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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 Supérieure des Mines de Saint-Etienne,FRANCE
SPIN, PROPICE / LGF, UMR CNRS 5307
28/10/2015
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Geometric Stochastic Modeling and objectives
Section 1
Geometric Stochastic Modeling and objectives
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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 + Ξi
(1)
xi ∈Φ
The Ξi are i.i.d.
Φ a point process
Law of Φ
Law of Ξ0
⇔
⇔
Spatial distribution
granulometry
Boolean model ⇒ Φ Poisson point process of intensity λ
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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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Geometric Stochastic Modeling and objectives
Characterization and description of the grain
For homothetic grains:
E[U(Ξ0 )]
2π
E[U(Ξ0 )]
=
4
E[A(Ξ0 )]
π
Disk of radius r : E[r ] =
& E[r 2 ] =
Square of side x :E[x]
& E[r 2 ] = E[A(Ξ0 )]
⇒ Parametric distribution of homothetic factor!
For non homothetic grains: rectangle, ellipse...
Same mean for area and perimeter (Minkowski densities)
⇒ insufficient to fully characterize Ξ0 ! What about the variations of
these geometrical characteristics?
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Theoretical aspects
Section 2
Theoretical aspects
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Theoretical aspects
From covariance of Ξ to variation of Ξ0
Mean covariogram:
γ̄Ξ0 (u) = E[A(Ξ0 ∩ Ξ0 + u)]
Covariance:
CΞ (u) = P(x ∈ (Ξ ∩ Ξ + u))
Relationship:
CΞ (u) − pΞ2
1
γ̄Ξ0 (u) = log 1 +
γ
(1 − pΞ )2
(2)
In addition:
R
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R2
γ̄Ξ0 (u)du = E[A(Ξ0 )2 ]
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Theoretical aspects
Stability by convex dilations
Ξ
Ξ⊕K
(a) grain Ξ0 , intensity λ
(b) grain Ξ0 ⊕ K , intensity λ
Where X ⊕ Y = {x + y |x ∈ X , y ∈ Y }
⇒ The Boolean model is stable under convex dilations
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Theoretical aspects
The proposed method
Consequently, for all r ≥ 0 we can estimate:
Z
2
ζ0,K (r ) = E[A(Ξ0 ⊕ rK ) ] =
E[γΞ0 ⊕rK (u)]du
R2
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Theoretical aspects
The proposed method
Consequently, for all r ≥ 0 we can estimate:
Z
2
ζ0,K (r ) = E[A(Ξ0 ⊕ rK ) ] =
E[γΞ0 ⊕rK (u)]du
R2
Steiner’s formula (mixed volumes)
A(Ξ0 ⊕ rK ) = A(Ξ0 ) + 2rW (Ξ0 , K ) + r 2 A(K )
The polynomial ζ0,K
ζ0,K (r ) = E[A20 ] + 4r E[A0 W (Ξ0 , K )] + r 2 (4E[W (Ξ0 , K )2 ] +
+ 2A(K )E[A0 ]) + 4r 3 A(K )E[W (Ξ0 , K )] + r 4 A(K )2
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Theoretical aspects
The proposed method
Consequently, for all r ≥ 0 we can estimate:
Z
2
ζ0,K (r ) = E[A(Ξ0 ⊕ rK ) ] =
E[γΞ0 ⊕rK (u)]du
R2
Steiner’s formula (mixed volumes)
A(Ξ0 ⊕ rK ) = A(Ξ0 ) + 2rW (Ξ0 , K ) + r 2 A(K )
The polynomial ζ0,K
ζ0,K (r ) = E[A20 ] + 4r E[A0 W (Ξ0 , K )] + r 2 (4E[W (Ξ0 , K )2 ] +
+ 2A(K )E[A0 ]) + 4r 3 A(K )E[W (Ξ0 , K )] + r 4 A(K )2
⇒ Estimation of E[A20 ], E[A0 W (Ξ0 , K )] and E[W (Ξ0 , K )2 ]
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Theoretical aspects
Generalization to nth order moments
The mean n-variogram
Tn−1
(n)
For n ≤ 2, γΞ0 (u1 , · · · un−1 ) = E[A( i=1
(Ξ0 − ui ) ∩ Ξ0 )]
Relation n-variogram → n point probability function (see proceding)
R
R
(n)
Of course R2 · · · R2 γΞ0 (u1 , · · · un−1 )du1 · · · dun−1 = E[A(Ξ0 )n ]
Then the development of E[A(Ξ0 ⊕ K )n ] by Steiner’s formula gives:
∀K convex, nth order moments of (A0 , W (Ξ0 , K ))
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Theoretical aspects
The interpretation of the mixed area
Definition
For Ξ0 and K convex, W (Ξ0 , K ) = 12 (A(Ξ0 ⊕ K ) − A(K ))
For unit ball :W (Ξ0 , B) = U(Ξ) the perimeter
