On the Implementation of the Algorithm for Calculating the Functionals of Minkowski Three-Dimensional Digital Space

Abstract

The paper is devoted to the implementation of an algorithm for calculating the functionals of the Minkowski set in a three-dimensional digital space. The algorithm is based on the calculation of the values of these functionals for various types of node neighborhoods into which a set in a digital space can be divided. The concept of Minkowski functionals appeared in the theory of convex sets in n-dimensional Euclidean space; they are coefficients in the expansion of the volume function ε of a neighborhood of a convex set in powers of ε. Subsequently, it turned out that the concept of functionals can be generalized to the case of sets with singularities, including the case of a set in a digital space. Minkowski functionals of a digital image representing a union cubic voxel intersecting along edges and vertices are statistical measures based on the Euler-Poincare characteristic of n-dimen-sional space, show sensitivity to the morphology of disordered structures, which is confirmed by applied research. They are used in calculating density measures for a number of unordered microstructural models; particle-based models, amorphous microstructures, cellular and foamlike structures. The results of calculations for various microstructures demonstrate a number of qualitative characteristics.
 The paper studies the implementation of the algorithm for finding the Minkowski functionals for a set in a threedimensional digital space.