Werner Nagel

Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration

Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space with the following stability property: their distribution is invariant with respect to the operation of iteration (or nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary tessellations is fully characterized.

Limits of sequences of stationary planar tessellations

In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.

THE ESTIMATION OF THE EULER-POINCARE CHARACTERISTIC FROM OBSERVATIONS ON PARALLEL SECTIONS

An unbiased estimator of the Euler‐Poincaré characteristic (Euler number) of an arbitrary object or a random structure, respectively, is given. The estimator is based on joint observations of pairs of parallel section profiles. Thus the present paper extends the use of Sterios disector from counting particles to determining the Euler number for a wide class of probes. The correctness of the given formulae is proved with mathematical strictness. Furthermore, the feasibility of the method is illustrated by an example from materials sciences.

The Euler Number Of Discretized Sets - On The Choice Of Adjacency In Homogeneous Lattices

Two approaches for determining the Euler-Poincare characteristic of a set observed on lattice points are considered in the context of image analysis { the integral geometric and the polyhedral approach. Information about the set is assumed to be available on lattice points only. In order to retain properties of the Euler number and to provide a good approximation of the true Euler number of the original set in the Euclidean space, the appropriate choice of adjacency in the lattice for the set and its background is crucial. Adjacencies are defined using tessellations of the whole space into polyhedrons. In R 3 , two new 14 adjacencies are introduced additionally to the well known 6 and 26 adjacencies. For the Euler number of a set and its complement, a consistency relation holds. Each of the pairs of adjacencies (14:1; 14:1), (14:2; 14:2), (6; 26), and (26; 6) is shown to be a pair of complementary adjacencies with respect to this relation. That is, the approximations of the Euler numbers are consistent if the set and its background (complement) are equipped with this pair of adjacencies. Furthermore, sufficient conditions for the correctness of the approximations of the Euler number are given. The analysis of selected microstructures and a simulation study illustrate how the estimated Euler number depends on the chosen adjacency. It also shows that there is not a uniquely best pair of adjacencies with respect to the estimation of the Euler number of a set in Euclidean space.

Length distributions of edges in planar stationary and isotropic STIT tessellations

Stationary and isotropic random tessellations of the euclidean plane are studied which have the characteristic property to be stable with respect to iteration (or nesting), STIT for short. Since their cells are not in a face-to-face position, three different types of linear segments appear. For all the types the distribution of the length of the typical segment is given.

The iteration of random tessellations and a construction of a homogeneous process of cell divisions

A random tessellation of ℝ d is said to be homogeneous if its distribution is invariant under all shifts of ℝ d . The iteration of homogeneous random tessellations is described in a new manner that makes it evident that the resulting random tessellation is homogeneous again. Furthermore, a tessellation-valued process is constructed, the random states of which are homogeneous random tessellations stable under iteration (STIT). It can be interpreted as a process of subsequent division of cells.

Measuring intrinsic volumes in digital 3d images

The intrinsic volumes – in 3d up to constants volume, surface area, integral of mean curvature, and Euler number – are a very useful set of geometric characteristics Combining integral and digital geometry we develop a method for efficient simultanous calculation of the intrinsic volumes of sets observed in binary images In order to achieve consistency in the derived intrinsic volumes for both foreground and background, suitable pairs of discrete connectivities have to be used To make this rigorous, the concepts discretization w.r.t an adjacency system and complementarity of adjacency systems are introduced.

Spatial STIT tessellations: distributional results for I-segments

In this paper we consider three-dimensional random tessellations that are stable under iteration (STIT tessellations). STIT tessellations arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the cell-dividing polygons are the so-called I-segments of the tessellation. The main result is an explicit formula for the distribution of the number of vertices in the relative interior of the typical I-segment. In preparation for its proof, we obtain other distributional identities for the typical I-segment and the length-weighted typical I-segment, which provide new insight into the spatiotemporal construction process.

dünne schnitte von stationaren räumlichen faserprozessen

A thin section of a fibre system in space is its subset located between two parallel planes The orthogonal projection of a thin section of a statinary spatial fibre process produces a stationary planar fibre jprocess.Its intensity as a function of direction of the section is refered to as rose of sections. With its intensities and the distributions of directions of both the processes can be calculated one from the others. This paper follows the ideas of MECKE/SOTOYAN und MECKE/ NAGEL

The β-mixing rate of STIT tessellations

We consider homogeneous STIT tessellations in the -dimensional Euclidean space and show that the (spatial) -mixing rate converges to zero.