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Wicksell's Problem in Local Stereology

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

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 

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