The extended persistent homology transform of manifolds with boundary

A shortcoming of persistent homology is that when two domains have different numbers of components or holes the persistence diagrams of any filtration will have an infinite distance between them. We address this issue by revisiting the theory of extended persistence, initially developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes for Morse functions on manifolds. We simplify the mathematical treatment of extended persistence by formulating it as a persistence module derived from a sequence of relative homology groups for pairs of spaces. Then, for n-manifolds with boundary embedded in Rⁿ, we use Morse theory to show that the extended persistent homology of a height function over M can be deduced from the extended persistent homology of the same height function restricted to ∂M. As an application, we describe the extended persistent homology transform (XPHT); 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 define a distance between two shapes by integrating over the sphere the distance between the respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different numbers of essential classes. 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.