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Relative Persistent Homology

Authors: Nello Blaser and Morten Brun

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

The alpha complex efficiently computes persistent homology of a point cloud X in Euclidean space when the dimension d is low. Given a subset A of X, relative persistent homology can be computed as the persistent homology of the relative Čech complex Č(X, A). But this is not computationally feasible for larger point clouds X. The aim of this note is to present a method for efficient computation of relative persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay-Čech complex DelČ(X, A) whose homology is the relative persistent homology. It is constructed from the Delaunay complex of an embedding of X in (d+1)-dimensional Euclidean space.

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Nello Blaser and Morten Brun. Relative Persistent Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 18:1-18:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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  author =	{Blaser, Nello and Brun, Morten},
  title =	{{Relative Persistent Homology}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{18:1--18:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.18},
  URN =		{urn:nbn:de:0030-drops-121762},
  doi =		{10.4230/LIPIcs.SoCG.2020.18},
  annote =	{Keywords: topological data analysis, relative homology, Delaunay-\v{C}ech complex, alpha complex}
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