Decomposing the Persistent Homology Transform of Star-Shaped Objects

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 R² can be derived from the degree-$0$ persistence diagrams of its sectors. Using this, we then establish sufficient conditions for star-shaped objects in R² so that they have ``trivial geometric monodromy''. Consequently, the PHT of such a shape can be decomposed as a union of curves parameterized by S¹, where the curves are given by the continuous movement of each point in the persistence diagrams that are parameterized by S¹. Finally, we discuss the current challenges of generalizing these results to higher dimensions.