Computing the multicover bifiltration

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

Abstract

Given a finite set A ⊂ ℝd, let Covr,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.

Original languageEnglish
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
ISBN (Electronic)9783959771849
DOIs
Publication statusPublished - 1 Jun 2021
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume189
ISSN (Print)1868-8969

Conference

Conference37th International Symposium on Computational Geometry, SoCG 2021
Country/TerritoryUnited States
CityVirtual, Buffalo
Period7/06/2111/06/21

Keywords

  • Bifiltrations
  • Denoising
  • Higher-order Delaunay complexes
  • Higher-order Voronoi diagrams
  • Multiparameter persistent homology
  • Nerves
  • Rhomboid tiling

ASJC Scopus subject areas

  • Software

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