Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T07:49:41.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of mean particle volume using the set covariance function

Part of: Sampling theory, sample surveys General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Annoesjka Cabo  and
Adrian Baddeley
Annoesjka Cabo*
Affiliation:
University of Western Australia
Adrian Baddeley*
Affiliation:
University of Western Australia
*
∗ Postal address: Nijendal 5, 3972 KC Driebergen, The Netherlands.
∗∗ Postal address: Department of Mathematics and Statistics, University of Western Australia, Nedlands WA 6009, Australia. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Our aim is to estimate the volume-weighted mean of the volumes of three-dimensional ‘particles’ (compact, not-necessarily-convex subsets) from plane sections of the particle population. The standard stereological technique is to place test lines in the plane section, and measure cubed intercept lengths with the two-dimensional particle profiles. This paper discusses more efficient estimators obtained by integrating over all possible placements of the test line. We prove that these estimators have smaller variance than the line transect estimators, and indeed are related to them by the Rao-Blackwell process. In the improved estimators, the cubed intercept length is replaced by a moment of the distance between two points in the section profile. This can be computed as a moment of the set covariance function, which in turn is computable using the fast Fourier transform. We also derive an isoperimetric-type inequality between the improved estimator and the area-weighted 3/2th moment of the profile areas. Finally, we present two practical applications to particles of silicon carbide and to synaptic boutons in brain tissue. We estimate the variance of the technique and the gain in efficiency over line transect techniques; the efficiency improvement appears to be as much as one order of magnitude.

Keywords

4-lincCampbell-Mecke formulachord lengthsdesign-based stereologyfast Fourier transformintegral geometryinterpoint distancesisoperimetric inequalityline transectslog-normal distributionmodel-based stereologyPalm distributionpoint-sampled interceptsRao-Blackwell processstereologyvertical sectionsvolume-weighted mean volumeweighted distributions

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry 62D05: Sampling theory, sample surveys
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 27 - 46
Copyright
Copyright © Applied Probability Trust 2003 

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.)

References

Artachó-Pérula, E. and Roldán-Villalobos, R. (1994). Volume-weighted mean particle volume estimation using different measurement methods. J. Microscopy 173, 73–78.CrossRefGoogle Scholar
Baddeley, A. J. and Cruz-Orive, L. M. (1995). The Rao–Blackwell theorem in stereology and some counterexamples. Adv. Appl. Prob. 27, 2–19.CrossRefGoogle Scholar
Baddeley, A. J., Gundersen, H. J. G. and Cruz-Orive, L. M. (1986). Estimation of surface area from vertical sections. J. Microscopy 142, 259–276.CrossRefGoogle ScholarPubMed
Blaschke, W. (1949). Vorlesungen über Integralgeometrie. Chelsea, New York.Google Scholar
Borel, E. (1925). Principes et Formules Classiques du Calcul des Probabilités. Gauthier-Villars, Paris.Google Scholar
Braendgaard, H. and Gundersen, H. J. G. (1986). The impact of recent stereological advances on quantitative studies of the nervous system. J. Neurosci. Meth. 18, 39–78.CrossRefGoogle ScholarPubMed
Brown, B. and Hettmansperger, T. (1996). Normal scores, normal plots and tests for normality. J. Amer. Statist. Assoc. 91, 1668–1675.Google Scholar
Cabo, A. J. (1994). Set functionals in stochastic geometry. , Technical University of Delft.Google Scholar
Cabo, A. J. and Baddeley, A. J. (1995). Line transects, covariance functions and set convergence. Adv. Appl. Prob. 27, 585–605.CrossRefGoogle Scholar
Carleman, T. (1919). Ueber eine isoperimetrische Aufgabe und ihre physikalischen Anwendungen. Math. Z. 3, 1–7.CrossRefGoogle Scholar
Chia, J. L. C. (2002). Data models and estimation accuracy in stereology. , University of Western Australia.Google Scholar
Crofton, M. W. (1885). Probability. In Encyclopaedia Britannica, Vol. 19, 9th edn, Stoddart, Philadelphia, PA, pp. 768–788.Google Scholar
D'Agostino, R. B., Belanger, A. and D'Agostino, R. B. Jr. (1990). A suggestion for using powerful and informative tests of normality. Amer. Statistician 44, 316–321.Google Scholar
Davy, P. J. and Miles, R. E. (1977). Sampling theory for opaque spatial specimens. J. R. Statis. Soc. B 39, 56–65.Google Scholar
Haas, A., Matheron, G. and Serra, J. (1967). Morphologie mathématique et granulométries en place. Ann. Mines 11, 736–753.Google Scholar
Haas, A., Matheron, G. and Serra, J. (1967). Morphologie mathématique et granulométries en place, II. Ann. Mines 12, 767–782.Google Scholar
Hadwiger, H. (1950). Neue Integralrelationen für Eikörperpare. Acta Sci. Math. (Szeged), 13, 252–257.Google Scholar
Jensen, E. B. and Gundersen, H. J. G. (1983). On the estimation of moments of the volume-weighted distribution of particle sizes. In Proc. Second Internat. Workshop Stereology Stoch. Geometry, Aarhus, eds Jensen, E. B. and Gundersen, H. J. G., University of Aarhus, Aarhus, pp. 81–103.Google Scholar
Jensen, E. B. and Gundersen, H. J. G. (1985). The stereological estimation of moments of particle volume. J. Appl. Prob. 22, 82–98.CrossRefGoogle Scholar
Jensen, E. B. V. (1998). Local Stereology. World Scientific, Singapore.CrossRefGoogle Scholar
Matérn, B., (1960). Spatial variation. Meddelanden från Statens Skogsforskningsinstitut 49, No. 5. Second edition: Springer, Berlin, 1986.Google Scholar
Mecke, J. (1967). Stationäre zufällige maße auf lokalkompakten abelschen gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Miles, R. E. (1979). Some new integral formulae, with stochastic applications. J. Appl. Prob. 16, 592–606.CrossRefGoogle Scholar
Miles, R. E. (1983). Contributed discussion (session on Stochastic Geometry). Bull. Internat. Statist. Inst. 44, 392.Google Scholar
Miles, R. E. (1983). Stereology for embedded aggregates of not-necessarily-convex particles. In Proc. Second Internat. Workshop Stereology Stoch. Geometry, Aarhus, eds Jensen, E. B. and Gundersen, H. J. G., Department of Theoretical Statistics, University of Aarhus, Aarhus, pp. 127–147.Google Scholar
Miles, R. E. (1985). A comprehensive set of stereological formulae for embedded aggregates of not-necessarily-convex particles. J. Microscopy 138, 115–125.CrossRefGoogle Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability (Encyclopedia Math. Appl. 1). Addison-Wesley, Reading, MA.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Shapiro, S. S. and Francia, R. S. (1972). Approximate analysis of variance test for normality. J. Amer. Statist. Assoc. 67, 215–216.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar