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The stereological estimation of moments of particle volume

Published online by Cambridge University Press:  14 July 2016

E. B. Jensen  and
H. J. G. Gundersen
E. B. Jensen*
Affiliation:
Aarhus University
H. J. G. Gundersen*
Affiliation:
Aarhus University
*
∗Postal address: Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗∗Postal address: Diabetes Research Laboratory, Bartholinbygningen, Aarhus University, DK-8000 Aarhus C, Denmark.
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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 the q th moment of the volume-weighted distribution of particle volume, based on point-sampling of particles and measurements on q -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 a q-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given for q = 1. For q = 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.

Keywords

CONVEXITYELLIPSOIDSGEOMETRIC PROBABILITYINTEGRAL GEOMETRYINTERCEPTSNON-CONVEXITYNUMBER DISTRIBUTIONPARTICLE SIZE DISTRIBUTIONSPOINT-SAMPLINGSTAR VOLUMEVOLUME-WEIGHTED DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 82 - 98
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

This paper was in part presented at the Second International Workshop on Stereology and Stochastic Geometry, October 1983, Aarhus, Denmark, and at the Sixth International Congress for Stereology, October 1983, Gainesville, Florida, USA.

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