Anisotropic Growth of Voronoi Cells

1994 ◽  
Vol 26 (01) ◽  
pp. 43-53 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Poisson Process ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Simple Extension ◽  
Anisotropic Growth ◽  
Homogeneous Poisson Process ◽  
Translation Invariant ◽  
Voronoi Cells ◽  
Invariant Distance ◽  
The Mean

This paper discusses a simple extension of the classical Voronoi tessellation. Instead of using the Euclidean distance to decide the domains corresponding to the cell centers, another translation-invariant distance is used. The resulting tessellation is a scaled version of the usual Voronoi tessellation. Formulas for the mean characteristics (e.g. mean perimeter, surface and volume) of the cells are provided in the case of cell centers from a homogeneous Poisson process. The resulting tessellation is stationary and ergodic but not isotropic.

Anisotropic Growth of Voronoi Cells

1994 ◽  
Vol 26 (1) ◽  
pp. 43-53 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Poisson Process ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Simple Extension ◽  
Anisotropic Growth ◽  
Homogeneous Poisson Process ◽  
Translation Invariant ◽  
Voronoi Cells ◽  
Invariant Distance ◽  
The Mean

This paper discusses a simple extension of the classical Voronoi tessellation. Instead of using the Euclidean distance to decide the domains corresponding to the cell centers, another translation-invariant distance is used. The resulting tessellation is a scaled version of the usual Voronoi tessellation. Formulas for the mean characteristics (e.g. mean perimeter, surface and volume) of the cells are provided in the case of cell centers from a homogeneous Poisson process. The resulting tessellation is stationary and ergodic but not isotropic.

Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

2000 ◽  
Vol 32 (01) ◽  
pp. 1-18 ◽  
Author(s):  
F. Baccelli ◽  
K. Tchoumatchenko ◽  
S. Zuyev
Keyword(s):  
Communication Networks ◽  
Mobile Networks ◽  
Path Length ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Voronoi Cells ◽  
Ergodic Properties ◽  
Potential Applications ◽  
The Mean ◽  
Mobile Communication Networks

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

2000 ◽  
Vol 32 (1) ◽  
pp. 1-18 ◽  
Author(s):  
F. Baccelli ◽  
K. Tchoumatchenko ◽  
S. Zuyev
Keyword(s):  
Communication Networks ◽  
Mobile Networks ◽  
Path Length ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Voronoi Cells ◽  
Ergodic Properties ◽  
Potential Applications ◽  
The Mean ◽  
Mobile Communication Networks

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

Equality of critical densities in continuum percolation

1995 ◽  
Vol 32 (01) ◽  
pp. 90-104 ◽  
Author(s):  
Ronald Meester
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Euclidean Distance ◽  
Integrability Condition ◽  
Continuum Percolation ◽  
Positive Probability ◽  
Homogeneous Poisson Process ◽  
Extra Point

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

Time spent below a random threshold by a Poisson driven sequence of observations

2003 ◽  
Vol 40 (03) ◽  
pp. 807-814 ◽  
Author(s):  
S. N. U. A. Kirmani ◽  
Jacek Wesołowski
Keyword(s):  
Poisson Process ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Nonhomogeneous Poisson Process ◽  
System Level ◽  
Homogeneous Poisson Process ◽  
Random Threshold ◽  
The Mean ◽  
Current Value ◽  
Independent And Identically Distributed

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

scholarly journals Gamma distributions for stationary Poisson flat processes

2009 ◽  
Vol 41 (4) ◽  
pp. 911-939 ◽  
Author(s):  
Volker Baumstark ◽  
Günter Last
Keyword(s):  
Poisson Process ◽  
Gamma Distribution ◽  
Shape Parameter ◽  
Voronoi Tessellation ◽  
Isotropic Case ◽  
Intensity Measure ◽  
First Case ◽  
Gamma Distributions ◽  
The Face ◽  
The Mean

We consider a stationary Poisson process X of k-flats in ℝd with intensity measure Θ and a measurable set S of k-flats depending on F1,…,Fn∈ X, x∈ℝd, and X in a specific equivariant way. If (F1,…,Fn,x) is properly sampled (in a ‘typical way’) then Θ(S) has a gamma distribution. This result generalizes and unifies earlier work by Miles (1971), Møller and Zuyev (1996), and Zuyev (1999). As a new example, we will show that the volume of the fundamental region of a typical j-face of a stationary Poisson–Voronoi tessellation is conditionally gamma distributed. This is true in the area-biased and the area-debiased cases. In the first case the shape parameter is not integer valued. As another new example, we will show that the generalized integral-geometric contents of the (area-biased and area-debiased) typical j-face of a Poisson hyperplane tessellation are conditionally gamma distributed. In the isotropic case the contents boil down to the mean breadth of the face.

Areas of components of a Voronoi polygon in a homogeneous Poisson process in the plane

2002 ◽  
Vol 34 (2) ◽  
pp. 281-291 ◽  
Author(s):  
A. Hayen ◽  
M. P. Quine
Keyword(s):  
Poisson Process ◽  
Voronoi Tessellation ◽  
Homogeneous Poisson Process ◽  
Voronoi Polygon

We study the contribution made by three or four points to certain areas associated with a typical polygon in a Voronoi tessellation of a planar Poisson process. We obtain some new results about moments and distributions and give simple proofs of some known results. We also use Robbins' formula to obtain the first three moments of the area of a typical polygon and hence the variance of the area of the polygon covering the origin.

scholarly journals Gamma distributions for stationary Poisson flat processes

2009 ◽  
Vol 41 (04) ◽  
pp. 911-939 ◽  
Author(s):  
Volker Baumstark ◽  
Günter Last
Keyword(s):  
Poisson Process ◽  
Gamma Distribution ◽  
Shape Parameter ◽  
Voronoi Tessellation ◽  
Isotropic Case ◽  
Intensity Measure ◽  
First Case ◽  
Gamma Distributions ◽  
The Face ◽  
The Mean

We consider a stationary Poisson process X of k-flats in ℝd with intensity measure Θ and a measurable set S of k-flats depending on F 1,…,F n ∈ X, x∈ℝd, and X in a specific equivariant way. If (F 1,…,F n ,x) is properly sampled (in a ‘typical way’) then Θ(S) has a gamma distribution. This result generalizes and unifies earlier work by Miles (1971), Møller and Zuyev (1996), and Zuyev (1999). As a new example, we will show that the volume of the fundamental region of a typical j-face of a stationary Poisson–Voronoi tessellation is conditionally gamma distributed. This is true in the area-biased and the area-debiased cases. In the first case the shape parameter is not integer valued. As another new example, we will show that the generalized integral-geometric contents of the (area-biased and area-debiased) typical j-face of a Poisson hyperplane tessellation are conditionally gamma distributed. In the isotropic case the contents boil down to the mean breadth of the face.

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