- AMSS-YMSC-BIMSA拓扑及应用进展联合讨论班

The Extended Persistent Homology Transform for Manifolds with Boundary

组织者

段海豹 , Fei Han , Yong Lin , 潘建中 , 魏国卫 , 吴杰 , 夏克林 , 丘成栋 , Chao Zhou

演讲者

Vanessa Robins

时间

2022年11月28日 11:00 至 12:30

地点

1120

线上

Zoom 518 868 7656 (BIMSA)

摘要

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.

演讲者介绍

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.