Here we present the theoretical background of Minkowski scalars and tensors as a versatile, sensitive and robust way to quantify shape.

Minkowski scalars, also known as intrinsic volumes, are robust and easy to apply shape measures. The Minkowski scalars in 2D can be interpreted as area, perimeter and the Euler characteristic of a body. By definition, the Minkowski scalars are rotation-invariant, which makes them not explicitly sensitive to directional and anisotropic features of morphology.

The tensorial generalization of Minkowski functionals are Minkowski tensors. A very easy way to quantify shape are Irreducible Minkowski Tensors (IMTs) which provide anisotropy indices q_s to quantify shape anisotropy of objects as well as the direction of the anisotropy.

The rich background in integral geometry of Minkowski functionals is explained in the background section.

Our software tool morphometer provides an easy way to analyse 2D objects by Minkowski shape descriptors.