Characterization and Estimation of the Variations of a Random Convex Set by it's Mean $n$Variogram : Application to the Boolean Model  JeanCharles Pinoli, Johan Debayle, Saïd Rahmani

Author(s): JeanCharles Pinoli, Johan Debayle, Saïd Rahmani
DOI URL: http://dx.doi.org/10.1007/9783319250403_33
Video: http://www.youtube.com/watch?v=2ebliJYcWEA
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTicEphTng2VE9wclU
Presentation: https://www.see.asso.fr/node/14287
Creative Commons AttributionShareAlike 4.0 InternationalAbstract:
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 nvariog
ram γ(n)Ξ0:(u1⋯un)↦E[νd(Ξ0∩(Ξ0−u1)⋯∩(Ξ0−un))] of a random convex set Ξ0 on ℝ d r
eveals information on the n th order structure of Ξ0. Especially we will show that considering the mean nvariograms of the dilated random sets Ξ0 ⊕ rK by an homothetic convex family rKr > 0, it’s possible to estimate some characteristic of the n th order structure of Ξ0. If we make a judicious choice of K, it provides relevant measures of Ξ0. Fortunately the germgrain model is stable by convex dilatations, furthermore the mean nvariogram of the primary grain is estimable in several type of stationary germgrain models by the so called npoints probability function. Here we will only focus on the Boolean model, in the planar case we will show how to estimate the n th order structure of the random vector composed by the mixed volumes t (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⊕˘Ξ0. 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 t (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, t((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 t (A(Ξ0),U(Ξ0)) and t((A(Ξ0),HΞ0(θ)).