The Extended Persistent Homology Transform for Manifolds with Boundary

The Persistent Homology Transform (PHT) is a topological transform introduced by Turner, Mukherjee and Boyer in 2014. Its input is a shape embedded in Euclidean space; then to each unit vector the transform assigns the persistence module of the height function over that shape with respect to that direction. The PHT is injective on piecewise-linear subsets of Euclidean space, and it has been demonstrably useful in diverse applications as it provides a landmark-free method for quantifying the distance between shapes. One shortcoming is that shapes with different essential homology (i.e., Betti numbers) have an infinite distance between them. The theory of extended persistence for Morse functions on a manifold was developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes. By using extended persistence modules of height functions over a shape, we obtain the extended persistent homology transform (XPHT) which provides a finite distance between shapes even when they have different Betti numbers. t may seem that the XPHT requires significant additional computational effort, but recent work by Katharine Turner and myself shows that when A is a compact n-manifold with boundary X, embedded in n-dimensional Euclidean space, the XPHT of A can be derived from the PHT of X, and a signature for each local minimum of the height function on X. James Morgan has implemented the required algorithms for 2-dimensional binary images as an R-package. This talk will provide an outline of our results and illustrate their application to shape clustering, and symmetry quantification. These applications were studied by former students Jency Jiang and Nicholas Bermingham.