Background

Here, we present more details on the rich theoretical background of Minkowski scalars and tensors in integral geometry.

First, we discuss some basic definitions and results about the so-called tensor valuations from integral geometry.

Then we provide the Cartesian Minkowski tensors (CMTs) in 2D and in 3D both for the translational invariant and translation covariant tensors.
Intuitively speaking these tensors measure the anisotropy with respect to the surface or volume distribution, respectively.

[Under construction: explicit formulas for the conversion of the Cartesian to the irreducible representation, that is, from the CMTs to the IMTs, and vice versa]

Finally, we compare various alternative scalar indices that quantify anisotropy with respect to different geometrical properties.