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

Abstract : 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.
Keywords :
Type de document :
Pré-publication, Document de travail
2016
Domaine :

https://hal.archives-ouvertes.fr/hal-01370902
Contributeur : Hermine Biermé <>
Soumis le : vendredi 23 septembre 2016 - 14:26:54
Dernière modification le : samedi 18 février 2017 - 01:20:23

### Fichier

LTC-EC-preprint.pdf
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### Identifiants

• HAL Id : hal-01370902, version 1

### Citation

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

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