Stereology: A Survey for Geometers

Abstract

The determination of characteristic geometric properties of (usually 3-dimensional) objects by investigations of sections, projections, intersections with test sets, or other transformed images is a problem inherent to most of the experimental sciences. Surprisingly enough, formulas for such purposes have been established by people working in quite different fields of life, materials and earth sciences for over a hundred years until, some twenty years ago, the common theoretical (and in fact mathematical) background was realized. The new discipline dealing with problems of this type was called stereology. It was only then that mathematicians pointed out that most of the stereological results stem from classical formulas in integral geometry and that the validity of the formulas presupposes certain random structures either of the underlying sets or of the images taken of them. The development of stereological models therefore was greatly influenced by the growth of stochastic geometry. On the other hand, parts of stochastic geometry (random sets, point processes of convex bodies) have their roots in stereological questions. It seems, however, that stereology as an important field of applications of (convex) geometry is not as well-known to geometers as it should be. This was the motivation for the following survey which, although it is of an introductory nature, will present some of the recent developments with detailed, yet not exhaustive references. Since there are several geometric models which can serve as a basis of stereology, a lot of cross connections with other mathematical fields, and a variety of stereological problems of especial type, we had to restrict ourselves to a part of stereology which allows a unified treatment within a justifiable extent. The model, we present here, is based on convex geometry (methods from differential geometry and geometric measure theory are mentioned only occasionally) and the integral geometry of convex bodies (and unions thereof) is a crucial part of our considerations. The theory of random sets and point processes is investigated mainly in view of random versions of the kinematic integral formulas. We have, however, tried to be more complete with the references and give short comments to further developments and the corresponding literature at the end of each section.KeywordsPoint ProcessConvex BodyUnbiased EstimatorIntegral GeometryStochastic GeometryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.