2D Functionals

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

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant. The vectors are closely related to the centers of mass in either solid or hollow bodies. Accordingly, the second-rank tensors correspond the tensors of inertia, or they can be interpreted as the moment tensors of the distribution of the normals on the boundary.

Minkowski Functionals

W_0 = \quad \int\limits_K \mathrm d A \propto V_2
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

\kappa =  curvature


Cartesian representation (Minkowski Tensors)

Using the position vector \textbf r and the normal vector \textbf n on the boundary, the Minkowski Vectors can be defined in the Cartesian representation.

The second-rank Minkowski tensors are defined using the symmetric tensor product (\textbf a\otimes \textbf a)_{ij} = a_i a_j.

Minkowski Vectors

W_0^{1,0} = \quad \int\limits_K \textbf r \, \mathrm d A\propto \Phi_2^{1,0}
W_1^{1,0} = \frac 12 \int\limits_{\partial K} \textbf r \, \mathrm d l\propto \Phi_1^{1,0}
W_2^{1,0} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf r \, \mathrm d l\propto \Phi_0^{1,0}

Minkowski Tensors

W_0^{2,0}  = \quad \int\limits_K \textbf r \otimes \textbf r \, \mathrm d A\propto \Phi_2^{2,0}
W_1^{2,0} = \frac 12 \int\limits_{\partial K} \textbf r \otimes \textbf r \, \mathrm d l\propto \Phi_1^{2,0}
W_2^{2,0} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf r \otimes \textbf r \, \mathrm d l\propto \Phi_0^{2,0}
W_1^{0,2} = \frac 12 \int\limits_{\partial K} \textbf n \otimes \textbf n \, \mathrm d l\propto \Phi_1^{0,2}
W_2^{0,2} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf n \otimes \textbf n \, \mathrm d l\propto \Phi_0^{0,2}


Irreducible representation (Circular Minkowski Tensors)


under construction

Consider a polygon with edges L:
\vec L_i = L_i \cdot \vec n_i

Density of normals:
\rho(\varphi) = \sum L_i \delta(\phi-\phi_i)

Fourier analysis:
\tilde\rho(l) = \int \textnormal d \varphi \, \textnormal{exp}(i l \varphi)  \, \rho(\varphi) = \sum L_i \textnormal{exp}(i l \varphi_i)

\tilde\rho(0) = L = W_1 (the circumference of the polygon)
\tilde\rho(1) = 0  (for a closed polygon)

The shape indexes q_l are defined as
q_l = \frac{|\tilde\rho(l)|}{\tilde\rho(0)} .

q_0 is the circumference of the object.
q_1 = 1 for closed polygons.
q_2 is the polar component of the normal density, it is related to \beta_1^{0,2} - 1
q_3 can detect anisotropy in a three-fold symmetric system
q_4 is the quadrupole component (suitable for detecting rectangles)
q_6 : …

Morphometric distance

A morphometric distance of a polygon K to a reference structure R can be quantified by considering the pseudo distance function

d(K) = \sqrt{\sum\limits_l [q_l(K) - q_l(R)]^2} .