The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D

The full moments expansion of the joint probability distribution of an isotropic random field, its gradient and invariants of the Hessian is presented in 2 and 3D. It allows for explicit expression for the Euler characteristic in ND and computation of extrema counts as functions of the excursion set threshold and the spectral parameter, as illustrated on model examples.