In this paper we introduce the transparent dead leaves (TDL) random field, a new germ-grain model in which the grains are combined according to a transparency principle. Informally, this model may be seen as the superposition of infinitely many semitransparent objects. It is therefore of interest in view of the modeling of natural images. Properties of this new model are established and a simulation algorithm is proposed. The main contribution of the paper is to establish a central limit theorem, showing that, when varying the transparency of the grain from opacity to total transparency, the TDL model ranges from the dead leaves model to a Gaussian random field.

We study stochastic properties of the empty space for stationary germ-grain models in Rd; in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞.

We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.

Consider an inhomogeneous germ-grain model with spherical grains whose radii depend on their positions through a rate function, possibly perturbed by a random noise. We find the critical rate function that separates the cases when the germ-grain model covers the whole space with a positive probability and when the total coverage occurs with probability zero.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

This paper considers a germ-grain model for a random system of non-overlapping spheres in ℝd for d = 1, 2 and 3. The centres of the spheres (i.e. the ‘germs’ for the ‘grains’) form a stationary Poisson process; the spheres result from a uniform growth process which starts at the same instant in all points in the radial direction and stops for any sphere when it touches any other sphere. Upper and lower bounds are derived for the volume fraction of space occupied by the spheres; simulation yields the values 0.632, 0.349 and 0.186 for d = 1, 2 and 3. The simulations also provide an estimate of the tail of the distribution function of the volume of a randomly chosen sphere; these tails are compared with those of two exponential distributions, of which one is a lower bound and is an asymptote at the origin, and the other has the same mean as the simulated distribution. An upper bound on the tail of the distribution is also an asymptote at the origin but has a heavier tail than either of these exponential distributions. More detailed information for the one-dimensional case has been found by Daley, Mallows and Shepp; relevant information is summarized, including the volume fraction 1 - e-1 = 0.63212 and the tail of the grain volume distribution e-yexp(e-y - 1), which is closer to the simulated tails for d = 2 and 3 than the exponential bounds.

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝd to Ξ. For almost all points x ε ℝd there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at the same speed. The notion of a typical point is made precise by using the generalized curvature measures of Ξ. The result generalizes a well known formula for the Boolean model. Specific examples are discussed in detail.