivatives are replaced by classical as well as the widely unknown nonstandard finite differences in the sense of Mickens. This allows a large class of space discretisations with two free parameters. Combining it with an explicit or semi-implicit time discretisation, we establish a general and easily applicable stability theory in terms of a decreasing Euclidean norm. Our framework comprises as many as seven existing space discretisations from the literature. However, we show that also novel schemes are possible that offer a better performance than existing ones. Our experimental evaluation confirms that the space discretisation can have a very substantial and often underestimated impact on the quality of anisotropic diffusion filters. © 2013 Springer-Verlag.