Minkowski Tensors of Anisotropic Spatial Structure

TL;DR

This paper introduces Minkowski tensors, tensor-valued measures for analyzing anisotropic spatial structures, with explicit algorithms for 3D shapes, bridging mathematical and physical approaches for applications in materials and biological sciences.

Contribution

It presents a novel tensor-based morphological analysis method with efficient algorithms, connecting mathematical theory with practical applications in physical sciences.

Findings

Explicit linear-time algorithms for 3D Minkowski tensors

Application to complex microstructured materials and biological networks

Bridging mathematical and physical approaches to Minkowski functionals

Abstract

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences.