Efficient Persistence Diagram Estimation for 3D Binary Images

Abstract

Persistent homology (PH) has become the most useful tool in topological data analysis, and is applied in a wide range of scientific and engineering fields. Furthermore, recent advances in imaging technology have made it crucial to develop efficient PH methods for images. In this research, we propose an efficient computational framework to estimate persistence diagrams (PDs) for 3D binary images. Our framework is based on approximating a 3D binary image by a Vietoris-Rips (VR) complex, and utilizes a popular PH library (GUDHI/Ripser) to compute its PD. Our VR-complex consists of a weighted proximity graph of connected regions in the 3D binary image, where its connectivity is determined by the geometric dual of the generalized Voronoi diagram obtained by the fast GPU-based method (PBA+) for the region boundaries, and half of the minimum distance between two regions is assigned to each edge weight. The vertex number of our graph is equal to the number of connected regions. Hence, the number of elements required to compute PD is significantly reduced compared with conventional representations such as alpha and cubical complexes consisting of dense voxels, which results in efficient PD estimations.