The workshops on stochastic geometry, stereology and image analysis have been held every second year since 1981. The first workshops were small meetings, but with prominent speakers, who have attended most of the workshops since then. Over the years the workshops have increased in size and in impact too.

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points), [4], [5], [8], [11], [12]. Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern, [1], [2], [3], [7], [9], [10]. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. Our purpose is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling.

Let (Φ1 Φ2) be a bivariate point process. Let be the probability that (Φ1 Φ2) - (0, s) (the process without the 0 and s points) verify U when we have a point of <1>1 in the origin and a point of Φ2 in s. This is the reduced cross Palm distribution. Some correlation measures for bivariate point processes based on this reduced cross Palm distribution are proposed. Their estimators and expressions under the independence and the random labelling hypothesis are considered. The differences and improvements with respect to the cross intensity function and its integrated version, the cross function (Stoyan et al. 1987), are studied. Some Monte Carlo tests for testing the independence and the random labelling hypothesis are proposed. They are applied to real bivariate point patterns: positions of hickories and maples in the Lansing Woods (Diggle 1983) and cases and controls of childhood leukaemia and lymphoma in North Humberside (Cuzick and Edwards 1990).

The study of networks with random conductivities or resistances is of considerable interest in physics, biology, geology and mathematics. The talk concentrated on a rather flexible class of parametric models for trees with random conductivities, based on properties of the inverse Gaussian distribution. For finite trees some exact distributional results were presented and extensions of the models to infinite trees were discussed. Joint work with Gerard Letac and Angelo E. Koudou (Toulouse), extending Barndorff-Nielsen (1994).

The shape of a rectangular prism in (d + 1) dimensions can be described by d quantities. In this talk, using a definition of shape which involves the ratios of edge lengths, we investigate shape distributions which are invariant when the prism is cut into two, also rectangular, prisms, with one prism retained for measurement and the other discarded. Interesting new distributions on [0, l]d arise.

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

This paper considers a structure, named a ‘random partition process’, which is a generalisation of a random tessellation. The cells, possibly multi-part and with holes, have a general topology summarised by the Euler characteristic. Vertices of all orders are allowed. Using the tools of ergodic theory, all of the formulae, from the traditional theory of random tessellations with convex cells, are generalised. Some motivating examples are given.

Consider A Stationary Line Segment Process In The Plane, Observed Through A Window W. Line Segments Which Cross The Boundary Of The Window Are Only Partially Observed. One Might Say That The Observed Lengths Of The Line Segments Are Then Censored. However, Classical Methods From The Theory Of Censored Survival Data (Kaplan-Meier Estimator) Can Be Seen To Be Inappropriate.

For a stationary particle process X with convex particles in ℝdd ≧ 2, a mean body M(X) can be defined by where h(M,·) denotes the support function of the convex body M, γ the intensity of X, and P0 is the distribution of the typical particle of X (a probability measure on the set of convex bodies with Steiner point at the origin). Replacing the support function h(M,·) by the surface area measure S(M,·) (see Schneider (1993), for the basic notions from convex geometry), we get the Blaschke body B(X) of X, After normalization, the left-hand side represents the mean normal distribution of X. The main problem discussed here is whether B(X) (respectively S(B(X), ·)) is uniquely determined by the mean bodies M(X ∩ E) in random planar sections X ∩ E of X. From more general results in Weil (1995), it follows that the expectation ES(M(X ∩ E), ·) (taken w.r.t. the uniform distribution of two-dimensional subspaces E in ℝd) equals the surface area measure of a section mean B2(B(X)) of B(X). Thus, the formulated stereological question can be reduced to the injectivity of the transform B2 : K ↦ B2(K).

In the statistical analysis of random closed sets the distribution of the distance from a fixed point to the nearest point of the random set (the empty space function) is an important tool, and therefore development of efficient estimators is crucial.

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of 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.

The classical Blaschke-Petkanschin formula is a formula in integral geometry givmg a geometric measure decomposition of the q-fold product of Lebesgue measure. The original versions are due to Blaschke and Petkanschin in the 1930s. In Zähle (1990) and Jensen and Kiêu (1992), generalized versions have been derived, where Lebesgue measure is replaced by Hausdorff measure.

This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3} is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi; determines a triple {X1 X2, X3} whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

For a stationary point process X of convex particles in ℝd the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.

A Boolean model in ℝd is the union of independent random compact subsets of ℝd (called ‘grains’) located according to a Poisson point process. A sequential algorithm is proposed to conditionally simulate a Boolean model, i.e. so that a finite set of specified points belongs to the grains and another set to the complementary phase (referred to as the ‘pores’).

Two sets of k labelled points, or configurations, in ℝm are defined to have the same shape if they differ only in translation, rotation and scaling. An important matter in practice is the estimation of the shape of the means; the shape determined by the means of data on the vertices of configurations. However, statistical models for vertices-based shapes always involve some unknown samplewise nuisance parameters associated with ambiguity of location, rotation and scaling. The use of procrustean mean shapes for a finite set of configurations, which are usually formulated directly in terms of their vertices, will enable one to eliminate these nuisance parameters.

The strength and range of interpoint interactions in a spatial point process can be quantified by the function J = (1 - G)/(1 - F), where G is the nearest-neighbour distance distribution function and F the empty space function of the process. J(r) is identically equal to 1 for a Poisson process; values of J(r) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a very large class of point processes, J(r) is constant for distances r greater than the range of spatial interaction. Hence both the range and type of interpoint interaction may be inferred from J without parametric model assumptions. We evaluate J(r) explicitly for a variety of point processes. The J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes.

The statistical study of the clustering patterns formed by the three-dimensional distribution of galaxies is one of the most important observational clues to discriminate between rival theories of structure formation in the Universe.