Numerical invariantization for morphological PDE schemes

Authors:
Martin Welk;Pilwon Kim;Peter J. Olver
Affiliations:
Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Saarbrücken, Germany;Department of Mathematics, Ohio State University, Columbus, Ohio;School of Mathematics, University of Minnesota, Minneapolis, MN
Venue:
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Year:
2007

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Abstract

Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this paper, we make the first step to apply this framework to the differential equations of image processing. We focus on the Hamilton-Jacobi equation governing dilation and erosion processes which displays morphological symmetry, i.e. is invariant under strictly monotonically increasing transformations of gray-values. Results demonstrate that invariantization is able to handle the specific needs of differential equations applied in image processing, and thus encourage further research in this direction.