A Faithful Discretization of the Verbose Persistent Homology Transform

The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT such that it faithfully represents the underlying shape. We provide a discretization that is exponential in the dimension of the shape. Moreover, we show that this discretization is stable with respect to various perturbations and we provide an algorithm for computing the discretization. Our approach relies only on knowing the heights and dimensions of topological events, which means that it can be adapted to provide discretizations of other dimension-returning topological transforms, including the Betti function transform. With mild alterations, we also adapt our methods to faithfully discretize the Euler characteristic function transform.