A Faithful Discretization of Verbose Directional Transforms
Abstract The persistent homology transform, Betti function transform, and Euler characteristic transform represent a shape with a multiset of persistence diagrams, Betti functions, or Euler characteristic functions, respectively, parameterized by the sphere of directions in the ambient space. In this work, we give the first explicit construction of finite sets of directions discretizing the verbose variants of these transforms and show that such discretizations faithfully represent the underlying shape. Our discretization, while exponential in the dimension of the shape, does not depend on any restrictions on the particular immersion beyond general position, and is stable with respect to various perturbations.
Authors
- Lucia Williams (ORCID: https://orcid.org/0000-0003-3785-0247)
- David L. Millman (ORCID: https://orcid.org/0000-0003-4112-3026)
- Samuel Micka (ORCID: https://orcid.org/0000-0003-3814-9727)
- Brittany Terese Fasy (ORCID: https://orcid.org/0000-0003-1908-0154)
- Anna Schenfisch (ORCID: https://orcid.org/0000-0003-2546-5333)
Institutions
- Western Colorado University (US)
- Montana State University (US)
- University of Montana (US)
- KTH Royal Institute of Technology (SE)
Publication Details
- Journal
- Discrete & Computational Geometry
- Published
- 2025-11-25
- DOI
- https://doi.org/10.1007/s00454-025-00791-w
- Primary Topic
- Topological and Geometric Data Analysis
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- National Science Foundation
- Montana State University
- Eidgenössische Technische Hochschule Zürich
- Nederlandse Organisatie voor Wetenschappelijk Onderzoek