Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures

Abstract

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.

Techniques for Computing Fabric Tensors: A Review

“…Recently, the Minkowski tensors have been proposed as an elegant a way to integrate boundary-and volume-based techniques [84,83]. Six linearly independent Minkowski tensors are defined in 3D:…”

“…These tensors are called the moment tensor solid, moment tensor hollow, moment tensor wireframe, moment tensor vertices, normal distribution and curvature distribution tensors respectively [84]. Notice that the moment tensor solid and the normal distribution tensor are closely related to the inertia tensor and the GST respectively.…”

“…This value is inversely proportional to the number of intercepts between a set of parallel lines and the interface between phases (see Figure 2). The MIL tensor is obtained either by applying ellipse/ellipsoid fitting [84,83] Diffusion tensor imaging (DTI) [80,8] Texture tensor [23] Skeleton-driven [41] Assessment of the power spectrum [7] algorithms to polar plots of the MIL computed in different orientations, also known as rose diagrams, or by computing a covariance matrix [64,86,38]. Although the orientation distribution of the MIL can also be approximated through higher-order fabric tensors [38], microstructural architecture of most materials can be accurately modeled by means of second-order tensors [93,49].…”

Tomographic analysis of jammed ellipsoid packings

“…The sizes, positions and orientations of the ellipsoids can be extracted from the volume, centers of mass and the moment tensors W 2,0 0 = grain r ⊗ r dV (similar to the tensor of inertia) where r is the position vector [18] of the labelled clusters.…”

Shape analysis of counts maps

“…The shape analysis based on the Minkowski functionals area A, perimeter P and Euler characteristic χ has been successfully applied in statistical physics [6][7][8][9], for pattern analysis [10][11][12] and in astronomy for point processes in cosmology [13][14][15] and the cosmic microwave background [16]. A fundamental theorem by Hadwiger ensures robustness and comprehensiveness of a morphology analysis based on Minkowski functionals, in the following sense [17]: any functional which is defined on unions of convex sets and which is motion invariant, additive and at least continuous on convex sets is a linear combination of Minkowski functionals; black and white pixel images can for example be represented as the union of quadratic bins, i.e convex sets.…”