Vector- and Tensor-Valued Descriptors for Spatial Patterns

Beisbart, Claus; Dahlke, Robert; Mecke, Klaus; Wagner, Hermann‐Josef

doi:10.1007/3-540-45782-8_10

Cited by 47 publications

(41 citation statements)

References 39 publications

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“…Finally in this Introduction, we mention that the Minkowski tensors have also begun to play a role (at least, up to rank two) in the applied sciences, as tools in the morphometry of spatial patterns; see [9,8] …”

Section: §1 Introductionmentioning

confidence: 99%

The space of isometry covariant tensor valuations

Hug

Schneider

Schuster

2007

St. Petersburg Math. J.

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Abstract. It is known that the basic tensor valuations which, by a result of S. Alesker, span the vector space of tensor-valued, continuous, isometry covariant valuations on convex bodies, are not linearly independent. P. McMullen has discovered linear dependences between these basic valuations and has implicitly raised the question as to whether these are essentially the only ones. The present paper provides a positive answer to this question. The dimension of the vector space of continuous, isometry covariant tensor valuations, of a fixed rank and of a given degree of homogeneity, is explicitly determined. The approach is constructive and permits one to provide a specific basis. §1. Introduction Alesker [6,7] determined these spaces and their dimensions explicitly.As a natural generalization of the motion invariant real valuations, McMullen [14] introduced isometry covariant tensor valuations, and he formulated the aim to find a characterization of such valuations, under continuity assumptions. To explain this, we denote by T r the vector space of symmetric tensors of rank r ∈ N 0 (the nonnegative integers) over R n (we use the scalar product of R n to identify R n with its dual space). The symmetric tensor product of symmetric tensors a, b is denoted by ab. We write x r for the r-fold symmetric tensor product of x ∈ R n . The normalization is chosen so that x r = x ⊗ · · · ⊗ x (with r factors x). A tensor valuation on K n is a valuation on K n 2000 Mathematics Subject Classification. Primary 52A20.

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Section: §1 Introductionmentioning

confidence: 99%

The space of isometry covariant tensor valuations

Hug

Schneider

Schuster

2007

St. Petersburg Math. J.

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“…These scalar functionals as well as their vector-valued counterparts have been investigated for a long time in mathematics [75,76] as well as in physics [77]. Their higher-rank tensorial generalizations have only recently become used in the context of physics and material science [57 -59,78-81] (see [57,78] for an overview).…”

Section: Minkowski Functionals and Tensorsmentioning

confidence: 99%

Tensorial Minkowski functionals of triply periodic minimal surfaces

2012

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A fundamental understanding of the formation and properties of a complex spatial structure relies on robust quantitative tools to characterize morphology. A systematic approach to the characterization of average properties of anisotropic complex interfacial geometries is provided by integral geometry which furnishes a family of morphological descriptors known as tensorial Minkowski functionals. These functionals are curvature-weighted integrals of tensor products of position vectors and surface normal vectors over the interfacial surface. We here demonstrate their use by application to non-cubic triply periodic minimal surface model geometries, whose Weierstrass parametrizations allow for accurate numerical computation of the Minkowski tensors.

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“…Minkowski tensors have already been shown to be relevant morphological parameters for simple models of fl uids of non-spherical particles, [ 22 , 23 ] of protein conformations, [ 42 ] and of transport of molecular motors. [ 51 ] They have been applied as morphology and anisotropy indices for neuronal cells, [ 52 ] galaxies, [ 53 ] jamming in granular bead packs, [ 54 ] fl uid models and Poisson point processes, [ 55 ] and Turing patterns. [ 56 ] As is the case for the scalar indices, Minkowski tensors are based on rigorous mathematical theory, [ 57 -60 ] with statements regarding morphological completeness equivalent to the scalar case.…”

Section: Minkowski Tensor Shape Analysis Of Cellular Granular and Pomentioning

confidence: 99%

Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures

et al. 2011

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Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline copper.

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Vector- and Tensor-Valued Descriptors for Spatial Patterns

Cited by 47 publications

References 39 publications

The space of isometry covariant tensor valuations

The space of isometry covariant tensor valuations

Tensorial Minkowski functionals of triply periodic minimal surfaces

Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures

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