The stereological estimation of moments of particle volume

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Abstract
In stereology or applied geometric probability quantitative characterization of aggregates of particles from information on lower-dimensional sections plays a major role. Most stereological methods developed for particle aggregates are based on the assumption that the particles are of the same, known (simple) shape. Information on the volume-weighted distribution of particle size may, however, be obtained under fairly general assumptions about particle shape if particle volume is chosen as size parameter. In fact, there exists in this case an unbiased stereological estimator of the first moment under the sole assumption that the particles are convex. In the present paper, we consider a particle aggregate in ℝ and derive estimators of theqth moment of the volume-weighted distribution of particle volume, based on point-sampling of particles and measurements onq-flats through sampled particles. The estimators are valid for arbitrarily shaped particles but if the particles are non-convex it is necessary for the determination of the estimators to be able to identify the different separated parts on aq-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given forq= 1. Forq= 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.