Mathematical Pipeline for Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors

Three-dimensional images of multiphase natural and engineered materials obtained by X-ray micro-computed tomography require quantitative processing methods that can describe not only phase volume but also connectivity, spatial heterogeneity, and anisotropy. In this article, X-ray micro-computed tomography is abbreviated as X-μCT. Scalar descriptors such as fractal dimension, Betti numbers, Euler characteristic, and Minkowski functionals provide compact phase-level summaries of segmented X-μCT data, but they do not encode where structural heterogeneity occurs, whether connectivity is directionally spanning, how finite sample boundaries affect topological measurements, or how surface-normal orientation is distributed. We propose a unified methodological framework that extends scalar topological and Minkowski-functional analysis of segmented multiphase 3D images by adding cut-response analysis, including its boundary-sensitivity interpretation, directional connectivity and orientation descriptors, and the rank-two surface Minkowski tensor W10,2. The framework is demonstrated on a previously published segmented geological X-μCT volume used as a benchmark multiphase geometry with four X-ray-density phases and on synthetic validation geometries with analytically known topology. The results show that the proposed extensions reveal spatial sensitivity, boundary-to-boundary connectivity, and surface fabric that are not captured by scalar phase-level invariants alone. The proposed framework can be used to analyze segmented 3D images of multiphase geological, porous, composite, and engineered samples, thereby expanding quantitative knowledge about their internal structure beyond scalar phase-level descriptors.