Mean Geometry for 2D random fields: level perimeter and level total curvature integrals
Résumé
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: 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.
Mots clés
stationary random field
shot noise random field
Total curvature
Gauss-Bonnet Theorem
Euler Characteristic
excursion sets
Gaussian random field
persistent homology
Perimeter
and phrases: Perimeter
Total curvature
Euler Char-
acteristic
excursion sets
stationary random field
Gaussian random
field
persistent homology
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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