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Wicksell's Problem in Local Stereology

Part of: Integral equations, integral transforms Inverse problems Nonparametric inference Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

Ó. Thórisdóttir  and
M. Kiderlen
Ó. Thórisdóttir*
Affiliation:
Aarhus University
M. Kiderlen*
Affiliation:
Aarhus University
*
∗ Postal address: Department of Mathematical Sciences, Aarhus University, Ny Munkegade 118, 8000 Aarhus, Denmark.
∗ Postal address: Department of Mathematical Sciences, Aarhus University, Ny Munkegade 118, 8000 Aarhus, Denmark.
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Abstract

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Wicksell's classical corpuscle problem deals with the retrieval of the size distribution of spherical particles from planar sections. We discuss the problem in a local stereology framework. Each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters are discussed. We also discuss the unfolding of the arising integral equations, uniqueness issues, and the domain of attraction relations. We illustrate the approach by providing reconstructions from simulated data using numerical unfolding algorithms.

Keywords

Wicksell's corpuscle problemlocal stereologyinverse problemnumerical unfoldingstereology of extremes

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 45Q05: Inverse problems
Secondary: 65R30: Improperly posed problems 62G05: Estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 925 - 944
Copyright
© Applied Probability Trust 

References

Anderssen, R. S. and De Hoog, F. R. (1990). Abel integral equations. In Numerical Solution of Integral Equations (Math. Concepts Meth. Sci. Eng. 42), ed. Golberg, M. A., Plenum, New York, pp. 373–410.Google Scholar
Anderssen, R. S. and Jakeman, A. J. (1975). Abel type integral equations in stereology. II. Computational methods of solution and the random spheres approximation. J. Microscopy 105, 135–153.Google Scholar
Anderssen, R. S. and Jakeman, A. J. (1975). Product integration for functionals of particle size distributions. Utilitas Math. 8, 111–126.Google Scholar
Baddeley, A. and Jensen, E. B. V. (2005). Stereology for Statisticians (Monogr. Statist. Appl. Prob. 103), Chapman & Hall, Boca Raton, FL.Google Scholar
Beneš, V., Bodlák, K. and Hlubinka, D. (2003). Stereology of extremes; bivariate models and computation. Methodology Comput. Appl. Prob. 5, 289–308.Google Scholar
Blödner, R., Mühlig, P. and Nagel, W. (1984). The comparison by simulation of solutions of Wicksell's corpuscle problem. J. Microscopy 135, 61–74.Google Scholar
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
Drees, H. and Reiss, R.-D. (1992). Tail behavior in Wicksell's corpuscle problem. In Probability Theory and Applications, eds Galambos, J. and Kátai, J., Kluwer, Dordrecht, pp. 205–220.CrossRefGoogle Scholar
Gorenflo, R. and Vessella, S. (1991). Abel Integral Equations. Analysis and Applications. Springer, Heidelberg.Google Scholar
Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Vol. 32. Mathematisch Centrum, Amsterdam.Google Scholar
Hlubinka, D. (2003). Stereology of extremes; shape factor of spheroids. Extremes 6, 5–24.CrossRefGoogle Scholar
Hlubinka, D. (2003). Stereology of extremes; size of spheroids. Math. Bohem. 128, 419–438.Google Scholar
Hlubinka, D. (2006). Extremes of spheroid shape factor based on two dimensional profiles. Kybernetika 42, 77–94.Google Scholar
Jensen, E. B. (1984). A design-based proof of Wicksell's integral equation. J. Microscopy 136, 345–348.CrossRefGoogle Scholar
Jensen, E. B. V. (1991). Recent developments in the stereological analysis of particles. Ann. Inst. Statist. Math. 43, 455–468.Google Scholar
Jensen, E. B. V. (1998). Local Stereology. (Adv. Ser. Statist. Sci. Appl. Prob. 5). World Scientific, Singapore.Google Scholar
Lukacs, E. (1960). Characteristic Functions. Griffin, London.Google Scholar
Malina, R. M., Bouchard, C. and Bar-Or, O. (2004). Growth, Maturation and Physical Activity. Human Kinetics, Champaign, IL.Google Scholar
Mecke, J. and Stoyan, D. (1980). Stereological problems for spherical particles. Math. Nachr. 96, 311–317.Google Scholar
Müller, C. (1966). Spherical Harmonics. Springer, Berlin.Google Scholar
Murakami, Y. and Beretta, S. (1999). Small defects and inhomogeneities in fatigue strength: experiments, models and statistical implications. Extremes 2, 123–147.Google Scholar
Ohser, J. and Mücklich, F. (2000). Statistical Analysis of Microstructures in Materials Science. John Wiley, New York.Google Scholar
Pawlas, Z. (2012). Local stereology of extremes. Image Anal. Stereology 31, 99–108.Google Scholar
Saltykov, S. A. (1974). Stereometrische Metallographie. VEB, Leipzig.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, New York.Google Scholar
Takahashi, R. (1987). Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution. Statist. Prob. Lett. 5, 197–200.Google Scholar
Takahashi, R. and Sibuya, M. (1996). The maximum size of the planar sections of random spheres and its application to metallurgy. Ann. Inst. Statist. Math. 48, 127–144.Google Scholar
Takahashi, R. and Sibuya, M. (1998). Prediction of the maximum size in Wicksell's corpuscle problem. Ann. Inst. Statist. Math. 50, 361–377.Google Scholar
Takahashi, R. and Sibuya, M. (2001). Prediction of the maximum size in Wicksell's corpuscle problem. II. Ann. Inst. Statist. Math. 53, 647–660.Google Scholar
Takahashi, R. and Sibuya, M. (2002). Metal fatigue, Wicksell transform and extreme values. Appl. Stoch. Models Business Industry 18, 301–312.Google Scholar
Taylor, C. C. (1983). A new method for unfolding sphere size distributions. J. Microscopy 132, 57–66.Google Scholar
Thórisdóttir, Ó. and Kiderlen, M. (2012). Wicksells problem in local stereology (extended version). Preprint, CSGB, Aarhus University. Available at http://csgb.dk/publications/csgbrr/2012/.Google Scholar
Weissman, I. (1978). Estimation of parameters and larger quantiles based on the k largest observations. J. Amer. Statist. Assoc. 73, 812–815.Google Scholar
Wicksell, S. D. (1925). The corpuscle problem. A mathematical study of a biometric problem. Biometrika 17, 84–99.Google Scholar