Lecture 19: Boolean Models and Random Voronoi Tessellations

Boolean Models

Recall the definition of a Boolean Model from last time:

Let $ = {n} {(X_n, Z_n)} $ be a stationary, independent germ-grain process such that the ground point process $ 0 = {n} _{X_n} $ is a stationary Poisson process (thus \(\widetilde{\Phi}\) is Poisson).

The coverage model \[ \widetilde{Z} := \bigcup_{n\in\mathbb{N}} (Z_n + X_n) \] is called a Boolean model.

The capacity functional of \(\widetilde{Z}\) is \[ T_{\widetilde{Z}}(C) := \mathbb{P}(\widetilde{Z} \cap C \neq \varnothing) \quad \text{for }\ C \in \mathcal{C}. \]

Let $ A_C := { (x, K) ^d ’: K + x _C} $ for \(C \in \mathcal{C}\). Then, \[ T_{\widetilde{Z}}(C) = \mathbb{P}\big( \widetilde{\Phi}(A_C) > 0 \big) = 1 - e^{-\widetilde{\Lambda}(A_C)}, \] where \(\widetilde{\Lambda}\) is the intensity measure of \(\widetilde{\Phi}\). Since \(\widetilde{\Phi}\) is stationary,

\[\begin{align*} \widetilde{\Lambda}(A_C) &= \lambda \int_{\mathcal{C}'}\int_{\mathbb{R}^d} \mathbf{1}_{\{ K + x \cap C \neq \varnothing \}}\, dx \, \mathbb{Q}(dK) \\ &= \lambda \int_{\mathcal{C}'} \mathrm{vol}(K - C)\, \mathbb{Q}(dK). \end{align*}\]

Since
$$
\mathbb{P}( d(0,\widetilde{Z}) > r ) 
= 1 - T_{\widetilde{Z}}(B(0,r)),
\qquad
\mathbb{P}(0 \notin \widetilde{Z}) 
= 1 - T_{\widetilde{Z}}(\{0\}),
$$
we obtain
$$
H_{B^d}(r)
=
1 - \exp\!\left(
    -\,\lambda
    \int_{\mathcal{C}'}
        \big( \mathrm{vol}(K + B(0,r)) - \mathrm{vol}(K) \big)
    \, \mathbb{Q}(dK)
\right).
$$

Voronoi Tessellations

Let \(X = \{ x_i \}_{i=1}^n\) be a locally finite subset of \(\mathbb{R}^d\) (with \(n=\infty\) allowed).

Equivalently, for \(x \neq y\), define \(H_y^{+}(x)\) as the closed halfspace containing \(x\) and bounded by the mid-hyperplane of \(x\) and \(y\).
Then \[ V(x_i, X) = \bigcap_{j \ne i} H_{x_j}^{+}(x_i). \]

Notes:

• \(V(x_i, X)\) is a closed convex polyhedron in \(\mathbb{R}^d\).
• \(\displaystyle \bigcup_{i=1}^n V(x_i, X) = \mathbb{R}^d\).