Stationary Tessellations Research Articles

Percolation on stationary tessellations: models, mean values, and second-order structure

We consider a stationary face-to-face tessellation X of Rd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes of Z ∩ W, where the observation window W is assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

Percolation on stationary tessellations: models, mean values, and second-order structure

We consider a stationary face-to-face tessellationXofRdand introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probabilitypand white otherwise. We are interested in geometric properties of the unionZof black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes ofZ∩W, where the observation windowWis assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes ofX. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

A general study of extremes of stationary tessellations with examples

Let m be a random tessellation in Rd, d≥1, observed in a bounded Borel subset W and f(⋅) be a measurable function defined on the set of convex bodies. A point z(C), called the nucleus of C, is associated with each cell C of m. Applying f(⋅) to all the cells of m, we investigate the order statistics of f(C) over all cells C∈m with nucleus in Wρ=ρ1/dW when ρ goes to infinity. Under a strong mixing property and a local condition on m and f(⋅), we show a general theorem which reduces the study of the order statistics to the random variable f(C), where C is the typical cell of m. The proof is deduced from a Poisson approximation on a dependency graph via the Chen–Stein method. We obtain that the point process {(ρ−1/dz(C),aρ−1(f(C)−bρ)),C∈m,z(C)∈Wρ}, where aρ>0 and bρ are two suitable functions depending on ρ, converges to a non-homogeneous Poisson point process. Several applications of the general theorem are derived in the particular setting of Poisson–Voronoi and Poisson–Delaunay tessellations and for different functions f(⋅) such as the inradius, the circumradius, the area, the volume of the Voronoi flower and the distance to the farthest neighbor.

THE CHARACTER OF PLANAR TESSELLATIONS WHICH ARE NOT SIDE-TO-SIDE

This paper studies stationary tessellations and tilings of the plane in which all cells are convex polygons. The focus is on the class of tessellations which are not side-to-side. The character of these tessellations is explored, with special attention paid to the relationship between edges of the tessellation and sides of the polygonal cells and to the combinatorial topology between the ‘adjacent’ geometric elements of the tessellation. Three new parameters, e0,e1 and e2 summing to unity, are introduced. These capture the essence of non side-to-side tessellations and play a role in understanding the adjacency of sides and cells. Examples illustrate the theory.

SECOND MOMENT MEASURE AND K-FUNCTION FOR PLANAR STIT TESSELLATIONS

For STIT tessellations – stationary tessellations that are stable under the operation iteration of tessellations – the second-ordermeasure of the edge system is studied. A result is that this measure coincides with that one of a Boolean segment process. In the isotropic case an explicit formula for the pair-correlation function is given. An estimator for the covariance function of the edge length measure is derived and adapted to digitized images of tessellations. For m pixels of an image the algorithm is of complexity O(mlogm).

Scaling limits for shortest path lengths along the edges of stationary tessellations

We consider spatial stochastic models, which can be applied to, e.g. telecommunication networks with two hierarchy levels. In particular, we consider Cox processes X L and X H concentrated on the edge set T (1) of a random tessellation T, where the points X L,n and X H,n of X L and X H can describe the locations of low-level and high-level network components, respectively, and T (1) the underlying infrastructure of the network, such as road systems, railways, etc. Furthermore, each point X L,n of X L is marked with the shortest path along the edges of T to the nearest (in the Euclidean sense) point of X H . We investigate the typical shortest path length C * of the resulting marked point process, which is an important characteristic in, e.g. performance analysis and planning of telecommunication networks. In particular, we show that the distribution of C * converges to simple parametric limit distributions if a scaling factor κ converges to 0 or ∞. This can be used to approximate the density of C * by analytical formulae for a wide range of κ.

Scaling limits for shortest path lengths along the edges of stationary tessellations

We consider spatial stochastic models, which can be applied to, e.g. telecommunication networks with two hierarchy levels. In particular, we consider Cox processes XL and XH concentrated on the edge set T(1) of a random tessellation T, where the points XL,n and XH,n of XL and XH can describe the locations of low-level and high-level network components, respectively, and T(1) the underlying infrastructure of the network, such as road systems, railways, etc. Furthermore, each point XL,n of XL is marked with the shortest path along the edges of T to the nearest (in the Euclidean sense) point of XH. We investigate the typical shortest path length C* of the resulting marked point process, which is an important characteristic in, e.g. performance analysis and planning of telecommunication networks. In particular, we show that the distribution of C* converges to simple parametric limit distributions if a scaling factor κ converges to 0 or ∞. This can be used to approximate the density of C* by analytical formulae for a wide range of κ.

Length distributions of edges in planar stationary and isotropic STIT tessellations

Stationary and isotropic random tessellations of the euclidean plane are studied which have the characteristic property to be stable with respect to iteration (or nesting), STIT for short. Since their cells are not in a face-to-face position, three different types of linear segments appear. For all the types the distribution of the length of the typical segment is given.

Stationary partitions and Palm probabilities

A stationary partition based on a stationary point process N in ℝ d is an ℝ d -valued random field π={π(x): x∈ℝ d } such that both π(y)∈N for each y∈ℝ d and the random partition {{y∈ℝ d : π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

Stationary partitions and Palm probabilities

A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration

Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space with the following stability property: their distribution is invariant with respect to the operation of iteration (or nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary tessellations is fully characterized.

Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration

Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space with the following stability property: their distribution is invariant with respect to the operation of iteration (or nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary tessellations is fully characterized.

Limit theorems for stationary tessellations with random inner cell structures

We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

Limit theorems for stationary tessellations with random inner cell structures

We consider stationary and ergodic tessellations X = Ξ n n≥1 in R d , where X is observed in a bounded and convex sampling window W p ⊂ R d . It is assumed that the cells Ξ n of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J 0 acting on R d . In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in W p , as W p ↑ R d . It turns out that this functional provides an unbiased estimator for the intensity vector associated with J 0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

Limits of sequences of stationary planar tessellations

In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.

Limits of sequences of stationary planar tessellations

In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.

Aggregate and fractal tessellations

Consider a sequence of stationary tessellations {Θ n }, n=0,1,…, of ℝ d consisting of cells {C n (x i n )}with the nuclei {x i n }. An aggregate cell of level one, C 0 1(x i 0), is the result of merging the cells of Θ1 whose nuclei lie in C 0(x i 0). An aggregate tessellation Θ0 n consists of the aggregate cells of level n, C 0 n (x i 0), defined recursively by merging those cells of Θ n whose nuclei lie in C n −1(x i 0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as n→∞ and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {Θ n }.

Interdependences of directional quantities of planar tessellations

For random planar stationary tessellations a parameter system of three mean values and two directional distributions of the edges is considered. To investigate the interdependences between these parameters their joint range is described, i.e. the set of values which can be realized by tessellations. The constructive proof is based on mixtures of stationary regular tessellations. To explore the range for the class of stationary ergodic tessellations a procedure is used which we refer to as iteration.

Interdependences of directional quantities of planar tessellations

For random planar stationary tessellations a parameter system of three mean values and two directional distributions of the edges is considered. To investigate the interdependences between these parameters their joint range is described, i.e. the set of values which can be realized by tessellations. The constructive proof is based on mixtures of stationary regular tessellations. To explore the range for the class of stationary ergodic tessellations a procedure is used which we refer to as iteration.

Contact and chord length distribution of a stationary Voronoi tessellation

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.