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April 2020 Mean geometry for 2D random fields: Level perimeter and level total curvature integrals
Hermine Biermé, Agnès Desolneux
Ann. Appl. Probab. 30(2): 561-607 (April 2020). DOI: 10.1214/19-AAP1508

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.

Citation

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Hermine Biermé. Agnès Desolneux. "Mean geometry for 2D random fields: Level perimeter and level total curvature integrals." Ann. Appl. Probab. 30 (2) 561 - 607, April 2020. https://doi.org/10.1214/19-AAP1508

Information

Received: 1 October 2017; Revised: 1 April 2019; Published: April 2020
First available in Project Euclid: 8 June 2020

zbMATH: 07236128
MathSciNet: MR4108116
Digital Object Identifier: 10.1214/19-AAP1508

Subjects:
Primary: 26B15, 60D05, 60E10, 60G17, 60G60
Secondary: 60E07, 60G10, 62M40

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 2 • April 2020
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