Anisotropy analysis

The Minkowski tensors contain information about both the preferred direction and the amplitude of the anisotropy. The latter can be conveniently extracted by scalar anisotropy indices.

Here, we present some common indices in both 2D and 3D.

Irreducible Minkowski indices qs
In 2D, q_s are the normalized Fourier coefficients of the density of the edge normals; for details see Circular Minkowski Tensors.

The shape indices q_s are defined as  q_s := \frac{|\psi_s|}{\psi_0}

All anisotropy indices q_s for s>0 vanish only for the circle (left). If there is only a single anisotropy index q_s non-zero, the choice of this rank s defines a convex shapes with an s-fold rotational symmetry.

q_0 = 1
q_1 = 0 for closed polygons.
q_2 is the quadrupole component of the normal density
q_3 can detect anisotropy in a three-fold symmetric system (e.g. suitable for detecting equilateral triangles)
q_4 can detect anisotropy in a four-fold symmetric system (e.g. suitable for detecting rectangles)

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_s(K) - q_s(R)]^2}

Ratio of Eigenvalues β
Ratio of Eigenvalues \beta_\nu^{r,s}

\displaystyle \beta_\nu^{r,s} := \frac{|\mu_{\mathrm{min}}|}{|\mu_{\mathrm{max}}|} \in [0,1]

\beta_\nu^{r,s} equal to 0 indicates a „flat“ body K.

\beta_\nu^{r,s} equal to 1 indicates an „isotropic“ body K, in the sense that it has a statistically identical mass distribution in any set of three orthogonal directions; this includes the sphere, but also regular polyhedra and the FCC, BCC and HCP Voronoi cells.

Isoperimetric ratio Q
The isoperimetric ratio Q is defined as the quotient of the area and the circumference of an object. It is normalized so that Q = 1 for a circle.

In 2D: Q = 4 \pi \frac{W_0}{(2\,W_1)^2}

In 3D: Q = 36 \pi \frac{W_0^2}{(3\,W_1)^3}