Tensor Minkowski Functionals for random fields on the sphere

Abstract

We generalize the translation invariant tensor-valued MinkowskiFunctionals which are defined on two-dimensional flat space to the unit sphere. Weapply them to level sets of random fields. The contours enclosing boundaries of levelsets of random fields give a spatial distribution of random smooth closed curves. Weoutline a method to compute the tensor-valued Minkowski Functionals numerically forany random field on the sphere. Then we obtain analytic expressionsfor the ensembleexpectation values of the matrix elements for isotropic Gaussian and Rayleigh fields.The results hold on flat as well as any curved space with affine connection. We elucidatethe way in which the matrix elements encode information about the Gaussian natureand statistical isotropy (or departure from isotropy) of the field. Finally, we apply themethod to maps of the Galactic foreground emissions from the 2015PLANCK data anddemonstrate their high level of statistical anisotropy and departure from Gaussianity. Tensor Minkowski Functionals for random fields on sphere