Abstract
This paper 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 generalizations 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 paper 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 formalism more readily accessible for future application in the physical sciences.
Highlights
DEFINITION AND FUNDAMENTAL PROPERTIES OF MINKOWSKI TENSORSThe definition of scalar Minkowski functionals and their generalizations, used in the mathematical disciplines of integral and convex geometry, is based on measure theory, see section I-D
The morphology of complex spatial microstructures is often classified qualitatively into types such as cellular, porous, network-like, fibrous, percolating, periodic, lamellar, hexagonal, disordered, fractal, etc
Because of their tensorial nature they are explicitly sensitive to anisotropic and orientational aspects of spatial structure, and robust measures of intrinsic anisotropy and alignment can be derived (A shape measure is intrinsic if, for a homogeneous body, its value is independent of the size and shape of the observation window)
Summary
The definition of scalar Minkowski functionals and their generalizations, used in the mathematical disciplines of integral and convex geometry, is based on measure theory, see section I-D. For a compact set K with nonempty interior, called a body, embedded in Euclidean space 3, with a sufficiently smooth bounding surface ∂K, the scalar Minkowski functionals are defined as. The normalizing prefactor is chosen such that for a sphere BR at the origin with radius R the scalar Minkowski functionals are Wν (BR) = κ3 R3−ν ; κ3 = 4π/3 is the volume of the 3-dimensional unit sphere. For a three-dimensional body, this definition yields ten Minkowski tensors (not counting the ones that vanish by definition for all bodies) This set is linearly independent in the vector space of isometry-covariant additive functionals.
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