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The Localized Union-Of-Balls Bifiltration

Authors: Michael Kerber and Matthias Söls

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point q, we focus our attention to a ball centered at q whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the q-ball, and a relative variant where the homology of the q-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not k-critical for any finite k. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for 2-parameter persistence (which usually assume k-criticality for some finite k) carry over to the ∞-critical case.

Cite as

Michael Kerber and Matthias Söls. The Localized Union-Of-Balls Bifiltration. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 45:1-45:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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  author =	{Kerber, Michael and S\"{o}ls, Matthias},
  title =	{{The Localized Union-Of-Balls Bifiltration}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{45:1--45:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.45},
  URN =		{urn:nbn:de:0030-drops-178953},
  doi =		{10.4230/LIPIcs.SoCG.2023.45},
  annote =	{Keywords: Topological Data Analysis, Multi-Parameter Persistence, Persistent Local Homology}
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