Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-06T04:08:25.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Counting figures in planar random configurations

Published online by Cambridge University Press:  14 July 2016

A. M. Kellerer
A. M. Kellerer*
Affiliation:
University of Würzburg
*
∗Postal address: Versbacher Str. 5, D-8700 Würzburg, Federal Republic of Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler–Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations.

Keywords

GEOMETRIC PROBABILITYRANDOM CLUMPINGPOISSON PROCESSIMAGE ANALYSISPARTICLE COUNTING
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 68 - 81
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

Research supported by EURATOM Contract B10–286–81 D and by GSI Contract 96731.

References

[1] Armitage, P. (1949) An overlap problem arising in particle counting. Biometrika 36, 257–266.CrossRefGoogle ScholarPubMed
[2] Davy, P. J. (1976) Projected thick sections through multi-dimensional particle aggregates. J. Appl. Prob. 13, 714–722.CrossRefGoogle Scholar
[3] Davy, P. J. (1978) Stereology — A Statistical Viewpoint. , Australian National University.CrossRefGoogle Scholar
[4] Kellerer, A. M. (1983) On the number of clumps resulting from the overlap of randomly placed figures in a plane. J. Appl. Prob. 20, 126–135.CrossRefGoogle Scholar
[5] Kellerer, H. G. (1984) Minkowski functions of Poisson processes. Z. Wahrscheinlichkeitsth. 67, 63–84.CrossRefGoogle Scholar
[6] Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
[7] Lindgren, B. W. (1976) Statistical Theory, 3rd edn. The Macmillan Company, New York.Google Scholar
[8] Mack, C. (1954) The expected number of clumps when convex laminae are placed at random and with random orientation on a plane area. Proc. Camb. Phil. Soc. 50, 581–585.CrossRefGoogle Scholar
[9] Mack, C. (1956) On clumps formed when convex laminae or bodies are placed at random in two or three dimensions. Proc. Camb. Phil. Soc. 52, 246–256.CrossRefGoogle Scholar
[10] Matheron, G. (1967) Eléments pour une théorie des milieux poreux. Masson, Paris.Google Scholar
[11] Matheron, G. (1974) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
[12] Morton, R. R. A. (1966) The expected number and angle of intersections between random curves in a plane. J. Appl. Prob. 3, 559–562.CrossRefGoogle Scholar
[13] Roach, S. A. (1968) The Theory of Random Clumping. Methuen, London.Google Scholar
[14] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, London.Google Scholar
[15] Weil, W. and Wieacker, J. A. (1984) Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.CrossRefGoogle Scholar