A Faithful Discretization of the Persistent Homology Transform and Other Topological Transforms

Topological descriptors, such as persistence diagrams and Euler characteristic curves, have been shown to be useful for summarizing and differentiating shapes, both empirically and theoretically. One common summary tool is the Persistent Homology Transform (PHT), which represents a shape with a multiset of persistence diagrams parameterized by directions in the ambient space. For practical applicability, we must bound the number of directions needed in order to ensure that the PHT is a faithful representation of a shape. In this work, we provide such a bound for geometric simplicial complexes in arbitrary finite dimension and only general position assumptions (as opposed to bounding curvature). Furthermore, through our choice of proof method, we also provide an algorithm to reconstruct simplicial complexes in arbitrary finite dimension using an oracle to query for augmented persistence diagrams. In the process, we also describe a discretization of other topological transforms, including the Betti Curve Transform and Euler Characteristic Curve Transform.