Minkowski Tensors of Anisotropic Spatial Structure

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.

References in corpus (14)

• Disordered, Quasicrystalline and Crystalline Phases of Densely Packed Tetrahedra
• Shortcomings of the Bond Orientational Order Parameters for the Analysis of Disordered Particulate Matter
• Onset of mechanical stability in random packings of frictional spheres
• Discrete rearranging disordered patterns, part I: Robust statistical tools in two or three dimensions
• Diffusion and mixing in gravity-driven dense granular flows
• Permeability of porous materials determined from the Euler characteristic
• An invariant distribution in static granular media
• Jammed Spheres: Minkowski Tensors Reveal Onset of Local Crystallinity
• Local and Global relations between the number of contacts and density in monodisperse sphere packs
• Jammed frictional tetrahedra are hyperstatic
• Topological estimation of percolation thresholds
• Local Anisotropy of Fluids using Minkowski Tensors
• Morphometric analysis in gamma-ray astronomy using Minkowski functionals - Source detection via structure quantification
• Motion by Stopping: Rectifying Brownian Motion of Non-spherical Particles

Cited by in corpus (45)

• Hypostatic jammed packings of frictionless nonspherical particles
• Analyzing X-Ray tomographies of granular packings
• Appearance of the Single Gyroid Network Phase in Nuclear Pasta Matter
• Refractive Index Matched Scanning and Detection of Soft Particle
• Non-universal Voronoi cell shapes in amorphous ellipsoid packings
• Characterization of maximally random jammed sphere packings: Voronoi correlation functions
• Machine Learning Characterization of Structural Defects in Amorphous Packings of Dimers and Ellipses
• Morphology of 21cm brightness temperature during the Epoch of Reioinization using Contour Minkowski Tensor
• Search for anomalous alignments of structures in Planck data using Minkowski Tensors
• Tensor Minkowski Functionals: first application to the CMB
• Minkowski Tensors in Two Dimensions - Probing the Morphology and Isotropy of the Matter and Galaxy Density Fields
• Low density nuclear matter with quantum molecular dynamics : The role of the symmetry energy
• Local Anisotropy of Fluids using Minkowski Tensors
• Minkowski Tensors in Three Dimensions - Probing the Anisotropy Generated by Redshift Space Distortion
• Pomelo, a tool for computing Generic Set Voronoi Diagrams of Aspherical Particles of Arbitrary Shape
• Morphometric analysis in gamma-ray astronomy using Minkowski functionals - Source detection via structure quantification
• Network-like propagation of cell-level stress in random foams
• Voronoi-based estimation of Minkowski tensors from finite point samples
• Microstructure-based prediction of hydrodynamic forces in stationary particle assemblies
• Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors
• Tracking down the origin of superbubbles and supergiant shells in the Magellanic Clouds with Minkowski tensor analysis
• Ensemble Average of Three-Dimensional Minkowski Tensors of a Gaussian Random Field in Redshift Space
• Application of Contour Minkowski Tensor and $\mathcal{D}$ Statistic to the Planck $E$-mode data
• Morphology of CMB fields -- effect of weak gravitational lensing
• Minkowski Tensors in Redshift Space -- Beyond the Plane Parallel Approximation
• Low-temperature statistical mechanics of the QuanTizer problem: fast quenching and equilibrium cooling of the three-dimensional Voronoi Liquid
• The geometrical meaning of statistical isotropy of smooth random fields in two dimensions
• Kinetics of Fluid Demixing in Complex Plasmas: Domain Growth Analysis using Minkowski Tensors
• Rotation covariant local tensor valuations on convex bodies
• Detecting structured sources in noisy images via Minkowski maps
• $\mathrm{SO}(n)$ covariant local tensor valuations on polytopes
• Structural similarity between dry and wet sphere packings
• Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative
• Coarse-grained curvature tensor on polygonal surfaces
• Estimation of Minkowski tensors from digital grey-scale images
• Morphometry on the sphere: Cartesian and irreducible Minkowski tensors explained and implemented
• Green-Kubo stress correlation function at the atomic scale and a long-range bond-orientational ordering in a model liquid
• Representative elementary volume via averaged scalar Minkowski functionals
• Scale-Free Crystallization of two-dimensional Complex Plasmas: Domain Analysis using Minkowski Tensors
• Intelligence of agents produces a structural phase transition in collective behaviour
• Statistical Isotropy of the CMB E-mode signal
• Combinatorial Methods for Minkowski Tensors of Polytopes
• Local tensor valuations
• Symmetry and the salience of textures
• Surface tensor estimation from linear sections