Minkowski tensors of anisotropic spatial structure

Abstract

This article describes the theoretical foundation of and explicit algorithms\nfor a novel approach to morphology and anisotropy analysis of complex spatial\nstructure using tensor-valued Minkowski functionals, the so-called Minkowski\ntensors. Minkowski tensors are generalisations of the well-known scalar\nMinkowski functionals and are explicitly sensitive to anisotropic aspects of\nmorphology, relevant for example for elastic moduli or permeability of\nmicrostructured materials. Here we derive explicit linear-time algorithms to\ncompute these tensorial measures for three-dimensional shapes. These apply to\nrepresentations of any object that can be represented by a triangulation of its\nbounding surface; their application is illustrated for the polyhedral Voronoi\ncellular complexes of jammed sphere configurations, and for triangulations of a\nbiopolymer fibre network obtained by confocal microscopy. The article further\nbridges the substantial notational and conceptual gap between the different but\nequivalent approaches to scalar or tensorial Minkowski functionals in\nmathematics and in physics, hence making the mathematical measure theoretic\nmethod more readily accessible for future application in the physical sciences.\n