Introduction into Integral Geometry and Stereology

Abstract

This chapter is a self-contained introduction into integral geometry and its applications in stereology. The most important integral geometric tools for stereological applications are kinematic formulae 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-formulae for stationary and isotropic Boolean models with convex particles are derived. In design-based stereology, Crofton’s formula leads to unbiased estimators 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.