Relative Persistent Homology

Authors Nello Blaser , Morten Brun

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Author Details

Nello Blaser
  • Department of Informatics, University of Bergen, Norway
Morten Brun
  • Department of Mathematics, University of Bergen, Norway

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • topological data analysis
  • relative homology
  • Delaunay-Čech complex
  • alpha complex


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