Minkowski Tensors: Geometry and Applications

• Minkowski tensors are tensor-valued extensions of Minkowski functionals that encode size, topology, orientation, symmetry, and anisotropy in convex bodies.
• They replace scalar integrands with tensor powers of position vectors and normals to provide detailed shape analysis via eigenvalue ratios and irreducible representations.
• Their wide applications range from morphology, stochastic geometry, and discrete algorithms to analyzing anisotropy in fluids, granular materials, and cosmological fields.

Minkowski tensors are tensor-valued extensions of the classical intrinsic volumes—also called Minkowski functionals or quermassintegrals—of convex bodies. By replacing scalar integrands with tensor powers of the position vector and the outer unit normal, and integrating against curvature or support measures, they encode not only size and topology but also orientation, symmetry, and anisotropy. In that sense they refine the scalar Minkowski functionals while preserving the valuation structure that makes integral geometry effective for morphology, stochastic geometry, and random-field analysis (Livchitz et al., 2020, Collischon et al., 2022).

1. Integral-geometric definition and notational conventions

A standard modern definition starts from the Steiner formula for a convex body $K\subset\mathbb{R}^d$,

$$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$

where $V_0,\dots,V_d$ are the intrinsic volumes. Minkowski tensors are tensor-valued analogues of these $V_j$, defined with support measures $\Lambda_j(K,\cdot)$ by

$$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$

for $j=0,\dots,d-1$, together with the volume tensors

$$\Phi_d^{r,0}(K)=\frac{1}{r!}\int_K x^{\otimes r}\,\lambda(dx).$$

Their rank is $r+s$, and the scalar intrinsic volumes are recovered as $\Phi_j^{0,0}(K)=V_j(K)$ (Livchitz et al., 2020).

A second, widely used notation—especially in two-dimensional morphology—writes volume- and surface-type tensors as

$$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$0

with $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$1 and $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$2. In this notation the scalar Minkowski functionals appear as special cases: $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$3 is area, $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$4 is proportional to perimeter, and $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$5 is proportional to Euler characteristic in two dimensions (Collischon et al., 2022).

The global theory is complemented by integral-geometric formulae. Kinematic and Crofton formulae express the integral mean of Minkowski tensors of intersections—either with a second convex body moved by a proper rigid motion or with an affine subspace—in terms of linear combinations of Minkowski tensors of the original objects (Hug et al., 2017).

2. Geometric information beyond scalar Minkowski functionals

The main distinction between scalar Minkowski functionals and Minkowski tensors is directional sensitivity. In two dimensions, the scalar functionals record area, perimeter, and Euler characteristic, but they are blind to orientation and detailed anisotropy. An elongated ellipse and a circle with the same area and perimeter can therefore be difficult to distinguish using the scalars alone. Tensorial insertions of $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$6 and $$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$7 remove this blindness by encoding where contributions occur and how boundary normals are distributed (Collischon et al., 2022).

The canonical rank-2 examples are the position–position tensor

$$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$8

and the normal–normal tensor

$$\lambda(K+\varepsilon B^d)=\sum_{j=0}^{d}\varepsilon^{d-j}\kappa_{d-j}V_j(K),$$9

For an isotropic shape such as a disk, $V_0,\dots,V_d$0 is proportional to the identity. For an elongated shape, its eigenvalues separate along principal directions. The eigenvectors then encode preferred directions, while the eigenvalues quantify the degree of anisotropy (Collischon et al., 2022).

This enlargement from scalar to tensor-valued valuations has a structural counterpart in valuation theory. Hadwiger’s theorem classifies additive, continuous, motion-invariant scalar valuations by the scalar Minkowski functionals, whereas Alesker’s theorem plays the corresponding role for tensor-valued valuations. In the formulation emphasized in the spherical shape-analysis literature, Minkowski tensors form a complete set of additive, continuous, rotation-covariant morphological measures (Collischon et al., 2022).

Rank also matters. Rank-2 tensors capture elongation and coarse anisotropy, but they do not exhaust shape information. In jammed sphere packings, two Voronoi cells can be isotropic in the rank-2 sense and still differ substantially in higher-order facet arrangement. The rank-four tensor

$V_0,\dots,V_d$1

resolves this degeneracy and provides crystalline fingerprints for fcc and hcp Voronoi cells via the eigenvalues of its $V_0,\dots,V_d$2 matrix representation (Kapfer et al., 2012).

