Interfacial Minkowski Tensors

• Interfacial Minkowski tensors are defined as rank-s tensor-valued measures that integrate symmetric products of boundary normals to capture detailed shape and orientation information.
• They generalize scalar Minkowski functionals and provide robust anisotropy indices, facilitating precise classification and discrimination in stochastic geometries and material structures.
• Applications span voxel-based imaging, random tessellations, and non-Gaussianity testing in cosmology, with mesh-based and local estimators offering reliable error control.

Interfacial Minkowski tensors are rank-$s$ tensor-valued measures defined on the boundary of compact sets in Euclidean space, which encode orientation and shape information by integrating symmetric tensor products of boundary normals. As a natural generalization of scalar Minkowski functionals, interfacial Minkowski tensors appear foundationally in integral geometry, stochastic geometry, and morphometric analysis of complex spatial structures. Their role is central in quantifying anisotropy, providing higher-order shape descriptors for random sets, tessellations, and fields, and underpinning advances in both theoretical classification theorems and robust applied estimators.

1. Definition and Principal Properties

For a compact set $K \subset \mathbb{R}^d$ with sufficiently regular boundary $\partial K$ (e.g., a finite union of sets of positive reach), the rank-$s$ interfacial Minkowski tensor is defined as

$$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$

where $\mathbf{n}(x)$ is the outward unit normal at $x \in \partial K$, $dS(x)$ is the $(d-1)$-dimensional Hausdorff measure, and $\otimes$ denotes the symmetric tensor product. In the most widely used case $K \subset \mathbb{R}^d$0, $K \subset \mathbb{R}^d$1 is a (symmetric) second-rank tensor, typically employed as the interfacial orientation tensor or surface-normal moment tensor.

For convex polytopes and bodies with piecewise smooth boundary, $K \subset \mathbb{R}^d$2 admits discrete representations as facet sums. For a surface mesh of planar facets $K \subset \mathbb{R}^d$3,

$K \subset \mathbb{R}^d$4

with $K \subset \mathbb{R}^d$5 the facet area and $K \subset \mathbb{R}^d$6 its unit normal (Klatt et al., 2016).

The trace of $K \subset \mathbb{R}^d$7 recovers the surface area: $K \subset \mathbb{R}^d$8. In two dimensions, the analogous object integrates $K \subset \mathbb{R}^d$9 over the boundary curve, with trace equal to the perimeter (Appleby et al., 2017).

2. Local and Global Tensor Valuations

Minkowski tensors generalize scalar intrinsic volumes via valuations on convex bodies and their boundaries. The global tensor for a convex $\partial K$0 is

$\partial K$1

where $\partial K$2 is the $\partial K$3-th support (curvature) measure, $\partial K$4 is the normal cycle, and $\partial K$5 the boundary normal (Hug et al., 2016, Hug et al., 31 Jan 2025). The interfacial tensor $\partial K$6 corresponds to $\partial K$7.

Local (measure-valued) versions assign tensor values to Borel subsets of $\partial K$8, with integral representations over local portions of $\partial K$9. Explicit polytope formulas localize sums over faces and account for face orientation and geometry, admitting full SO($s$0)-covariant bases (Hug et al., 2016).

These tensor valuations are translation- and rotation-covariant. For polytopes in dimensions 2 and 3, the classification theorems yield explicit bases—including extra orientation-sensitive tensors when restricting covariance to SO($s$1) (Hug et al., 2016).

3. Anisotropy Indices and Shape Discrimination

Diagonalization of $s$2 yields real eigenvalues $s$3. In practical applications, the anisotropy or elongation index

$s$4

quantifies deviation from isotropy: $s$5 for a sphere or regular polytope, with smaller values indicating increased anisotropy (Klatt et al., 2016, Hug et al., 31 Jan 2025).

Averaged over an ensemble of cells or grains, $s$6 provides a global interfacial anisotropy metric. Coupled to isoperimetric-type indices such as $s$7 (which equals 1 for a sphere), joint plots $s$8 effectively discriminate among stochastic tessellation models including Poisson–Voronoi, Gibbs, STIT, and hyperplane tessellations, as well as equilibrium sphere packings or foams (Klatt et al., 2016).

