Numerical Invariantization for Morphological PDE Schemes

  • Martin Welk
  • Pilwon Kim
  • Peter J. Olver
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4485)


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.


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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Martin Welk
    • 1
  • Pilwon Kim
    • 2
  • Peter J. Olver
    • 3
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, 66041 SaarbrückenGermany
  2. 2.Department of Mathematics, Ohio State University, Columbus, Ohio 43210U.S.A.
  3. 3.School of Mathematics, University of Minnesota, Minneapolis, MN 55455U.S.A.

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