3. Scalar invariants, irreducible representations, and symmetry measures

Practical use almost always passes through scalar invariants derived from tensors. For a rank-2 symmetric Minkowski tensor, the most common anisotropy index is the eigenvalue ratio

$V_0,\dots,V_d$3

or equivalently $V_0,\dots,V_d$4. Values near $V_0,\dots,V_d$5 indicate isotropy; strong deviations indicate elongation or anisotropy. In two-dimensional Voronoi analysis this ratio is often used as a local order parameter, and in cosmological excursion-set studies the same principle is applied to the eigenvalues of $V_0,\dots,V_d$6 or $V_0,\dots,V_d$7 to characterize global anisotropy or mean object shape (Kapfer et al., 2010, Appleby et al., 2017).

A more refined organization uses irreducible representations. For planar interfacial tensors, the extended Gaussian image of the normal distribution admits a Fourier expansion, and its coefficients $V_0,\dots,V_d$8 serve as irreducible Minkowski tensors. The normalized magnitudes

$V_0,\dots,V_d$9

are rotationally invariant anisotropy measures of order $V_j$0. In this representation, higher-rank tensors indeed contain additional anisotropy information compared to a rank-two tensor. The Gaussian-random-field analysis of rank-$V_j$1, rank-$V_j$2, and higher tensors makes this explicit and shows that each rank probes a distinct angular sector of the normal distribution (Klatt et al., 2021).

For Gaussian random fields there is, however, an important simplification. The expected interfacial Minkowski tensors of arbitrary even rank are fixed by the covariance matrix of the field gradient, and the higher-rank anisotropy information can therefore be predicted from the second-rank tensor under the Gaussian hypothesis. That relation is independent of model details and was proposed as a null-hypothesis test for non-Gaussianity in anisotropic random fields (Klatt et al., 2021).

On curved data sets, Cartesian tensors can be replaced or supplemented by irreducible constructions. A spherical framework introduced irreducible Minkowski tensors specifically to avoid the redundancies of previous Cartesian representations, while retaining scalar anisotropy measures and preferred directions in the tangent plane (Collischon et al., 2024).

4. Computation on polytopes, pixelized fields, and discrete geometries

For convex polytopes, Minkowski tensors admit exact combinatorial expressions. In the surface-tensor case $V_j$3, one has

$V_j$4

so surface tensors become sums over facets weighted by facet volume tensors and facet normals. For simplicial polytopes, generating functions encode all surface tensors, and the numerator obtained after putting the generating function over a common denominator is the surface adjoint. This gives explicit combinatorial formulas in terms of facets, normals, and elementary symmetric functions of vertex coordinates (Livchitz et al., 2020).

In digital image analysis and random-field morphology, boundary reconstruction is typically performed with marching squares. In two-dimensional density-field applications, a pixelized excursion set is converted into a polygonal approximation of $V_j$5; the scalar functionals and the translationally invariant tensor $V_j$6 are then accumulated from edge segments. The same framework can be applied to individual connected regions and holes in order to estimate both global anisotropy and mean object shape from tensor eigenvalues (Appleby et al., 2017).

For spherical data in HEALPix format, the same logic was adapted to local quadrilateral patches on the sphere. Threshold crossings are linearly interpolated along pixel edges, local contour segments are approximated in the tangent plane or as short great-circle arcs, and tensor contributions are accumulated within local neighborhoods. Additivity,

$V_j$7

is the key property that makes localized “Minkowski maps” possible: local tensors are computed in sliding windows and then summed or averaged over larger regions (Collischon et al., 2022).

The same emphasis on additivity reappears in local tensor-valuation theory. Local Minkowski tensors generalize intrinsic volumes, curvature measures, and the isometry-covariant Minkowski tensors, and locally defined tensor measures on convex bodies that share isometry covariance and weak continuity admit a complete classification (Hug et al., 2013).

5. Applications in disordered matter, fluids, and phase transitions

One of the most direct applications is local anisotropy of Voronoi or free-volume cells. In simple fluids, hard-disk and hard-sphere ensembles, and Lennard–Jones systems, anisotropy indices derived from rank-2 Minkowski tensors provide a robust characterization of local structure. The abstract summary of the fluid study reports that the local anisotropy index ranges from $V_j$8 for vapor phases to $V_j$9 for ordered solids, and that these indices decrease monotonously with increasing free volume and randomness of particle positions (Kapfer et al., 2010).

