Abstract:
We generalize the translation invariant tensor-valued Minkowski Functionals
which are defined on two-dimensional flat space to the unit sphere. We apply them to level
sets of random fields. The contours enclosing boundaries of level sets of random fields give a
spatial distribution of random smooth closed curves. We outline a method to compute the
tensor-valued Minkowski Functionals numerically for any random field on the sphere. Then
we obtain analytic expressions for the ensemble expectation 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 elucidate the way in which the matrix elements encode
information about the Gaussian nature and statistical isotropy (or departure from isotropy)
of the field. Finally, we apply the method to maps of the Galactic foreground emissions
from the 2015 PLANCK data and demonstrate their high level of statistical anisotropy and
departure from Gaussianity.
