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The Extended Persistent Homology Transform of Manifolds With Boundary
(2022)
Katharine Turner, Vanessa Robins, James Morgan
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.