Integral geometry of tensor valuations

Hug, Daniel; Schneider, Rolf; Schuster, Ralph

doi:10.1016/j.aam.2008.04.001

Cited by 63 publications

(86 citation statements)

References 15 publications

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“…The tensor

C_{m}

is equal to a linear combination of the Minkowski tensor

W_{1}^{0, 2}

(which is defined below, see Tables I and II) and the unit tensor multiplied by the surface area (or the perimeter for d = 2). The coefficients are explicitly given by so‐called Crofton formulas for Minkowski tensors . For local versions of these Crofton formulas, see Ref.…”

Section: Mean Intercept Length Of Anisotropic Boolean Modelsmentioning

confidence: 99%

See 1 more Smart Citation

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

2017

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Our analysis reveals several shortcomings of the mean intercept length tensor analysis that pose conceptual problems and limitations on the information content of this commonly used analysis method. We suggest the Minkowski tensors from integral geometry as alternative sensitive measures of anisotropy. The Minkowski tensors allow for a robust, comprehensive, and systematic approach to quantify various aspects of structural anisotropy. We show the Minkowski tensors to be more sensitive, in the sense, that they can quantify the remnant anisotropy of structures not captured by the mean intercept length analysis. If applied to porous tissue and microstructures, this improved structure characterization can yield new insights into the relationships between geometry and material properties.

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“…The tensor

C_{m}

is equal to a linear combination of the Minkowski tensor

W_{1}^{0, 2}

Section: Mean Intercept Length Of Anisotropic Boolean Modelsmentioning

confidence: 99%

“…The coefficients are explicitly given by so-called Crofton formulas for Minkowski tensors. 41,42 For local versions of these Crofton formulas, see Ref. [43].…”

Section: F Generalized Anisotropy Measuresmentioning

confidence: 99%

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

2017

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“…Now we shall derive a local Steiner formula and support measures, following [23] to obtain local support measures. The definition of MF and MT then follows directly as a special case [33,62]. The intuitive idea underlying the definition of a local parallel set is described in two steps.…”

Section: Definition Based On Fundamental Measure Theorymentioning

confidence: 99%

Minkowski tensors of anisotropic spatial structure

et al. 2013

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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 socalled Minkowski tensors. Minkowski tensors are generalizations of the wellknown 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.

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“…Some other important particular cases of valuations are given, for instance, when considering the vector space of symmetric tensors (see [2,9,20,30,37] for more information on tensor-valued valuations), or (K(V ), +) with + the Minkowski sum between two convex bodies (i.e. K + L = {x + y: x ∈ K, y ∈ L}).…”

Section: Introductionmentioning

confidence: 99%