For a segment: W (Ξ0 , Sθ ) = HΞ0 (θ) the Féret’s diameter
HΞ0 (θ)
Ξ0
Ox
θ
For a polygon W (Ξ,
LN
i=1 αi Sθi )
=
PN
i=1 αi HΞ0 (θi )
⇒ ∀N, ∀(θ1 , · · · θN ) all moments of (HΞ0 (θ1 ), · · · HΞ0 (θN ))
⇒ Characterization of the random process HΞ0
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Theoretical aspects
The Féret’s diameter random process HΞ0
trajectory of HΞ0 is the support function of the realization Ξ0 ⊕ Ξ˘0
Ξ0
θ
HΞ0 (θ)
Feret diameter HΞ0 (θ)
8
7
6
5
4
0
π/2
π
3π/2
2π
orientation θ
The process HΞ0 describes and characterizes Ξ0 ⊕ Ξ˘0
NB: Ξ0 isotropic ⇔ HΞ0 strong stationary
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Practical aspects
Section 3
Practical aspects
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Practical aspects
The simplest cases
Estimation of 1st and 2nd-order moments
E[A(Ξ0 )] and E[W (Ξ0 , K )]
E[A(Ξ0 )2 ], E[A(Ξ0 )W (Ξ0 , K )] and E[W (Ξ0 , K )2 ]
Disk
Segment
E[A(Ξ0 )2 ]
E[A(Ξ0 )2 ]
E[A(Ξ0 )U(Ξ0 )]
E[HΞ0 (θ)2 ]
E[U(Ξ0 )2 ]
E[A(Ξ0 )HΞ0 (θ)]
Parallelogram
E[A(Ξ0 )2 ]
E[HΞ0 (θ)2 ]
E[A(Ξ0 )HΞ0 (θ)]
E[HΞ0 (θ1 )HΞ0 (θ2 )]
additional quantity of interest
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Practical aspects
The simplest cases
Estimation of 1st and 2nd-order moments
E[A(Ξ0 )] and E[W (Ξ0 , K )]
E[A(Ξ0 )2 ], E[A(Ξ0 )W (Ξ0 , K )] and E[W (Ξ0 , K )2 ]
Disk
Segment
E[A(Ξ0 )2 ]
E[A(Ξ0 )2 ]
E[A(Ξ0 )U(Ξ0 )]
E[HΞ0 (θ)2 ]
E[U(Ξ0 )2 ]
E[A(Ξ0 )HΞ0 (θ)]
Parallelogram
E[A(Ξ0 )2 ]
E[HΞ0 (θ)2 ]
E[A(Ξ0 )HΞ0 (θ)]
E[HΞ0 (θ1 )HΞ0 (θ2 )]
additional quantity of interest
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Practical aspects
Procedure
Realizations
Ξ(ω) ∩ M
r1 , · · · rn
K
Dilations
(Ξ(ω) ⊕ ri K ) ∩ (M ri K )
Covariances
CΞ⊕ri K
Mean Covariograms
γ̄Ξ0 ⊕ri K
Integration
Polynomial fitting
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2
E[A(Ξ0 ⊕ ri K ) ] =
Z
γ̄Ξ0 ⊕ri K (u)du
E[A(Ξ0 )2 ], E[A(Ξ0 )W (Ξ0 , K )], E[W (Ξ0 , K )2 ]
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Practical aspects
statistical aspects
The following estimator for n point probability function is unbiased and
strong consistent as M → ∞.
(n)
ĈΞ,M (x1 , · · · xn ) =
A((Ξ ∩ M) {0, x1 − xn , · · · xn−1 − xn })
A(M {0, x1 − xn , · · · xn−1 − xn })
Then it follows a consistent estimator for n-variogram and thus the
moments of (A(Ξ0 ), W (Ξ0 , K )), but not necessarily unbiased.
⇒ Small bias for M bigger than Ξ0 .(check by simulation)
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Test by simulation
Section 4
Test by simulation
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Test by simulation
Experiments
Several realizations of the following Boolean model
a ∼ N(40, 10),
b ∼ N(30, 10)
M : 500 × 500
100
λ=
500 × 500
r = 0, 1, · · · 10
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Test by simulation
Results
Dilation with a segment
Dilation with a disk
30
E[Hθ (Ξ0 )2 ]
E[A(Ξ0 )Hθ (Ξ0 )]
E[A(Ξ0 )2 ]
20
10
relative error (%)
relative error (%)
30
E[U(Ξ0 )2 ]
E[A(Ξ0 )U(Ξ0 )]
E[A(Ξ0 )2 ]
20
10
5
5
2
2
0
500
0
1,000 1,500 2,000
1,000
number of realizations
number of realizations
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500
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Test by simulation
Conclusions and prospects
Conclusions
Theoretical estimator for nth order moments of the process HΞ0
Practical estimation of 1st and 2nd-order moments of:
t (A(Ξ ), U(Ξ )) and t (A(Ξ ), H (θ))
0
0
0
Ξ0
⇒ Characterization of a random particle depending on 2 parameters:
rectangle, ellipse...
Prospects
Describing complex random convex by first and second order
characteristics of the process HΞ0 (Ex:Gaussian process).
quantifying the anisotropy of the grain.
Bias corrector.
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Test by simulation
Thanks for listening!
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