Research output: Working paper › Research

- Imf thiele 2010 03
Submitted manuscript, 386 KB, PDF document

- T.N. Thiele Centre
- Department of Mathematical Sciences

This text is the extended version of two talks held at the Summer Academy Stochastic Geometry, Spatial Statistics and Random Fields in the Soellerhaus, Germany, in September 2009. It forms (with slight modifications) a chapter of the Springer lecture notes Lectures on Stochastic Geometry, Spatial Statistics and Random Fields and is a self-containing introduction into integral geometry and its applications in stereology.

The most important integral geometric tools for stereological applications are kinematic formulas and results of Blaschke-Petkantschin type. Therefore, Crofton's formula and the principal kinematic formula for polyconvex sets are stated and shown using Hadwiger's characterization of the intrinsic volumes. Then, the linear Blaschke-Petkantschin formula is proved together with certain variants for flats containing a given direction (vertical flats) or contained in an isotropic subspace. The proofs are exclusively based on invariance arguments and an axiomatic description of the intrinsic volumes.

These tools are then applied in model-based stereology leading to unbiased estimators of specific intrinsic volumes of stationary random sets from observations in a compact window or a lower dimensional flat. Also, Miles-formula for stationary and isotropic Boolean models with convex particles are derived. In design-based stereology, Crofton's formula leads to an unbiased estimator of intrinsic volumes from isotropic uniform random flats. To estimate the Euler characteristic, which cannot be estimated using Crofton's formula, the disector design is presented. Finally we discuss design-unbiased estimation of intrinsic volumes from vertical and from isotropic sections.

The most important integral geometric tools for stereological applications are kinematic formulas and results of Blaschke-Petkantschin type. Therefore, Crofton's formula and the principal kinematic formula for polyconvex sets are stated and shown using Hadwiger's characterization of the intrinsic volumes. Then, the linear Blaschke-Petkantschin formula is proved together with certain variants for flats containing a given direction (vertical flats) or contained in an isotropic subspace. The proofs are exclusively based on invariance arguments and an axiomatic description of the intrinsic volumes.

These tools are then applied in model-based stereology leading to unbiased estimators of specific intrinsic volumes of stationary random sets from observations in a compact window or a lower dimensional flat. Also, Miles-formula for stationary and isotropic Boolean models with convex particles are derived. In design-based stereology, Crofton's formula leads to an unbiased estimator of intrinsic volumes from isotropic uniform random flats. To estimate the Euler characteristic, which cannot be estimated using Crofton's formula, the disector design is presented. Finally we discuss design-unbiased estimation of intrinsic volumes from vertical and from isotropic sections.

Original language | English |
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Place of publication | Århus |

Publisher | Thiele Centre, Institut for Matematiske Fag, Aarhus Universitet |

Number of pages | 25 |

Publication status | Published - 3 Feb 2010 |

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