A general method for solving stereological problems for particle systems is applied to polyhedron structures. We suggested computing the kernel function of the respective stereological integral equation by means of computer simulation. Two models of random polyhedrons are investigated. First, regular prisms are considered which are described by their size and shape. The size-shape distribution of a stationary and isotropic spatial ensemble of regular prisms can be estimated from the size-shape distribution of the polygons observed in a section plane. Secondly, random polyhedrons are constructed as the convex hull of points which are uniformly distributed on surfaces of spheres. It is assumed that the size of the polyhedrons and the number of points (i.e. the number of vertices) are random variables. Then the distribution of a spatially distributed ensemble of polyhedrons is determined by its size-number distribution. The corresponding numerical density of this bivariate size-number distribution can be stereologically determined from the estimated numerical density of the bivariate size-number distribution of the intersection profiles.

In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

A biphase image, representing the normal and degenerated fibres in a vertical cross-section of a nerve, is considered. A random set model based on a Gibbs point process is proposed for the union of the two phases. A kind of independence between the degeneration process and the original fibres is defined and tested.

Simulation procedures for typical Johnson-Mehl crystals generated under various models for random nucleation are proposed. These procedures include algorithms for simulating spatio-time-inhomogeneous Poisson processes. Empirical results for a particular class of Johnson-Mehl tessellations in two and three dimensions show remarkably different crystals.

Formulas for anisotropic stereology of fibre and surface processes are presented. They concern the relation between second-order quantities of the original process and its projections and sections. Various mathematical tools for handling these formulas are presented, including stochastic optimization. Finally applications in stereology are discussed, relating to intensity estimators using anisotropic sampling designs. Variances of these estimators are expressed and evaluated for processes with the Poisson property.

Quite often in the statistical analysis of medical and biological problems, data are images corresponding to entire objects that include smaller objects within them. In these cases, we need models of random closed sets (RACS) confined to compact subsets of the plane. There is no room for stationarity hypotheses and the increase of statistical information comes from independent replicates of the same phenomena rather than increasing our sample window. We investigate practical methods of modelling RACS by means of circumscribed balls, leading to a natural definition of location, size and shape. We discuss the possibilities of using these random variables in order to define statistical spaces of RACS that will allow us to use maximum likelihood methods.

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.

A stationary (but not necessarily isotropic) Boolean model Y in the plane is considered as a model for overlapping particle systems. The primary grain (i.e. the typical particle) is assumed to be simply connected, but no convexity assumptions are made. A new method is presented to estimate the intensity y of the underlying Poisson process (i.e. the mean number of particles per unit area) from measurements on the union set Y. The method is based mainly on the concept of convexification of a non-convex set, it also produces an unbiased estimator for a (suitably defined) mean body of Y, which in turn makes it possible to estimate the mean grain of the particle process.

Unlike the means of distributions on a euclidean space, it is not entirely clear how one should define the means of distributions on the size-and-shape or shape spaces of k labelled points in ℝm since these spaces are all curved. In this paper, we discuss, from a shape-theoretic point of view, some questions which arise in practice while using procrustean methods to define mean size-and-shapes or shapes. We obtain sufficient conditions for such means to be unique and for the corresponding generalized procrustean algorithms to converge to them. These conditions involve the curvature of the size-and-shape or shape spaces and are much less restrictive than asking for the data to be concentrated.

Non-parametric estimators of the distribution of the grain of the Boolean model are considered. The technique is based on the study of point processes of tangent points in different directions related to the Boolean model. Their second- and higher-order characteristics are used to estimate the mean body and the distribution of the typical grain. Central limit theorems for the improved estimator of the intensity and surface measures of the Boolean model are also proved.

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.

The typical cell of a stationary Poisson hyperplane tessellation in the d-dimensional Euclidean space is called the Poisson polytope, and the cell containing the origin is called the Poisson 0-polytope. The intention of the paper is to show that the cells of the anisotropic tessellations are in some sense larger than those of the isotropic tessellations. Under the condition of equal intensities, it is proved that the moments of order n = 1, 2, … for the volume of the Poisson 0-polytope in the anisotropic case are not smaller than the corresponding moments in the isotropic case. Similar results are derived for the Poisson polytope. Finally, generalizations are mentioned.

This paper is the third in a series with the general title ‘How to look at the five-dimensional shape space Σ4,3. Parts I and II were concerned with distributions and diffusions respectively. Here we ‘look at’ geodesics.

A version of the Rao–Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; random s-dimensional samples are worse than random r-dimensional samples for s < r. Furthermore, the standard stereological ratio estimators of different dimensions are canonically related to each other by the Rao–Blackwell process. However, there are realistic cases where sampling with a lower-dimensional probe increases efficiency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatistics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context.

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a 1,· ··, an. Furthermore, let Δ(Xn ): = Vol (Bd Xn ) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn )/Ε (Δ (Xn )) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn , the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

Three different stereological methods for the determination of second-order quantities of planar fibre processes which have been suggested in the literature are considered. Proofs of the formulae are given (also by using a new integral geometric formula), relations between the methods are derived and the prerequisites are discussed. Furthermore, edge-corrected unbiased estimators for the second-order quantities are given.

We solve a problem proposed by V. Klee (1969). He asked for a calculation of κ, the expected value of V, the volume of a daughter tetrahedron whose vertices are chosen at random (i.e. independently and uniformly) in the interior of a parent tetrahedron of unit volume. We discover:

We also calculate the second, fourth and sixth moments of V.

The mean number of edges of a randomly chosen neighbouring cell of the typical cell in a planar stationary tessellation, under the condition that it has n edges, has been studied by physicists for more than 20 years. Experiments and simulation studies led empirically to the so-called Aboav's law. This law now plays a central role in Rivier's (1993) maximum entropy theory of statistical crystallography. Using Mecke's (1980) Palm method, an exact form of Aboav's law is derived. Results in higher-dimensional cases are also discussed.

We study convergence in total variation of non-stationary Markov chains in continuous time and apply the results to the image analysis problem of object recognition. The input is a grey-scale or binary image and the desired output is a graphical pattern in continuous space, such as a list of geometric objects or a line drawing. The natural prior models are Markov point processes found in stochastic geometry. We construct well-defined spatial birth-and-death processes that converge weakly to the posterior distribution. A simulated annealing algorithm involving a sequence of spatial birth-and-death processes is developed and shown to converge in total variation to a uniform distribution on the set of posterior mode solutions. The method is demonstrated on a tame example.