Simulation of the Typical Poisson-Voronoi Cell

 Suppose that we want to verify the hypothesis whether an observed tessellation is a realization of a Poisson-Voronoi tessellation. If we can observe not only the edges of the tessellation but also the nuclei of the cells, then we can use the techniques presented in Section 2.2 in order to check the hypothesis whether the observed point pattern of nuclei is sampled from a homogeneous Poisson process. If the nuclei of the cells are not observable or if we even do not know whether the observed set of edges is a Voronoi tessellation induced by a certain point pattern, we could try to reconstruct the nuclei from the observed edges. However, another type of testing the Poisson-Voronoi hypothesis can be based on relationships between characteristics of the typical cell, for example on (5.10).

The radial simulation procedure of the homogeneous Poisson process X introduced in Section 4.3 can be used for simulating the typical cell of the Poisson-Voronoi tessellation induced by X. As we already mentioned in Section 5.4, (3.9) implies that we can simulate the Voronoi cell C₀ with nucleus at the origin which is induced by the (Palm) point process $X \cup \{o\}$, where a point at the origin has been added to X.

Let $x=\{x_1,x_2,\ldots\}$ be a realization of the radially generated Poisson process $X=\{X_1,X_2,\ldots\}$ given by (4.27). Then with probability 1 the Voronoi cell $C_0(x_1,\ldots,x_n)$with nucleus at the origin which is generated by the finite set of nuclei $\{o,x_1,\ldots,x_n\}$, is bounded for all sufficiently large n. Let l_n denote the distance to the furthest vertex of $C_0(x_1,\ldots,x_n)$ provided that $C_0(x_1,\ldots,x_n)$ is bounded and let  

$$n_0 = \min \{ n \, : \, \vert x_{n + 1}\vert \gt 2 l_n \}.$$ (11)

Note that  

$$C_0(x_1,x_2,\ldots) = C_0(x_1,x_2,\ldots,x_n)$$ (12)

for all $n \geq n_0$ where (4.27) implies that the minimum n₀ is finite with probability 1. Thus, by (5.11) a stopping rule is given which says when the realization $C_0(x_1,x_2,\ldots)$ of C₀ is completely generated. Namely, generate the realizations $x_1,\ldots,x_{n+1}$ of $X_1,\ldots,X_{n+1}$ until $C_0(x_1,\ldots,x_n)$ is bounded and $n \geq n_0$.

Sometimes, for example if we want to determine the spherical contact distribution function for the edge-set E corresponding to $C=\{C_1,C_2,\ldots\}$, we need to simulate the cell $\tilde{C}_0$induced by X which contains the origin, rather than to simulate the (typical) cell C₀. For the radially generated Poisson process X, we have $\tilde{C}_0 = C_1$ which can be simulated in a similar way as C₀. Namely, generate the realizations $x_1,\ldots,x_{n+1}$of $X_1,\ldots,X_{n+1}$ until the cell C₁(n) with nucleus x₁, which is induced by the finite set of nuclei $\{x_1,\ldots,x_n\}$,is bounded and the distance from x₁ to the furthest vertex of C₁(n) is less than $\frac{\vert x_{n+1}\vert}{2} - \vert x_1\vert$. Then $C_1(n) = C_1(x_1,x_2,\ldots)$.

We also remark that an effective simulation procedure for the typical cell of a Poisson-Johnson-Mehl tessellation has been given in Møller (1995). Like in the case of a Voronoi tessellation discussed above, this procedure is based on the radial simulation of a homogeneous Poisson process.