The Extended Persistent Homology Transform of manifolds with boundary

TL;DR

The paper introduces the Extended Persistent Homology Transform (XPHT), a topological shape descriptor that captures shape features with finite distances even for shapes with different Betti numbers, using Morse theory and efficient algorithms.

Contribution

It extends the Persistent Homology Transform to include boundary information and develops an algorithm for efficient computation of XPHT for shapes and images.

Findings

XPHT provides finite shape distances with different Betti numbers.

Morse theory relates boundary persistence to the full shape.

Efficient algorithm exploits boundary curve relationships.

Abstract

The Extended Persistent Homology Transform (XPHT) is a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We can define a distance between two shapes by integrating over the sphere the distance between their respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different Betti numbers. We use Morse theory to show that the extended persistence of a height function over a manifold with boundary can be deduced from the extended persistence for that height function restricted to the boundary, alongside labels on the critical points as positive or negative critical. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.