A four-pixel scheme for singular differential equations

Authors:
Martin Welk;Joachim Weickert;Gabriele Steidl
Affiliations:
Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Bldg. 27, Saarland University, Saarbrücken, Germany;Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Bldg. 27, Saarland University, Saarbrücken, Germany;Faculty of Mathematics and Computer Science, University of Mannheim, Mannheim, Germany
Venue:
Scale-Space'05 Proceedings of the 5th international conference on Scale Space and PDE Methods in Computer Vision
Year:
2005

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Abstract

Singular diffusion equations such as total variation (TV) and balanced forward–backward (BFB) diffusion are appealing: They have a finite extinction time, and experiments show that piecewise constant structures evolve. Unfortunately, their implementation is awkward. The goal of this paper is to introduce a novel class of numerical methods for these equations in the 2D case. They are simple to implement, absolutely stable and do not require any regularisation in order to make the diffusivity bounded. Our schemes are based on analytical solutions for 2×2-pixel images which are combined by means of an additive operator splitting (AOS). We show that they may also be regarded as iterated 2D Haar wavelet shrinkage. Experiments demonstrate the favourable performance of our numerical algorithm.