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.

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.