Mean geometry for 2D random fields: Level perimeter and level total curvature integrals

We introduce the level perimeter integral and the total curvature integral associated with a real valued function f defined on the plane R^2 as integrals allowing to compute the perimeter of the excursion set of f above level t and the total (signed) curvature of its boundary for almost every level t. Thanks to the Gauss-Bonnet theorem, the total curvature is directly related to the Euler Characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be explicitly computed in two different frameworks: piecewise constant functions (also called here elementary functions) and smooth (at least C^2) functions. Considering 2D random fields (in particular considering shot noise random fields), we compute their mean perimeter and total curvature integrals, and this provides new explicit computations of the mean perimeter and Euler Characteristic densities of excursion sets, beyond the Gaussian framework.