Mathematical morphology for shape description

Abstract
We first examine the measurements one can perform on the space of compact and convex sets. A famous theorem, due to Hadwiger (1957), shows that any measurement with nice properties, namely additivity, is a linear combination of Minkowski functionals. Then, some useful formulae, linking measurements in different dimensions of space are derived.

In the second step, we make use of the morphological operations transforming sets into sets. These sets are then measured using the previous measurements. The most famous attempt yields the concept of granulomerties and their extensions.

In the last part, we examine a way to build morphological random sets which are compatible with morphological operators like erosions or openings and apply it to the most famous example in morphology, the Boolean model, describing objects located at random.