A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Foam models, especially random tessellations, are powerful tools to study the relations between the geometric structure of foams and their physical properties. In this paper, we propose the use of random Laguerre tessellations, weighted versions of the well-known Voronoi tessellations, as models for the microstructure of foams. Based on geometric characteristics estimated from a tomographic image of a closed-cell polymer foam, we fit a Laguerre tessellation model to the material. It is shown that this model allows for a better fit of the geometric structure of the foam than some classical Voronoi tessellation models

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Volume image analysis provides a number of methods for the characterization of the microstructure of open foams. Mean values of characteristics of the edge system are measured directly from the volume image. Further characteristics like the intensity and mean size of the cells are obtained using model assumptions where the edge system of the foam is interpreted as a realization of a random closed set. Macroscopically homogeneous random tessellations provide a suitable model for foam structures. However, their cells often lack the degree of regularity observed in real data. In this respect some deterministic models seem to be closer to realistic structures, although they do not capture the microscopic heterogeneity of real foams. In this paper, the influence of the model choice on the obtained mean values is studied. Moreover, a method for reconstruction of the cells of an open foam from its edge system is described and tested for the tessellations under consideration

SummaryIn this paper, a ceramic open foam is modelled using a two-step procedure. First, a random Laguerre tessellation is fitted to the edge system of the polyurethane foam forming the core of the ceramic foam. In a second phase, a model of the ceramic foam is obtained using dilations with balls of locally varying size. The model fitting is based on geometric characteristics of both polyurethane and ceramic foam estimated from reconstructed tomographic images of these structures.