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

Abstract

We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, 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 computed in two different frameworks: smooth (at least $C^{2}$) functions and piecewise constant functions (also called here elementary functions). Considering 2D random fields (in particular 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: for piecewise constant shot noise random fields, we give some examples of completely explicit formulas, and for smooth shot noise random fields the provided examples are only partly explicit, since the formulas are given under the form of integrals of some special functions.