Minkowski Scalars

For sufficiently smooth bodies K, the Minkowski Scalars can be intuitively defined via (weighted) integrals over the volume or boundary of the body K.

2D

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant (for compact bodies it is given by the number of components minus the number of holes).

Area
W_0 = \quad \int\limits_K \mathrm d A \propto V_2
Perimeter
W_1 = \frac 12 \int\limits_{\partial K} \mathrm d l \propto V_1
Euler characteristic
W_2 = \frac 12 \int\limits_{\partial K} \kappa\, \mathrm d l \propto V_0

with
\kappa = curvature

3D

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant (for compact bodies it is given by the number of components minus the number of rings plus the number of cavities).

Volume
W_0 = \quad \int\limits_K \mathrm d V \propto V_3
Surface area
W_1 = \frac 13 \int\limits_{\partial K} \mathrm d \mathcal O \propto V_2
Integrated mean curvature
(Mean width for convex bodies)
W_2 = \frac 13 \int\limits_{\partial K} H \mathrm d \mathcal O \propto V_1
Euler Characteristic
W_3 = \frac 13 \int\limits_{\partial K} G \mathrm d \mathcal O\propto V_0

with
H= \frac 12 (\kappa_1 + \kappa_2)
G= \kappa_1 \kappa_2
\kappa_i = principal curvature