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.

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.