4/1/2024 0 Comments Nearest neighbor stereology![]() ![]() ![]() ![]() In design-based stereology, Crofton’s formula leads to unbiased estimators of intrinsic volumes from isotropic uniform random flats. Also, Miles-formulae for stationary and isotropic Boolean models with convex particles are derived. 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. The proofs are exclusively based on invariance arguments and an axiomatic description 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. Therefore, Crofton’s formula and the principal kinematic formula for polyconvex sets are stated and shown using Hadwiger’s characterization of the intrinsic volumes. The most important integral geometric tools for stereological applications are kinematic formulae and results of Blaschke–Petkantschin type. This chapter is a self-contained introduction into integral geometry and its applications in stereology. ![]()
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