Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Arya, Shreya, Giunti, Barbara, Hickok, Abigail, Kanari, Lida, Scullen, Sarah McGuire, Turner, Katharine
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: PostDoctoral Research Fellowship, Statistical shape analysis using persistent homology, Travel: Third Workshop for Women in Computational Topology, Collaborative Research: AF: Medium: A Unified Framework for Geometric and Topological Signature-Based Shape Comparison, Multi-parameter Persistent Homology, G7820977245, Finding emergent structure in multi-sample biological data with the dual geometry of cells and features
OpenAlex ID: W4402705071