In jammed monodisperse sphere packings, rank-four surface tensors of Voronoi cells were used as crystalline order metrics. The distance of the six eigenvalues of $\Lambda_j(K,\cdot)$0 from the fcc or hcp reference spectra defines $\Lambda_j(K,\cdot)$1 and $\Lambda_j(K,\cdot)$2, and the resulting analysis identifies an abrupt onset of local crystallinity at $\Lambda_j(K,\cdot)$3. The same work shows that the commonly used bond-orientational order metric $\Lambda_j(K,\cdot)$4 produces false positives, whereas the Minkowski-tensor criterion excludes cells whose facet geometry is not genuinely crystalline (Kapfer et al., 2012).

In two-dimensional complex plasmas, Minkowski tensors were used to test the Fractal-Domain-Structure theory during recrystallization. Rank-2 and rank-4 descriptors of Voronoi cells define defect fractions and crystalline fractions, and the area–boundary relation of crystalline domains yields a robust exponent $\Lambda_j(K,\cdot)$5. The same study reports that Minkowski-tensor-based defect measures extend the verified power-law relation between defect fraction and system energy by about one order of magnitude toward higher energies compared to the bond-order parameter $\Lambda_j(K,\cdot)$6 (Böbel et al., 2018).

These applications illustrate a common pattern. Rank-2 tensors are effective for elongation and local anisotropy; rank-4 tensors resolve higher-order symmetry classes and suppress false positives when different structures share similar second-order moments; and tensor additivity makes local-to-global aggregation straightforward in particulate and cellular data (Kapfer et al., 2012, Böbel et al., 2018).

6. Sphere, curved surfaces, cosmology, and current directions

The extension from Euclidean space to curved manifolds is not formal only. On the sphere there is no global translation invariance, boundary geometry is intrinsically curved, and geodesic curvature replaces planar curvature. Spherical Minkowski analysis therefore treats excursion sets $\Lambda_j(K,\cdot)$7 with

$\Lambda_j(K,\cdot)$8

and uses tangent normals rather than Euclidean outward normals. The 2022 HEALPix algorithm emphasized that this extension is an active research topic and used localized tensor sums to build Minkowski maps for searches for CMB anisotropies and non-Gaussianities at varying scales (Collischon et al., 2022).

In cosmology, Minkowski tensors have been used in both Euclidean slices and full-sky analyses. In two-dimensional slices of the matter and galaxy density fields, the translationally invariant tensor $\Lambda_j(K,\cdot)$9 remains proportional to the identity for statistically isotropic Gaussian fields, while redshift-space distortions generate clear eigenvalue asymmetries and thus a probe of the large-scale velocity field (Appleby et al., 2017). In three dimensions, the analogous tensors $$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$0 and $$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$1 behave the same way: for isotropic Gaussian random fields their ensemble averages are proportional to the identity, whereas linear redshift-space distortion produces unequal diagonal components aligned with the line of sight (Appleby et al., 2018).

Direct spherical CMB applications also separate global from local effects. A direct-sphere analysis of Planck temperature maps found very good agreement between the coadded SMICA map and simulations within approximately $$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$2, with no significant difference across frequencies except the $$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$3 GHz channel, where the reported discrepancy is approximately $$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$4 and was attributed most likely to inaccurate instrumental-beam estimation (K. et al., 2018). A later spherical framework with Cartesian and irreducible Minkowski tensors introduced localized Minkowski sky maps, handled masking explicitly, and reported an anomalous region close to the well-known Cold Spot as well as a second spot near $$\Phi_j^{r,s}(K)=\frac{\omega_{d-j}}{r!\,s!\,\omega_{d-j+s}}\int_{\mathbb{R}^d\times S^{d-1}}x^{\otimes r}u^{\otimes s}\,\Lambda_j(K,d(x,u)),$$5; the accompanying software package is litchi (Collischon et al., 2024).

Beyond the sphere, surface Minkowski tensors now extend to general curved surfaces. A 2025 construction defines them through a modified transport of the boundary co-normal to a reference point that compensates the angular defect produced by classical parallel transport. The associated irreducible surface Minkowski tensors classify embedded shapes by normalized eigenvalues, and the paper analyzes approximation error, perturbation stability, and applications to cells in a curved epithelial monolayer (Happel et al., 16 Jun 2025).

Several points remain structurally important. Scalar Hadwiger-type classification exists on the sphere, but an analogue of Alesker’s classification theorem for tensor valuations on the sphere is not currently known. Pixelization, interpolation, masks, and smoothing remain leading sources of systematic error in numerical work on HEALPix maps. At the same time, the combination of valuation theory, discrete algorithms, irreducible representations, and localized map constructions has made Minkowski tensors a standard high-information morphology formalism for convex geometry, random fields, and shape analysis on both flat and curved spaces (Collischon et al., 2024, Collischon et al., 2022).