# Bicovariograms and Euler characteristic of Regular sets

Abstract : We establish an expression of the \EC~of a $r$-regular planar set in function of some variographic quantities. The usual $\mathcal{C} ^{2}$ framework is relaxed to a $\mathcal{C} ^{1,1}$ regularity assumption, generalising existing local formulas for the \EC. We give also general bounds on the number of connected components of a measurable set of $\mathbb{R}^{2}$ in terms of local quantities. These results are then combined to yield a new expression of the mean \EC~of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non-connected polyrectangular grains in $\mathbb{R}^{2}$. Applications to excursions of smooth bivariate random fields are derived in the companion paper \cite{LacEC2}, and applied for instance to $\C^{1,1}$ Gaussian fields, generalising standard results.
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https://hal.archives-ouvertes.fr/hal-01207501
Contributor : Raphael Lachieze-Rey <>
Submitted on : Thursday, March 9, 2017 - 4:17:37 PM
Last modification on : Friday, April 10, 2020 - 5:22:20 PM
Document(s) archivé(s) le : Saturday, June 10, 2017 - 2:47:16 PM

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### Identifiers

• HAL Id : hal-01207501, version 3
• ARXIV : 1510.00501

### Citation

Raphaël Lachièze-Rey. Bicovariograms and Euler characteristic of Regular sets. Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2018, 291 (2-3), pp.398-419. ⟨hal-01207501v3⟩

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