Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T05:00:25.087Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Blaschke, W. (1935a) Integralgeometrie 1. Ermittlung der Dichten für lineare Underräume im En. Actualités Sci. Indust. 252, 1–22.Google Scholar
Blaschke, W. (1935b) Integralgeometrie 2. Zu Ergebnissen von M. W. Crofton. Bull. Math. Soc. Roumaine Sci. 37, 3–11.Google Scholar
Cruz-Orive, L. M. (1976) Particle size-shape distributions: the general spheroid problem I. Mathematical model. J. Microsc. 107, 235–253.CrossRefGoogle Scholar
Cruz-Orive, L. M. (1978) Particle size-shape distributions: the general spheroid problem II. Stochastic model and practical guide. J. Microsc. 112, 153–167.Google Scholar
Cruz-Orive, L. M. (1980) On the estimation of particle number. J. Microsc. 120, 15–27.Google Scholar
Davy, P. and Miles, R. E. (1977) Sampling theory for opaque spatial specimens. J. R. Statist. Soc. B 39, 56–65.Google Scholar
Gundersen, H. J. G. and Jensen, E. B. (1983) Particle sizes and their distributions estimated from line- and point-sampled intercepts. Including graphical unfolding. J. Microsc. 131, 291–310.CrossRefGoogle Scholar
Jensen, E. B. (1983) Random test systems and their use in the sampling of intercepts through non-convex bodies in Euclidean space. Research Report No. 94, Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, Denmark.Google Scholar
Jensen, E. B. and Gundersen, H. J. G., (Eds) (1983) Proceedings of the Second International Workshop on Stereology and Stochastic Geometry. Aarhus 1983., Memoirs No. 6, Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, Denmark.Google Scholar
Matheron, G. (1967) Eléments pour une théorie des milieux poreux. Masson, Paris.Google Scholar
Miles, R. E. (1971) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
Miles, R. E. (1979) Some new integral geometric formulae, with stochastic applications. J. Appl. Prob. 16, 592–606.Google Scholar
Miles, R. E. (1983) Stereology for embedded aggregates of not-necessarily-convex particles. Pages 127–147 of Jensen and Gundersen (1983).Google Scholar
Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n -dimensionalen Raum. Abh. Math. Sem. Univ. Hamburg 11, 249–310.CrossRefGoogle Scholar
Röthlisberger, H. (1955) An adequate method of grain-size determination in sections. J. Geol. 63, 579–584.CrossRefGoogle Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its Applications, volume 1. Addison-Wesley, Reading. Mass. Google Scholar
Serra, J. (1982) Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Sterio, D. C. (1984) The unbiased estimation of number and sizes of arbitrary particles using the disector. J. Microsc. 134, 127–136.Google Scholar