Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T23:18:55.833Z Has data issue: false hasContentIssue false
  • English
  • Français

The Estimation of mean shape and mean particle number in overlapping particle systems in the plane

Part of: Stochastic processes General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Wolfgang Weil
Wolfgang Weil*
Affiliation:
Universität Karlsruhe
*
* Postal address: Mathematisches Institut II, Universität Karlsruhe, 76128 Karlsruhe, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

A stationary (but not necessarily isotropic) Boolean model Y in the plane is considered as a model for overlapping particle systems. The primary grain (i.e. the typical particle) is assumed to be simply connected, but no convexity assumptions are made. A new method is presented to estimate the intensity y of the underlying Poisson process (i.e. the mean number of particles per unit area) from measurements on the union set Y. The method is based mainly on the concept of convexification of a non-convex set, it also produces an unbiased estimator for a (suitably defined) mean body of Y, which in turn makes it possible to estimate the mean grain of the particle process.

Keywords

BOOLEAN MODELANISOTROPYINTENSITYMEAN GRAINCONVEXIFICATION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 102 - 119
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

Artstein, Z. and Vitale, R. A. (1975) A strong law of large numbers for random compact sets. Ann. Prob. 5, 879–882.Google Scholar
Aumann, R. J. (1965) Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12.CrossRefGoogle Scholar
Ayala, G., Ferrandiz, J. and Montes, F. (1989a) Two methods of estimation in Boolean models. Acta Stereologica 8, 629–634.Google Scholar
Ayala, G., Ferrandiz, J. and Montes, F. (1989b) On parametric estimation in Boolean models. Rass. Met. Statist. Appl. 8, 1–17.Google Scholar
Betke, U. and Weil, W. (1991) Isoperimetric inequalities for the mixed area of plane convex sets. Arch. Math. 57, 501–507.CrossRefGoogle Scholar
Bonnesen, T. and Fenchel, W. (1934) Theorie der konvexen Körper. Springer-Verlag, Berlin.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
Davy, P. (1976) Projected thick sections through multidimensional particle aggregates. J. Appl. Prob. 13, 714–722. Correction: J. Appl. Prob. 15 (1978), 456.CrossRefGoogle Scholar
Davy, P. (1978) Stereology—a statistical viewpoint. Thesis, Australian National University, Canberra.CrossRefGoogle Scholar
Giné, E., Hahn, M. C. and Zinn, J. (1983) Limit theorems for random sets: an application of probability in Banach space results. In Lecture Notes in Mathematics 990, pp. 112–135. Springer-Verlag, New York.Google Scholar
Goodey, P. and Weil, W. (1992) The determination of convex bodies from the mean of random sections. Math. Proc. Camb. Phil. Soc. 112, 419–430.Google Scholar
Groemer, H. (1978) On the extension of additive functionals on classes of convex sets. Pacific J. Math. 75, 397–410.Google Scholar
Karr, A. F. (1986) Point Processes and their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kellerer, A. M. (1983) On the number of clumps resulting from the overlap of randomly placed figures in the plane. J. Appl. Prob. 20, 126–135.CrossRefGoogle Scholar
Kellerer, A. M. (1985) Counting figures in planar random configurations. J. Appl. Prob. 22, 68–81.Google Scholar
Kellerer, H. G. (1984) Minkowski functionals of Poisson processes. Z. Wahrscheinlichkeitsth. 67, 63–84.CrossRefGoogle Scholar
Leichtweiss, K. (1980) Konvexe Mengen. Springer-Verlag, Berlin.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Mcmullen, P. and Schneider, R. (1983) Valuations on convex bodies. In Convexity and Its Applications , ed. Gruber, P. and Wills, J. M., pp. 170–247. Birkhäuser, Basel.CrossRefGoogle Scholar
Miles, R. E. (1976) Estimating aggregate and overall characteristics from thick sections by transmission microscopy. J. Microsc. 107, 227–233.CrossRefGoogle Scholar
Molchanov, I. S. (1994) Set-valued estimators for mean bodies related to Boolean models, (submitted).Google Scholar
Molchanov, I. S. (1995) Statistics of the Boolean model: from the estimation of means to the estimation of distributions. Adv. Appl. Prob. 27, 63–86.CrossRefGoogle Scholar
Molchanov, I. S. and Stoyan, D. (1994a) Directional analysis of fibre processes related to Boolean models. Metrika 41, 183–199.CrossRefGoogle Scholar
Molchanov, I. S. and Stoyan, D. (1994b) Asymptotic properties of estimators for parameters of the Boolean models. Adv. Appl. Prob. 26, 301–323.Google Scholar
Rataj, J. (1994) Estimation of oriented direction distribution of a planar body, (submitted).Google Scholar
Rataj, J. and Saxl, I. (1989) Analysis of planar anisotropy by means of the Steiner compact. J. Appl. Prob. 26, 490–502.Google Scholar
Schmitt, M. (1991) Estimation of the density in a stationary Boolean model. J. Appl. Prob. 28, 702–708.CrossRefGoogle Scholar
Schneider, R. (1993) Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (1992) Integralgeometrie. Teubner, Stuttgart.Google Scholar
Schröder, M. (1992) Schätzer für Boolesche Modelle im ℝ2 und ℝ3 . Diplomarbeit, Universität Karlsruhe.Google Scholar
Serra, J. (1982) Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Akademie-Verlag, Berlin.Google Scholar
Vitale, R. A. (1988) An alternate formulation of the mean value for random geometric figures. J. Microsc. 151, 197–204.Google Scholar
Weil, W. (1984) Densities of quermassintegrals for stationary random sets. In Stochastic Geometry, Geometric Statistics, Stereology , ed, Ambartzumian, R. V. and Weil, W., pp. 233–247. Teubner, Leipzig.Google Scholar
Weil, W. (1988) Expectation formulas and isoperimetric properties for non-isotropic Boolean models. J. Microsc. 151, 235–245.CrossRefGoogle Scholar
Weil, W. (1990a) Iterations of translative integral formulae and nonisotropic Poisson processes of particles. Math. Z. 205, 531–549.CrossRefGoogle Scholar
Weil, W. (1990b) Lectures on translative integral geometry and stochastic geometry of anisotropic random geometric structures. Atti del Primo Convegno Italiano di Geometria Integrale. Rend. Sem. Mat. Messina, Ser. II, 13, 79–97.Google Scholar
Weil, W. (1993) The determination of shape and mean shape from sections and projections. Acta Stereologica 12, 73–84.Google Scholar
Weil, W. (1994a) Support functions on the convex ring in the plane and densities of random sets and point processes. Suppl. Rend. Circ. Mat. Palermo (II) 35, 323–344.Google Scholar
Weil, W. (1994b) On the mean shape of particle processes. (in preparation).Google Scholar
Weil, W. and Wieacker, J. A. (1984) Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.CrossRefGoogle Scholar
Zähle, M. (1986) Curvature measures and random sets, II. Prob. Theory Rel. Fields 71, 37–58.Google Scholar