Relative Persistent Homology

Abstract

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 Čech persistent homology can be computed as the persistent homology of the relative Čech complex C ˇ (X, A). However, 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 Čech persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay–Čech complex Del C ˇ (X, A) whose homology is the relative Čech persistent homology. It is constructed from the Delaunay complex of an embedding of X in (d+ 1) -dimensional Euclidean space.