The anisotropy of 2D or 3D Gaussian random fields through their Lipschitz-Killing curvature densities

We are interested here in modeling and estimating the anisotropy of 2D and 3D Gaussian random fields through the geometry of their excursion sets. In order to do this, we use Lipschitz-Killing curvatures of the level sets as functions of the levels and see them as generalized processes for which we are able to obtain a joint functional Central Limit Theorem. For 2D and 3D stationary Gaussian fields we provide explicit formulas for the Lipschitz-Killing curvature densities from which we can deduce geometrical equivalent of second spectral moments and anisotropy ratios that allow the estimation of the anisotropy of the underlying Gaussian field.