In Boolean models, anisotropy in the aggregate is captured by the eigenstructure of the global density tensor $s$9, reflecting orientational order and the degree of alignment of grains. For isotropic models, tensor densities collapse to scalar multiples of intrinsic volumes, while in non-isotropic settings, eigenvalue splitting grows with alignment (Hörrmann et al., 2013).

4. Application Domains and Estimation Methodologies

The broad applicability of interfacial Minkowski tensors encompasses:

• Random Tessellations and Foams: Used to analyze shape statistics of cells in stochastic geometries and material microstructures. Software such as "papaya" (2D) and "karambola" (3D) provides implementation for surface/voxel data (Klatt et al., 2016).
• Statistical Physics and Random Fields: Interfacial tensors quantify morphological anisotropy in level sets of Gaussian and non-Gaussian fields. Closed-form expressions are available for expected tensors in Gaussian random fields, depending only on the gradient covariance matrix. Higher-rank tensor information is fully determined by the second-rank tensor in the Gaussian case (Klatt et al., 2021).
• Voxelized and Imaging Data: For digital images and voxel-based data, robust, asymptotically unbiased estimators using Voronoi tensor sums and linear extrapolation schemes accurately recover interfacial tensors for unions of sets of positive reach, with dimension-independent error control (Hug et al., 31 Jan 2025).
• Composite Materials and Nanostructures: Applied to AFM/SEM scans of metallic grains and nanorough surfaces, interfacial tensors distinguish distributions of orientation and correlations with surface roughness (Hug et al., 31 Jan 2025).

In numerical practice, calculation is via mesh-based (facet sum) formulas in explicit geometry or local estimators on pixel/voxel grids, with recent estimator frameworks providing error guarantees and applicability across arbitrary dimension (Hug et al., 31 Jan 2025).

5. Theoretical Advances: Classification and Isotropy

Under SO($$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$0)-covariance, the full classification of local tensor valuations is established for polytopes, with distinct bases in low dimensions. In 3D, this includes edge-type tensors involving the cross product of edge tangents and normals, relevant for granular networks and foams (Hug et al., 2016). For smooth convex grains, interfacial tensor densities are directly related to the mean curvature radius function, with reconstruction achievable via harmonic decomposition in 2D (Hörrmann et al., 2013).

For isotropic ensembles, all information is encoded by scalar intrinsic volumes and the metric tensor; for non-isotropic models, tensor densities reveal the orientation distribution and symmetry breaking, enabling statistical estimation of alignment parameters.

6. Quantitative Morphological Analysis and Null-Hypothesis Testing

A pivotal feature of interfacial Minkowski tensors is their deployment in quantitative morphology and non-Gaussianity testing. In cosmology, for example, the translation-invariant interfacial tensor $$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$1 detects global anisotropies in excursion sets of the galaxy-density field or CMB maps. Global eigenvalue ratios ($$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$2) and averaged local shape indices ($$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$3) provide robust probes of symmetry breaking due to physical effects such as redshift-space distortions or non-linear structure formation (Appleby et al., 2017).

Crucially, in Gaussian random fields, a fundamental result is that all higher-rank interfacial Minkowski tensors are determined by the second-rank tensor; measuring deviations at higher order enables model-free null-hypothesis tests for non-Gaussian features (Klatt et al., 2021).

7. Summary Table: Key Quantities in Interfacial Minkowski Tensor Analysis

Quantity	Definition/Computation	Interpretation
$$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$4	$$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$5	Surface orientation tensor
$$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$6	$$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$7 of $$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$8	Anisotropy/elongation index
$$W_1^{0,s}(K) = \int_{\partial K}\, \mathbf{n}(x)^{\otimes s}\; dS(x),$$9	$\mathbf{n}(x)$0	Isoperimetric-type shape index
$\mathbf{n}(x)$1	Global density tensor in random/Boolean models	Aggregate orientation information
$\mathbf{n}(x)$2	$\mathbf{n}(x)$3 via irreducible tensor decomposition	Degree of interface anisotropy (2D)

Computation methods, shape indices, and tensor densities together enable the rigorous analysis and discrimination of stochastic spatial processes, with applications spanning materials science, cosmology, statistical physics, and image analysis (Klatt et al., 2016, Appleby et al., 2017, Hug et al., 2016, Hörrmann et al., 2013, Klatt et al., 2021, Hug et al., 31 Jan 2025).