Computing extended persistent homology of radial distance filtrations of Euclidean shapes

Abstract

We study the extended persistent homology of the radial filtration of a shape $M\subseteq {\mathbb{R}}^n$. A radial filtration is formed by choosing a center point $c$ and taking points of $M$ within distance $r$ of $c$. We show that, under mild assumptions, we can recover the radial extended persistence of a manifold with boundary from the radial persistence of its boundary. We also establish algorithms to compute radial extended persistence when $M$ is a set of pixels in a 2D digital grid. The methods are similar to those used to compute the extended persistent homology of a height filtration of a shape embedded in Euclidean space. We envisage these results will be useful in biomedical image analysis settings where it is natural to consider a radial filtration with respect to a fixed center, for example in the study of neuronal structures.