Characterization and Estimation of the Variations of a Random Convex Set by Its Mean n-Variogram: Application to the Boolean Model

In this paper we propose a method to characterize and estimate the variations of a random convex set Ξ0 in terms of shape, size and direction. The mean n-variogram γ Ξ0 : (u1 · · ·un) 7→ E[νd(Ξ0∩(Ξ0−u1) · · ·∩ (Ξ0 − un))] of a random convex set Ξ0 on R reveals information on the n order structure of Ξ0. Especially we will show that considering the mean n-variograms of the dilated random sets Ξ0 ⊕ rK by an homothetic convex family rKr>0, it’s possible to estimate some characteristic of the n order structure of Ξ0. If we make a judicious choice of K, it provides relevant measures of Ξ0 . Fortunately the germ-grain model is stable by convex dilatations, furthermore the mean n-variogram of the primary grain is estimable in several type of stationary germ-grain models by the so called n-points probability function. Here we will only focus on the Boolean model, in the planar case we will show how to estimate the n order structure of the random vector composed by the mixed volumes (A(Ξ0),W (Ξ0,K)) of the primary grain, and we will describe a procedure to do it from a realization of the Boolean model in a bounded window. We will prove that this knowledge for all convex body K is sufficient to fully characterize the so called difference body of the grain Ξ0 ⊕ Ξ̆0. we will be discussing the choice of the element K, by choosing a ball, the mixed volumes coincide with the Minkowski’s functional of Ξ0 therefore we obtain the moments of the random vector composed of the area and perimeter (A(Ξ0), U(Ξ)) . By choosing a segment oriented by θ we obtain estimates for the moments of the random vector composed by the area and the Ferret’s diameter in the direction θ, ((A(Ξ0), HΞ0(θ)). Finally, we will evaluate the performance of the method on a Boolean model with rectangular grain for the estimation of the second order moments of the random vectors (A(Ξ0), U(Ξ0)) and ((A(Ξ0), HΞ0(θ)).