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Computing the Multicover Bifiltration

Authors René Corbet , Michael Kerber , Michael Lesnick , Georg Osang



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Author Details

René Corbet
  • Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
Michael Kerber
  • Institute of Geometry, Graz University of Technology, Austria
Michael Lesnick
  • Department of Mathematics and Statistics, University at Albany, SUNY, NY, USA
Georg Osang
  • IST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria

Acknowledgements

The authors want to thank the reviewers for many helpful comments and suggestions.

Cite AsGet BibTex

René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the Multicover Bifiltration. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 27:1-27:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.27

Abstract

Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Computational geometry
Keywords
  • Bifiltrations
  • nerves
  • higher-order Delaunay complexes
  • higher-order Voronoi diagrams
  • rhomboid tiling
  • multiparameter persistent homology
  • denoising

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