Introduction to several models from stochastic geometry Pierre - PowerPoint PPT Presentation

Introduction to several models from stochastic geometry Pierre Calka Computational Geometry Week 2015 Eindhoven , 25 June 2015

default Plan From game to theory: Buffon, integral geometry, random tessellations From game to theory: 150 years of random convex hulls Addendum: some more models

default Plan From game to theory: Buffon, integral geometry, random tessellations Buffon’s needle problem Example of a formula from integral geometry Poisson point process Poisson line tessellation Poisson-Voronoi tessellation From game to theory: 150 years of random convex hulls Addendum: some more models

default Roots of geometric probability Georges-Louis Leclerc, Comte de Buffon (1733) Probability p that a needle of length ℓ dropped on a floor made of parallel strips of wood of same width D > ℓ will lie across a line? D

default Roots of geometric probability Georges-Louis Leclerc, Comte de Buffon (1733) Probability p that a needle of length ℓ dropped on a floor made of parallel strips of wood of same width D > ℓ will lie across a line? D l

default Roots of geometric probability R Θ ℓ/ 2 − π 2 , π R and Θ independent r.v., uniformly distributed on ]0 , D � � 2 [ and . 2 There is intersection when 2 R ≤ ℓ cos(Θ). π ℓ 2 cos( θ ) 2 π = 2 ℓ � � d r d θ 2 p = D π D θ = − π r =0 2

default Roots of geometric probability p = p ([0 , ℓ ]) = 2 ℓ π D D l

default Roots of geometric probability Same question when dropping a polygonal line? D

default Roots of geometric probability Same question when dropping a convex body K ? D K

default Roots of geometric probability p ( ∂ K ) = per ( ∂ K ) where per ( ∂ K ) : perimeter of ∂ K π D D K

default Roots of geometric probability Notation • p k ( C ) probability to have exactly k intersections of C with the lines • f ( C ) = � k ≥ 1 kp k ( C ) mean number of intersections Several juxtaposed needles • f ([0 , ℓ ]), ℓ > 0, additive and increasing so f ([0 , ℓ ]) = αℓ , α > 0 • Similarly, f ( C ) = α per ( C ) • f (Circle of diameter D ) = 2 = απ D • If C is the boundary of a convex body K with diam ( K ) < D , f ( C ) = 2 p ( C )

default Extensions in integral geometry K convex body of R 2 L p ,θ = p (cos( θ ) , sin( θ )) + R ( − sin( θ ) , cos( θ )), p ∈ R , θ ∈ [0 , π ) L p ,θ p θ � π � + ∞ per ( ∂ K ) = 1 ( L p ,θ ∩ K � = ∅ ) dpd θ θ = 0 p = −∞

default Extensions in integral geometry K convex body of R 2 L p ,θ = p (cos( θ ) , sin( θ )) + R ( − sin( θ ) , cos( θ )), p ∈ R , θ ∈ [0 , π ) diam θ ( K ) L p ,θ p θ K θ � π � + ∞ per ( ∂ K ) = 1 ( L p ,θ ∩ K � = ∅ ) dpd θ θ = 0 p = −∞ Cauchy-Crofton formula � π per ( ∂ K ) = diam θ ( K ) d θ θ = 0

default Random points • W convex body • µ probability measure on W • ( X i , i ≥ 1) independent µ -distributed variables �� �� �� �� � � � �� �� � �� �� �� �� E n = { X 1 , · · · , X n } ( n ≥ 1) � � � � �� �� B 4 B 1 � � � • #( E n ∩ B 1 ) number of points in B 1 � � � �� �� �� �� �� �� ◮ #( E n ∩ B 1 ) binomial variable � B 2 � � n � µ ( B 1 ) k (1 − µ ( B 1 )) n − k , P (#( E n ∩ B 1 ) = k ) = � � B 3 � k � � � �� �� � 0 ≤ k ≤ n � ◮ #( E n ∩ B 1 ) , · · · , #( E n ∩ B n ) not independent ( B 1 , · · · , B n ∈ B ( R 2 ), B i ∩ B j = ∅ , i � = j )

default Poisson point process Poisson point process with intensity measure µ : �� �� locally finite subset X of R d such that �� �� � � � �� �� � �� �� �� �� ◮ #( X ∩ B 1 ) Poisson r.v. of mean µ ( B 1 ) � � � � �� �� B 4 B 1 P (#( X ∩ B 1 ) = k ) = e − µ ( B 1 ) µ ( B 1 ) k � � � k ! , k ∈ N � � � �� �� �� �� �� �� ◮ #( X ∩ B 1 ) , · · · , #( X ∩ B n ) independent � � B 2 � � ( B 1 , · · · , B n ∈ B ( R d ), B i ∩ B j = ∅ , i � = j ) B 3 � � � � � �� �� �

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Poisson line tessellation ◮ X Poisson point process in R 2 of intensity measure d p d θ ◮ For ( p , θ ) ∈ X , polar line L p ,θ = p (cos( θ ) , sin( θ )) + (cos( θ ) , sin( θ )) ⊥ ◮ Tessellation: set of connected components of R d \ � L p ,θ ( p ,θ ) ∈ X Properties : invariance under translations and rotations References : Meijering (1953), Miles (1964), Stoyan et al. (1987)

default Questions of interest ◮ Asymptotic study of the population of cells (means, extremes): number of vertices, edge length in a window... ◮ Study of a particular cell zero-cell C 0 containing the origin typical cell C chosen uniformly at random Means, moments and distribution of functionals of the cell (area, perimeter...), asymptotic sphericality J. Møller (1986), I. N. Kovalenko (1998), D. Hug, M. Reitzner & R. Schneider (2004)

default Mean number of vertices per cell • Each vertex from the tessellation is contained in exactly 4 cells. • Each vertex is the highest point from a unique cell with probability 1. • There are as many vertices as there are cells. Conclusion. The mean number of vertices of a typical cell is 4.

default Probability to belong to the zero-cell C 0 K 0 Consequence of the Cauchy-Crofton formula : K convex body containing 0, C 0 cell of the tessellation containing 0 � �� � P ( K ⊂ C 0 ) = exp − 1 ( L p ,θ ∩ K � = ∅ ) d p d θ = exp( − per ( ∂ K )) Remark. In higher dimension, the perimeter is replaced by the mean width.

default Poisson-Voronoi tessellation ◮ X Poisson point process in R 2 of intensity measure d x ◮ For every nucleus x ∈ X , the cell associated is C ( x | X ) := { y ∈ R 2 : � y − x � ≤ � y − x ′ � ∀ x ′ ∈ X } ◮ Tessellation: set of cells C ( x | X ) Properties : invariance under translations and rotations References : Descartes (1644), Gilbert (1961), Okabe et al. (1992)

default Deterministic Voronoi grids

default Mean number of vertices per cell • Each vertex from the tessellation is contained in exactly 3 cells. • Each vertex is the highest or lowest point from a unique cell with probability 1. • There are twice as many vertices as there are cells. Conclusion. The mean number of vertices of a typical cell is 6.

default Probability to belong to the zero-cell F 0 ( K ) C 0 0 0 K K convex body containing 0, C 0 Voronoi cell C (0 | X ∪ { 0 } ) P ( K ⊂ C 0 ) = exp( − V d ( F 0 ( K ))) where V d is the volume and F 0 ( K ) = ∪ x ∈ K B ( x , � x � ) flower of K