Minkowski Scalars – morphometry.org

Minkowski scalars are robust and intuitive shape measures. For sufficiently smooth bodies $K$ , the Minkowski Scalars are just (weighted) integrals over the volume or boundary of the body $K$ . There are three linearly independent Minkowski scalars in two dimensions, and four in three dimensions. Minkowski scalars are also known as intrinsic volumes.

The Minkowski scalars can be interpreted as area, perimeter, and the Euler characteristic. The latter is a topological measure; for compact bodies it is given by the number of components minus the number of cavities. All three can be written as integrals, weighted (blue color) by the area, the contour length, and the local curvature of the contour:

Body $K$

Area $A(K) = \quad \int\limits_K \mathrm d A$

Perimeter $P(K) = \int\limits_{\partial K} \mathrm d r$

Euler characteristic $\chi(K) = \frac 1{2\pi} \int\limits_{\partial K} \kappa(\vec r)\, \mathrm d r$

where $\kappa(\vec r)$ is the curvature of the contour at point $\vec r$ . The Euler characteristic is given by the number of components minus the number of cavities:

Hadwiger’s characterization theorem

Hadwiger’s characterization theorem [bibcite key=hadwiger1957vorlesungen] 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) $\begin{equation*} F(K) = \sum\limits_{i=0}^{d} \alpha_i W_i(K). \end{equation*}$

Here, $\alpha_i$ are constants independent of $K$ . The shape information about $K$ is completely contained in the $W_i(K)$ , which are $W_0 = A$ , $W_1 = P/2$ , and $W_2 = \pi\chi/2$ . In this sense, the Minkowski scalars are the relevant shape indices for physical functionals with the additivity property.