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Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization

Authors Dominique Attali, André Lieutier

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Dominique Attali
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
André Lieutier
  • Dassault systèmes, Aix-en-Provence, France

Acknowledgements

We are grateful to the anonymous referees for carefully reading the paper and many helpful suggestions.

Cite AsGet BibTex

Dominique Attali and André Lieutier. Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.8

Abstract

In this paper, we focus on one particular instance of the shape reconstruction problem, in which the shape we wish to reconstruct is an orientable smooth submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the submanifold (such as the Čech complex or the Rips complex), we recast the reconstruction problem as a 𝓁₁-norm minimization problem in which the optimization variable is a chain of K. Providing that K satisfies certain reasonable conditions, we prove that the considered minimization problem has a unique solution which triangulates the submanifold and coincides with the flat Delaunay complex introduced and studied in a companion paper [D. Attali and A. Lieutier, 2022]. Since the objective is a weighted 𝓁₁-norm and the contraints are linear, the triangulation process can thus be implemented by linear programming.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • manifold reconstruction
  • Delaunay complex
  • triangulation
  • sampling conditions
  • optimization
  • 𝓁₁-norm minimization
  • simplicial complex
  • chain
  • fundamental class

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