Level Total Curvature integral: Euler characteristic and 2D random fields

Hermine Biermé (1) , Agnès Desolneux (2, 3)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]
2 CNRS - Centre National de la Recherche Scientifique
3 CMLA - Centre de Mathématiques et de Leurs Applications

We introduce the level total curvature function associated with a real valued function f defined on the plane R^2 as the function that, for any level t ∈ R, computes the total (signed) curvature of the boundary of the excursion set of f above 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 total curvature function 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 will compute their mean total curvature function, and this will provide new explicit computations of the mean Euler Characteristic of excursion sets, beyond the Gaussian framework.

Mots clés: Total curvature; Gauss-Bonnet Theorem; Euler Characteristic; excursion sets; stationary random field; shot noise random field; Gaussian random field; persistent homology

Domaines: Probabilités [math.PR]

Hermine Biermé, Agnès Desolneux. Level Total Curvature integral: Euler characteristic and 2D random fields. 2016. ⟨hal-01370902v1⟩