Numerical invariantization for morphological PDE schemes

Martin Welk; Pilwon Kim; Peter J. Olver

doi:10.1007/978-3-540-72823-8_44

Numerical invariantization for morphological PDE schemes

Martin Welk, Pilwon Kim, Peter J. Olver

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

8 Scopus citations

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.

Original language	English (US)
Title of host publication	Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Proceedings
Publisher	Springer Verlag
Pages	508-519
Number of pages	12
ISBN (Print)	9783540728221
DOIs	https://doi.org/10.1007/978-3-540-72823-8_44
State	Published - 2007
Event	1st International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2007 - Ischia, Italy Duration: May 30 2007 → Jun 2 2007

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	4485 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	1st International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2007
Country/Territory	Italy
City	Ischia
Period	5/30/07 → 6/2/07

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10.1007/978-3-540-72823-8_44

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Welk, M., Kim, P., & Olver, P. J. (2007). Numerical invariantization for morphological PDE schemes. In Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Proceedings (pp. 508-519). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4485 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-540-72823-8_44

Numerical invariantization for morphological PDE schemes. / Welk, Martin; Kim, Pilwon; Olver, Peter J.
Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Proceedings. Springer Verlag, 2007. p. 508-519 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4485 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Welk, M, Kim, P & Olver, PJ 2007, Numerical invariantization for morphological PDE schemes. in Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4485 LNCS, Springer Verlag, pp. 508-519, 1st International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2007, Ischia, Italy, 5/30/07. https://doi.org/10.1007/978-3-540-72823-8_44

Welk M, Kim P, Olver PJ. Numerical invariantization for morphological PDE schemes. In Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Proceedings. Springer Verlag. 2007. p. 508-519. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-540-72823-8_44

Welk, Martin ; Kim, Pilwon ; Olver, Peter J. / Numerical invariantization for morphological PDE schemes. Scale Space and Variational Methods in Computer Vision, First International Conference, SSVM 2007, Proceedings. Springer Verlag, 2007. pp. 508-519 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Numerical invariantization for morphological PDE schemes

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