In this thesis we develop statistical methodology for stereological estimation problems which in particular appear in the field of materials science. We study mathematical models which may describe materials microstructures, and we develop statistical methods for estimating the parameter(s) of these models in a stereological setting. That is, typically we consider a 3D model, and instead of directly observing data generated by this model we only observe a 2D planar section. In many cases we even consider (𝑑 −1)-dimensional sections of a 𝑑-dimensional model (𝑑 = 2, 3, . . . ), because it turns out this is often not more complicated than the 𝑑 = 3 case. In order to formally define and study these models we mainly rely on concepts from stochastic geometry and point process theory. This thesis is divided into two parts. In part I we consider a model of randomly positioned and -oriented particles of the same shape, but varying in size, and in part II we consider a space-filling model by studying a random tessellation (mosaic). In these models, the particles of a particle process, or the cells of a random tessellation, may represent the so-called grains of a materials microstructure....

Obtaining information about the 3D grain size distribution of metallic microstructures is crucial for understanding the mechanical behavior of metals. This paper addresses the problem of estimating the 3D grain size distribution from 2D cross sections. This is a well-known stereological problem and different estimators have been proposed in the literature. We propose a statistical estimation procedure that provides consistent estimates without relying on arbitrary binning choices. When applying this procedure to space filling structures, we investigate the impact of the choice of grain shape and propose a heuristic to choose the best grain shape. To validate our approach, we employ simulations using Laguerre–Voronoi diagrams and apply our methodology to a sample of Interstitial-Free steel, obtained via EBSD.

In this paper, we consider statistical inference for Poisson-Laguerre tessellations in (Formula presented.). The object of interest is a distribution function (Formula presented.) which describes the distribution of the arrival times of the generator points. The function (Formula presented.) uniquely determines the intensity measure of the underlying Poisson process. Two nonparametric estimators for (Formula presented.) are introduced, which depend only on the points of the Poisson process that generate non-empty cells and the actual cells corresponding to these points. The proposed estimators are proven to be strongly consistent as the observation window expands unboundedly to the whole space. We also consider a stereological setting, where one is interested in estimating the distribution function associated with the Poisson process of a higher-dimensional Poisson-Laguerre tessellation, given that a corresponding sectional Poisson-Laguerre tessellation is observed.

Consider an opaque medium that contains 3D particles. All particles are convex bodies of the same shape, but they vary in size. The particles are randomly positioned and oriented within the medium and cannot be observed directly. Taking a planar section of the medium we obtain a sample of observed 2D section profile areas of the intersected particles. In this paper, the distribution of interest is the underlying 3D particle size distribution for which an identifiability result is obtained. Moreover, a non-parametric estimator is proposed for this size distribution. The estimator is proven to be consistent and its performance is assessed in a simulation study.

In various stereological problems ann-dimensional convex body is intersected with an(n−1)-dimensionalIsotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associatedwith the(n−1)-dimensional volume of such a random section is studied. This distribution is also knownas chord length distribution and cross section area distribution in the planar and spatial case respectively.For various classes of convex bodies it is shown that these distribution functions are absolutely continuouswith respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating thecorresponding probability density functions.