Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T15:39:25.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Stereological estimation of particle size distributions

Part of: Numerical approximation and computational geometry Applications Inference from stochastic processes Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Shigeru Mase
Shigeru Mase*
Affiliation:
Hiroshima University
*
* Postal address: Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima City, 724 Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

The corpuscle problem of Wicksell is discussed. We give a numerical quadrature of Gauss–Chebyshev type for Wicksell's integral equation which combines a size distribution of discs on a sectional plane with that of spheres. We also give an estimation procedure of three-dimensional size distributions based on this quadrature and examine its theoretical properties. In practice, we need a smoothing technique for empirical distribution functions before applying this estimator. Simulation results are given. Our idea also is applied to the thick section case and an analysis of microscopic data is given.

Keywords

WICKSELL'S EQUATIONSTEREOLOGYGAUSS–CHEBYSHEV QUADRATURESIZEDISTRIBUTIONSECTIONAL PLANECORPUSCLE PROBLEMSMOOTHING

MSC classification

Primary: 62G07: Density estimation
Secondary: 62P10: Applications to biology and medical sciences 62G20: Asymptotic properties 62M30: Spatial processes 65D30: Numerical integration
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 350 - 366
Copyright
Copyright © Applied Probability Trust 1995 

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

The original version of this paper was presented at the International Workshop on Stochastic Geometry, Stereology and Image Analysis held at the Universidad Internacional Menendez Pelayo, Valencia, Spain, on 21–24 September 1993.

References

Cruz-Orive, L. M. (1983) Distribution-free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods. J. Microscopy 131, 265–290.Google Scholar
Friedman, J. H. (1984) A variable span smoother. Tech. Rept. No. 5, Laboratory for Computational Statistics, Stanford University.Google Scholar
Es, A. J. Van and Hoogendoorn, A. (1990) Kernel estimation in Wicksell's corpuscle problem. Biometrika 77, 139–145.Google Scholar
Goldsmith, P. L. (1967) The calculation of true particle size distributions from the sizes observed in a thick section. Br. J. Appl. Phys. 18, 813–830.Google Scholar
Mecke, J. and Stoyan, D. (1980) Stereological problems for spherical particles. Math. Nachr. 96, 311–317.Google Scholar
Rivlin, T. J. (1990) Chebyshev Polynomials, 2nd edn. Wiley, New York.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
Silverman, B. W., Jones, M. C., Wilson, J. D., and Nychka, D. W. (1990) A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography. J. R. Statist. Soc. B 52, 271–324.Google Scholar
Stoyan, D., Kendall, W. S., and Mecke, J. (1987) Stochastic Geometry and Its Applications. Wiley, New York.Google Scholar
Suzaki, T. and Ando, M. (1992) A computer program for estimating size distributions of spherical organelles from electron micrographs (in Japanese). Mem. Fac. Integrated Arts Sci. Hiroshima Univ. Sci. Rep. 18, 35–42.Google Scholar
Watson, G. S. (1971) Estimating functionals of particle size distributions. Biometrika 58, 483–490.Google Scholar
Weibel, E. R. (1979) Stereological Methods. Academic Press, New York.Google Scholar
Wicksell, S. D. (1925) The corpuscle problem. Biometrika 17, 84–99.Google Scholar