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 persistent homology can be computed as the persistent homology of the relative \v{C}ech complex. But this is not computationally feasible for larger point clouds. 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 \v{C}ech complex whose homology is the relative persistent homology. It can be constructed from the Delaunay complex of an embedding of the point clouds in $(d+1)$-dimensional Euclidean space.