The Extended Persistent Homology Transform for Manifolds with Boundary
Speaker: Vanessa Robins
Time: 11:00-12:30 2022/11/28
Venue: BIMSA 1120
Zoom: 787 662 9899 （PW: BIMSA）
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 ofthe 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 functionsover a shape, we obtain the extended persistent homology transform (XPHT) which provides a finite distance between shapes even when they have different Betti numbers.
It 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 manifold with boundary X, embedded in Euclidean space, the XPHT of A can be derivedfrom the PHT of 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 an illustration of their application to shape clustering.
Vanessa Robins is an associate professor in the Research School of Physics at the Australian National University. She develops theory and algorithmsfor the quantification of shape in data. Her major contributions include fundamental mathematical results for persistent homology, algorithm and software development for computing topological information from digital images, and their application to the characterisationof porous and granular materials.
[AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its Applications]