Classification of Topological Contacts Between Phases in 3D Multiphase Systems by Betti Number Dynamics

A methodological approach to the quantitative description of the spatial relations between phases in multicomponent systems is developed, based on the comparison of Betti numbers of united phases. Two disjoint phases Xi and Xj in three-dimensional space are considered, each consisting of a finite number of connected components. Reference points R1 (absolute minimum), R2 = min (βk), R3 = max (βk), and R4 = ∑βk are introduced, with respect to which the value βk(Xi ∪ Xj) is analyzed. For each Betti number (k = 0,1,2) the possible types of topological contacts (unifying, filling, generating) are defined, and rigorous interpretation rules are established, linking the position of βk(Xi ∪ Xj) relative to the reference points to the presence and quantity of contacts. All interpretations are proved and summarized in tables. The obtained results create a theoretical basis for automated analysis of any three-dimensional distributed data, e.g., the tomographic, material science, and biomedical images, where it is necessary to characterize the spatial arrangement of phases without exhaustive pairwise comparison of objects. The ways of adaptation to real data (accounting for discreteness, boundary effects, and segmentation uncertainty) and directions for further development (generalization to more than two phases, use of persistent homology, and relation to physical properties) are outlined. The approach will widen and automatize the interpretation of a 3D structure of any volume of images in medical, geological, and material sciences.