Decomposing the Persistent Homology Transform of Star-Shaped Objects

• Main Authors: Arya, Shreya, Giunti, Barbara, Hickok, Abigail, Kanari, Lida, McGuire, Sarah, Turner, Katharine
• Format: Journal Article
• Language: English
• Published: 27.08.2024
• Subjects: Mathematics - Algebraic Topology
• DOI: 10.48550/arxiv.2408.14995

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.