Decomposing the Persistent Homology Transform of Star-Shaped Objects

TL;DR

This paper analyzes how the persistent homology transform of star-shaped objects in 2D can be decomposed into simpler components, revealing structural properties and conditions for trivial monodromy, with implications for higher dimensions.

Contribution

It introduces a geometric decomposition of the degree-0 PHT for star-shaped objects and establishes conditions for trivial monodromy, advancing understanding of shape analysis via persistent homology.

Findings

Persistence diagrams of sectors determine the whole shape's diagram.

Conditions for trivial geometric monodromy are established.

Decomposition of PHT into parameterized curves is demonstrated.

Abstract

In this paper, we study the geometric decomposition of the degree-$0$ Persistent Homology Transform (PHT) as viewed as a persistence diagram bundle. We focus on star-shaped objects as they can be segmented into smaller, simpler regions known as ``sectors''. Algebraically, we demonstrate that the degree-$0$ persistence diagram of a star-shaped object in $\mathbb{R}^2$ can be derived from the degree-$0$ persistence diagrams of its sectors. Using this, we then establish sufficient conditions for star-shaped objects in $\mathbb{R}^2$ so that they have ``trivial geometric monodromy''. Consequently, the PHT of such a shape can be decomposed as a union of curves parameterized by $S^1$, where the curves are given by the continuous movement of each point in the persistence diagrams that are parameterized by $S^{1}$. Finally, we discuss the current challenges of generalizing these results to higher dimensions.