Cartesian Minkowski Tensors

The Minkowski tensors can be intuitively defined via weighted volume or surface integrals in the Cartesian representation. In praticular, this definition is an intuitive generalization of the Minkowski scalars. For example, the perimeter can be generalized to the moment tensor of the orientation of the interface (surface area measure).

2D T-Inv Tensors

In two dimensions, the second-rank translation invariant (T-Inv) tensors are defined in the Cartesian representation as follows using the normal vector \textbf n on the boundary:

W_1^{0,2} = \frac 12 \int\limits_{\partial K} \textbf n \otimes \textbf n \, \mathrm d l
W_2^{0,2} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf n \otimes \textbf n \, \mathrm d l

with \kappa being the curvature and using the symmetric tensor product (\textbf n\otimes \textbf n)_{ij} = n_i n_j, which can be represented by a matrix
with the entry n_in_j in row i and column j.

Higher rank translation invariant (T-Inv) tensors

W_1^{0,2} = \frac 12 \int\limits_{\partial K} \textbf n^s \, \mathrm d l
W_2^{0,2} = \frac 12 \int\limits_{\partial K} \kappa \, \textbf n^s \, \mathrm d l

where \textbf{n}^s=\bigotimes_{i=1}^s\textbf{n} is the tensor product.

Relation to IMTs
These Cartesian Minkowski Tensors (CMT) and the Irreducible Minkowski Tensors (IMT) are, of course, only different representations of the same Minkowski tensors and can therefore be related to each other.

W_1^{0,s}(K) =\frac 12 \int\limits_{0}^{2\pi}\mathbf{u}^s(\varphi) \rho_K(\varphi)\, \text{d}\varphi =\frac 12 \sum\limits_{k=-\infty}^{\infty} \psi_k \int\limits_{0}^{2\pi}\mathbf{u}^s(\varphi)\frac{\exp(ik\varphi)}{2\pi} \, \text{d}\varphi,

where \mathbf{u}(\varphi):=\left( \begin{array}{c} \cos\varphi\\ \sin\varphi \end{array} \right). Because all summands with |k|>s vanish due to the vanishing the Fourier components of \mathbf{u}^s, we obtain

W_1^{0,s}(K) =\frac 12 \sum\limits_{k=-s}^{s} \psi_k \int\limits_{0}^{2\pi}\mathbf{u}^s(\varphi)\frac{\exp(ik\varphi)}{2\pi}\, \text{d}\varphi.

Note that the Fourier coefficients of \mathbf{u}^s are independent of the body K. For s=2, they are given by

\int\limits_{0}^{2\pi}\mathbf{u}^2(\varphi)\frac{\exp(ik\varphi)}{2\pi}\,\text{d}\varphi= \begin{cases} \frac{1}{2}\cdot \mathbf{1} & \text{if } k=0\;,\\ \frac{1}{4} \cdot \left({\boldsymbol\sigma_3} + \mathrm i {\boldsymbol\sigma_1} \right) & \text{if } k=2\;,\\ \frac{1}{4} \cdot \left({\boldsymbol\sigma_3} - \mathrm i {\boldsymbol\sigma_1} \right) & \text{if } k=-2\;,\\ \mathbf{0} & \text{else}\;, \end{cases}

where \mathbf{1} is the unit tensor, \mathbf{0} is the zero tensor, and the {\boldsymbol\sigma_j} are Pauli matrices with

{\boldsymbol\sigma_3}=\left( \begin{array}{c c} 1 & 0\\ 0 & -1 \end{array} \right) and {\boldsymbol\sigma_1}=\left( \begin{array}{c c} 0 & 1\\ 1 & 0 \end{array} \right).

Thus, the second-rank CMT W_1^{0,2} can be expressed by the IMTs \psi_0, \psi_1, and \psi_2:

W_1^{0,2} = \tfrac 14\, \psi_0\, \mathbf{1} + \tfrac 14\, \mathrm{Re}\{\psi_2\}\, {\boldsymbol\sigma_3} - \tfrac 14\, \mathrm{Im}\{\psi_2\} \,{\boldsymbol\sigma_1}

and vice versa:
\psi_0 = 2(W_1^{0,2})_{xx}+2(W_1^{0,2})_{yy} = 2 W_1
\mathrm{Re}\{\psi_2\} = 2(W_1^{0,2})_{xx}-2(W_1^{0,2})_{yy}
\mathrm{Im}\{\psi_2\} = -4(W_1^{0,2})_{xy}.

Thus the Minkowski structure metric q_2 can be expressed by the eigenvalues \delta_1 and \delta_2 of the Minkowski tensor W_1^{0,2}:
q_2=\frac{|\delta_1 - \delta_2|}{W_1}=\frac{|\delta_1 - \delta_2|}{\delta_1 + \delta_2}.

Similarly the fourth rank index q_4 can be expressed by the components of the Minkowski tensor W_1^{0,4}, where for convenience the coordinate axes are assumed to coincide with the eigenvectors of W_1^{0,2}:

2D T-Cov Tensors

While the IMT offer an intuitive quantification of the degree of anisotropy for arbitrary ranks of the tensors, the Cartesian representation offers an intuitive definition of not only of the transliation-invariant (T-Inv) Minkowski tensors but also of the translation-covariant (T-Cov) tensors. The latter are required for a complete additive characterization (according to Alesker’s theorem).

Using the position vector \textbf r the Minkowski Vectors are defined as:

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

and the second-rank translation covariant (T-Cov) Minkowski tensors as:

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

They thus generalize the area or perimeter to the second moment of the mass distribution of a solid or hollow body.
They thus contain the information of the tensor of inertia of such massive bodies.

The T-covariant tensors for short, depend on the chosen origin and transform as follows when the object K is being translated by a vector \textbf t:
W_{\nu}^{r,s}(K+\textbf t)=\sum_{k=0}^{r}{r\choose k}\textbf{t}^k\otimes W_{\nu}^{r-k,s}(K).

3D T-Inv and T-Cov Tensors

Second-rank T-Inv Minkowski Tensors

W_1^{0,2} = \frac 13 \int\limits_{\partial K} \textbf n \otimes \textbf n \, \mathrm d \mathcal O
W_2^{0,2} = \frac 13 \int\limits_{\partial K} H \, \textbf n \otimes \textbf n \, \mathrm d \mathcal O

Minkowski Vectors

W_0^{1,0} = \quad \int\limits_K \textbf r \, \mathrm d V
W_1^{1,0} = \frac 13 \int\limits_{\partial K} \textbf r \, \mathrm d \mathcal O
W_2^{1,0} = \frac 13 \int\limits_{\partial K} H \, \textbf r \, \mathrm d \mathcal O
W_3^{1,0} = \frac 13 \int\limits_{\partial K} G \, \textbf r \, \mathrm d \mathcal O

Second-rank T-Cov Minkowski Tensors

W_0^{2,0} = \quad \int\limits_K \textbf r \otimes \textbf r \, \mathrm d V
W_1^{2,0} = \frac 13 \int\limits_{\partial K} \textbf r \otimes \textbf r \, \mathrm d \mathcal O
W_2^{2,0} = \frac 13 \int\limits_{\partial K} H \, \textbf r \otimes \textbf r \, \mathrm d \mathcal O
W_3^{2,0} = \frac 13 \int\limits_{\partial K} G \, \textbf r \otimes \textbf r \, \mathrm d \mathcal O

dD Tensors

Similar integrals can be defined for smooth bodies in arbitrary dimensions using the elementary symmetric function of the principal curvatures.