Bicovariograms and Euler characteristic I. Regular sets

TL;DR

This paper derives new formulas for the Euler characteristic of regular planar sets under relaxed smoothness conditions, providing bounds and expressions for random sets, with applications to Gaussian fields and Boolean models.

Contribution

It introduces generalized local formulas for the Euler characteristic under ,1 regularity and connects these to the analysis of random regular sets and level sets.

Findings

New expression for Euler characteristic of ,1 regular sets

Bounds on the number of connected components based on local quantities

Explicit formulas for mean Euler characteristic of random 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.