Integral geometric formulae are the basis of unbiased estimators in stochastic geometry and stereology , for instance to estimate volume or surface area of an object in n -dimensional space. We review some current developments in integral geometry and illustrate the concepts with an application in biology.

After giving an overview of classical Crofton formulae for intrinsic volumes and certain other valuations on convex bodies, we formulate known and new rotational Crofton formulae for these functionals . From a special case of these results one can obtain an unbiased estimator of surface area in local stereology . The calculation of this estimator is based on a refinement of the classical tangent count method for IUR (isotropic uniform random) planes.

As an illustrating example, we estimate the surface area of the nuclei of giant-cell glioblastoma from microscopy images.

This talk is based on joint work with E.B.V.Jensen, A.H. Rafati and Ó. Thórisdóttir.