Quantitative description of internal 3D structure of a geological sample using algebraic topology methods

Description of the internal spatial structure and texture of geological objects is an important element in their study. A qualitative description of the internal morphology of samples, despite its clarity and wide applicability, does not have a sufficient degree of accuracy and detail. Quantitative description would provide the information in a compact form. This information can be used straightforwardly for classification and comparison with other samples and for evaluation of effective physical characteristics. Integral geometry and algebraic topology provide a set of powerful tools for the quantitative characterization of complex spatial structures regardless of their nature. An approach to quantitative characterization of the spatial structure and texture using the example of segmented rock sample has been developed in this paper. Segmentation in our paper means distinguishing X-ray density phases. The approach is based on integral geometry and algebraic topology. The example of a meimechite sample from the Kontozero complex demonstrates the applicability of this approach for the quantitative analysis of structures in geological sample. A classifier for objects of selected X-ray density phases of geological samples based on their integral-geometric characteristics has been developed, and the stability of its operation has been demonstrated. Detailed analysis of the algebraic-topological characteristics for the selected X-ray density phases and their combinations has been carried out. The obtained results have been meaningfully interpreted in geological-genetic terms. The developed approach will allow describing the volume distribution of more than two phases in any natural and synthesized substance or even space of properties. In this case, the phase can also be understood as the distribution of properties, which makes the approach widely applicable in many science sections.