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arXiv · 2408.14995

Decomposing the Persistent Homology Transform of Star-Shaped Objects

Shreya Arya·Barbara Giunti·Abigail Hickok·Lida Kanari·Sarah McGuire Scullen·Katharine Turner
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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 $\R^2$ can be derived from the degree-$0$ persistence diagrams of its sectors. Using this, we then establish sufficient conditions for star-shaped objects in $\R^2$ so that they have ``trivial geometric monodromy''. Consequently, the PHT of such a shape can be decomposed as a union of curves parameterized by $\bS^1$, where the curves are given by the continuous movement of each point in the persistence diagrams that are parameterized by $\bS^{1}$. Finally, we discuss the current challenges of generalizing these results to higher dimensions.

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Shreya Arya, Barbara Giunti, Abigail Hickok, Lida Kanari, Sarah McGuire Scullen, Katharine Turner. 2024-08-27. Decomposing the Persistent Homology Transform of Star-Shaped Objects. https://arxiv.org/abs/2408.14995

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