Topological Delaunay Graph for Efficient 3D Binary Image Analysis

Topological data analysis (TDA) based on persistent homology (PH) has become increasingly popular in automation technology. Recent advances in imaging and simulation techniques demand TDA for 3D binary images, but it is not a trivial task in practice, especially in terms of the computational speed of PH. This paper proposes a simple and efficient computational framework to extract topological features of 3D binary images by estimating persistence diagrams (PDs) for 3D binary images. The proposed framework is based on representing a 3D binary image by constructing a topological Delaunay graph with distance edge weights as a Rips complex, and it utilizes PD computation libraries for the constructed graph. The vertices, edges, and edge weights of the proposed graph correspond to connected-components (CCs) in the 3D binary image, Delaunay edges of the generalized Voronoi diagram for the CC boundaries, and minimum distances between adjacent CCs, respectively. Thus, the number of elements required to compute PD is significantly reduced for large objects in 3D binary images compared with conventional representations such as cubical complexes, which results in efficient topological feature estimations.