Two Measures for the Homology Groups of Binary Volumes

Aldo Gonzalez-Lorenzo (1, 2, 3) , Alexandra Bac (2, 3) , Jean-Luc Mari (2, 3) , Pedro Real (1)

1 Universidad de Sevilla / University of Sevilla
2 GMOD-LSIS - GMOD-LSIS
3 AMU - Aix Marseille Université

Given a binary object (2D or 3D), its Betti numbers characterize the number of holes in each dimension. They are obtained algebraically, and even though they are perfectly defined, there is no unique way to display these holes. We propose two geometric measures for the holes, which are uniquely defined and try to compensate the loss of geometric information during the homology computation: the thickness and the breadth. They are obtained by filtering the information of the persistent homology computation of a filtration defined through the signed distance transform of the binary object.

Keywords: binary object; distance transform; persistent homology; geometric information; holes

Domains: Computer Science [cs]; Computer Science [cs]; Graphics [cs.GR]; Computer Science [cs]; Computational Geometry [cs.CG]; Computer Science [cs]; Other [cs.OH]

Aldo Gonzalez-Lorenzo, Alexandra Bac, Jean-Luc Mari, Pedro Real. Two Measures for the Homology Groups of Binary Volumes. 19th IAPR International Conference on Discrete Geometry for Computer Imagery (DGCI 2016), Apr 2016, Nantes, France. pp.154-165, ⟨10.1007/978-3-319-32360-2_12⟩. ⟨hal-01341015⟩