Minkowski Scalars

Minkowski scalars, also known as intrinsic volumes, are robust and easy to apply shape measures. For sufficiently smooth bodies K, the Minkowski Scalars can be intuitively defined via (weighted) integrals over the volume or boundary of the body K. There are three independent Minkowski scalars in 2D, and four independent in 3D.

Hadwiger’s theorem states that any additive, continuous, and motion-invariant functional F of a convex body K can be expressed as a linear combination of the d+1 Minkowski functionals W_i of the body K:

(1)   \begin{equation*} F(K) = \sum\limits_{i=0}^{d} \alpha_i W_i. \end{equation*}

In this sense, mathematical theory imposes that scalar Minkowski functionals are the relevant shape indices w.r.t. to physical properties represented by such additive functionals.

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
Perimeter
W_1 = \frac 12 \int\limits_{\partial K} \mathrm d l
Euler characteristic
W_2 = \frac 12 \int\limits_{\partial K} \kappa\, \mathrm d l

with
\kappa = curvature

3D

UNDER CONSTRUCTION

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
Surface area
W_1 = \frac 13 \int\limits_{\partial K} \mathrm d \mathcal O
Integrated mean curvature
(Mean width for convex bodies)
W_2 = \frac 13 \int\limits_{\partial K} H \mathrm d \mathcal O
Euler Characteristic
W_3 = \frac 13 \int\limits_{\partial K} G \mathrm d \mathcal O

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