Bicovariograms and Euler characteristic of regular sets

Abstract

We establish an expression of the Euler characteristic of a -regular planar set in function of some variographic quantities. The usual  2 framework is relaxed to a  1,1 regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of ℝ 2 in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic 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 ℝ 2 . Applications to excursions of smooth bivariate random fields are derived in the companion paper [25], and applied for instance to  1,1 Gaussian fields, generalising standard results. K E Y W O R D S Euler characteristic, intrinsic volumes, shot noise processes, boolean model M S C ( 2 0 1 0 ) 28A75, 52A22, 60D05, 60G10It is more generally an indicator of the regularity of the set, as an irregular structure is more likely to be shredded in many small pieces, or pierced by many holes, which results in a large value for | ( )|.As an integer-valued quantity, the Euler characteristic can be easily measured and used in estimation and modelisation procedures. It is an important indicator of the porosity of a random media [7,19,33], it is used in brain imagery [23,37], astronomy, [15,27,31], and many other disciplines. See also [1] for a general review of applied algebraic topology. In the study of parametric random media or graphs, a small value of | ( )| indicates the proximity of the percolation threshold, when that makes sense. See [30], or [13] in the discrete setting.The mathematical additivity property is expressed, for suitable sets and , by the formula ( ∪ ) = ( ) + ( ) − ( ∩ ), which applies recursively to finite such unions. In the validity domain of this formula, the Euler characteristic of a set can therefore be computed by summing local contributions. The Gauss-Bonnet theorem formalises this notion for  2 manifolds, stating that the Euler characteristic of a smooth set is the integral along the boundary of its Gaussian curvature. Exploiting the local nature of the curvature in applications seems to be a geometric challenge, in the sense that it is not always 398

A central limit theorem for Lipschitz–Killing curvatures of Gaussian excursions

“…The excursion set of X for the level u ∈ R is the random set X −1 ([u, ∞)) = {t ∈ R d | X(t) ≥ u}, whose properties are an active area of research, cf. [2], [4], [20], [1] among others. As a stochastic model, random fields have many applications, for instance in human brain mapping ( [8]), astrophysics ( [23]) and optics ( [6]).…”

Bicovariograms and Euler characteristic of random fields excursions

“…Our results exploit the findings of [21] connecting smooth sets Euler characteristic and variographic tools. For some λ ∈ R and a bi-variate function f , define for x ∈ R 2 the event…”

“…The results of [21] also yield that the Lipschitzness of ∇f is sufficient for the digital approximation of χ({f λ}) to be valid. It seems therefore that the C 1,1 assumption is the minimal one ensuring the Euler characteristic to be computable in this fashion.…”

“…For such a function f in dimension 2, it is shown in [21] that the Euler characteristic of its excursion set F = F λ (f ) ∩ W can be expressed by means of its bicovariograms, defined in (3). For ε > 0 sufficiently small…”

“…It is likely that the results concerning the planar Euler characteristic could be extended to higher dimensions. See for instance [25], that paves the way to an extension of the results of [21] to random fields on spaces with arbitrary dimension. Also, the uniform bounded density hypothesis is relaxed and allows for the density of the (d + 1)-tuple (f (x), ∂ 1 f (x), .…”