  Higher-rank Minkowski valuations are efficient means for describing the
geometry and connectivity of spatial patterns. We show how to extend the
framework of the scalar Minkowski valuations to vector- and tensor-valued
measures. The properties of these extensions are described in detail. We show
the versatility of these measures by using simple toy models as well as real
data. Our applications cover the morphology of galaxy clusters, the structure
of spiral galaxies, and the geometry of molecules. Furthermore, we consider a
physical ansatz closely related to higher-rank Minkowski valuations, the
Rosenfeld functional known from density functional theory.
