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Characterization and Estimation of the Variations of a
Random Convex Set by its Mean n-Variogram :
Application to the Boolean Model
S.Rahmani, J-C.Pinoli & J.Debayle
Ecole Nationale Sup´erieure des Mines de Saint-Etienne,FRANCE
SPIN, PROPICE / LGF, UMR CNRS 5307
28/10/2015
SR (ENSM-SE / LGF-PMDM) GSI 2015 28/10/2015 1 / 22

Geometric Stochastic Modeling and objectives
Section 1
Geometric Stochastic Modeling and objectives
SR (ENSM-SE / LGF-PMDM) GSI 2015 28/10/2015 2 / 22

Geometric Stochastic Modeling and objectives
Stochastic materials
Material modelling
Material characterization
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Geometric Stochastic Modeling and objectives
Germ-Grain model [Matheron 1967]
Definition
Ξ = [
xi∈Φ
xi+ Ξi(1)
The Ξiare i.i.d.
Φ a point process
Law of Φ ⇔Spatial distribution
Law of Ξ0⇔granulometry
Boolean model ⇒ΦPoisson point process of intensity λ
SR (ENSM-SE / LGF-PMDM) GSI 2015 28/10/2015 4 / 22

Geometric Stochastic Modeling and objectives
Objectives and state of the art
Geometrical characterization of Ξ0
from measurements in a bounded window Ξ ∩M
No assumption on Ξ0’s shape.
Describing Ξ0.
State of the art
Miles formulae [Miles 1967]
Tangent points method [Molchanov 1995]
Minimum contrast method[ Dupac & Digle 1980]
⇒Mean geometric parameter λ,E[A(Ξ0)], E[U(Ξ0)]
Formula for distribution for model of disk [Emery 2012]
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