Combinatorial Persistent Homology Transform

Abstract

The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on ${\mathbb{S}}^n$ to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes.

This is the final version of the paper

References in corpus (5)

• Sheaves, Cosheaves and Applications
• Edit Distance and Persistence Diagrams Over Lattices
• Spatio-temporal Persistent Homology for Dynamic Metric Spaces
• How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms
• Multiparameter persistent homology via generalized Morse theory

Cited by in corpus (1)

• A Faithful Discretization of the Verbose Persistent Homology Transform