Characterization of Anisotropic Gaussian Random Fields by Minkowski Tensors

Gaussian random fields are among the most important models of amorphous spatial structures and appear across length scales in a variety of physical, biological, and geological applications, from composite materials to geospatial data. Anisotropy in such systems can sensitively and comprehensively be characterized by the so-called Minkowski tensors from integral geometry. Here, we analytically calculate the expected Minkowski tensors of arbitrary rank for the level sets of Gaussian random fields. The explicit expressions for interfacial Minkowski tensors are confirmed in detailed simulations. We demonstrate how the Minkowski tensors detect and characterize the anisotropy of the level sets, and we clarify which shape information is contained in the Minkowski tensors of different rank. Using an irreducible representation of the Minkowski tensors in the Euclidean plane, we show that higher-rank tensors indeed contain additional anisotropy information compared to a rank two tensor. Surprisingly, we can nevertheless predict this information from the second-rank tensor if we assume that the random field is Gaussian. This relation between tensors of different rank is independent of the details of the model. It is, therefore, useful for a null hypothesis test that detects non-Gaussianities in anisotropic random fields.