Christian Jung

• Studying the microstructure of industrial materials allows for a better understanding of their mechanical properties. Random tessellations are a tool predestined to be used for modelling microstructures such as cellular materials, solid particles and crack structures in concrete. This thesis is split up into two parts. In Part I we consider models based on random tessellations and several applications. First, we establish a model for crack structures in concrete. Cracks are assumed to take the path of least resistance when propagating through concrete. The model is based on the idea of weighting the facets of a Voronoi tessellation. Bounded by a contour, a connected set of facets of minimal weight is computed by solving a binary integer program. The point process models that generate the Voronoi diagrams are interchangeable such that the model realizations show several levels of regularity. This makes the model highly flexible and allows for the generation of a whole range of different crack structures. For embedding the model realizations into real computed tomography images of concrete, a dedicated image processing pipeline is developed. Second, germ-grain processes based on the cells of Gibbs-Laguerre tessellations are used for modelling solid particles in active protective coatings. Our goal is to compute realizations of Gibbs-Laguerre tessellations whose distributions of cell shape and size are similar to those of the observed particles. Several techniques are proposed to also account for the correlation between these characteristics. Third, we present an analytical description of the 2d generalized balanced power diagram. Its edges are conic sections for which we derive a parametric representation. Its vertices are the intersections of two conics. This analytical description allows for the formulation of a novel algorithm to simulate the generalized balanced power diagram and for the derivation of closed formulas to compute cell areas and perimeters. Part II of this thesis deals with the segmentation of cracks in 3d computed tomography images of concrete. To this end, several segmentation techniques from classical image processing and machine learning are implemented and compared to each other. The evaluation is based on semi-synthetic crack images. From the classical methods, Hessian-based percolation is identified as the best-performing method. It is then adapted to and validated on real images of cracked concrete. To develop a better understanding of the algorithm‘s performance and robustness with respect to branching, multi-scale cracks and noise, a new image data set is generated based on the crack modelling method from Part I. The performance of Hessian-based percolation is then evaluated objectively based on the annotated labels that are a byproduct of the crack generation procedure.