Sandbjerg Estate, Sønderborg, Denmark

The modern theory of random sets is strongly based on results in geometric measure theory and has important applications in digital stereology. The aim of the winter school is to give an overview of this area that ranges from classical generalizations of differential geometry over stochastic geometry to recent applications in the analysis of digital images.

The winter school is addressed to PhDs, PostDocs and other researchers in mathematics and statistics who want to get introduced in the field. Scientists from the natural sciences with an interest in mathematics are also welcome.

The theory of random sets describes and examines random geometric structures which may appear in microscopical images of materials and biological tissues and macroscopically e.g. in soil and geological sediments. In the course, the notion of a random closed set will be made precise, statistical models like the fundamental Boolean model will be introduced, and the estimation of parameters and geometric characteristics will be discussed. For a deeper understanding of the latter, it is necessary to specify regularity of the underlying structures and to study the geometric properties by measure theoretical means. Therefore, tools from geometric measure theory will be presented, as well.

In applications, the geometric structures are often not observable directly, but samples (e.g. planar sections) are available, that only give partial information on the structure. Stereology is the discipline that derives estimators of geometric functions, like volume or surface area, from this partial information. The increasing use of digital devices lead to an exceptional recent interest to derive also estimation procedures based on digital (0-1) images of those structures. The course will give an overview as well as recent results in this field.

The invited lecturers are leading researchers in the field: Jan Rataj is a specialist in geometric measure theory and has recently (together with V. Beneš) published a state-of-the-art book on stochastic geometry. Ilya Molchanov, an acknowledged expert in the field, has written several books on random sets, and Markus Kiderlen has very recently contributed decisively to the emerging discipline of digital stereology.