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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 3, pp 359–379 | Cite as

Laplace–Beltrami Operator on Digital Surfaces

  • Thomas CaissardEmail author
  • David Coeurjolly
  • Jacques-Olivier Lachaud
  • Tristan Roussillon
Article
  • 300 Downloads

Abstract

This article presents a novel discretization of the Laplace–Beltrami operator on digital surfaces. We adapt an existing convolution technique proposed by Belkin et al. (in: Teillaud (ed) Proceedings of the 24th ACM symposium on computational geometry, College Park, MD, USA, pp 278–287, 2008,  https://doi.org/10.1145/1377676.1377725) for triangular meshes to topological border of subsets of \(\mathbb {Z}^n\). The core of the method relies on first-order estimation of measures associated with our discrete elements (such as length, area etc.). We show strong consistency (i.e., pointwise convergence) of the operator and compare it against various other discretizations.

Keywords

Laplace–Beltrami operator Digital surface Discrete geometry Differential geometry 

Notes

Supplementary material

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Authors and Affiliations

  1. 1.Univ Lyon, CNRS, INSA-Lyon, LIRIS, UMR 5205VilleurbanneFrance
  2. 2.Laboratoire de Mathématiques (LAMA), UMR 5127 CNRSUniversité Savoie Mont BlancChambéryFrance

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