Symmetries of the EIT Problem

The EIT Forward Map commutes with several transformations. This is useful both for analysis and for designing neural networks that respect the physics (see Invariant and Equivariant Functions).

Rigid motions (Euclidean group $E(n)$). Let $T(x)=Qx+b$ with $Q\in O(n)$. If $u$ solves $\nabla \cdot (\gamma \nabla u)=0$ in $\Omega$, then $u\circ T^{-1}$ solves the equation with conductivity $\gamma \circ T^{-1}$ in $T(\Omega )$. The normal derivative is preserved because $T$ is an isometry, so

$${\Lambda }_{\gamma \circ T^{-1}}(f\circ T^{-1})=\left({\Lambda }_{\gamma }f\right)\circ T^{-1}.$$

Rotating or reflecting the body and the boundary data together rotates or reflects the measured currents. The forward map is equivariant, not invariant: an interior feature is reconstructed the same way wherever it sits, provided the whole setup (domain and electrodes) is transformed with it. On a square domain the relevant finite subgroup is the Dihedral Group D4.

Scaling. For $x\mapsto cx$ ($c>0$), gradients scale by $1/c$, so the DtN map of the dilated problem is $\frac{1}{c}$ times the transported one. Multiplying the conductivity by a constant gives ${\Lambda }_{c\gamma }=c{\Lambda }_{\gamma }$.

Conformal maps (2D). For a conformal map $\Phi$, $D\Phi D{\Phi }^{\top }=|\det D\Phi |I$. The push-forward of an isotropic conductivity is therefore again isotropic, ${\Phi }_{\ast }\gamma =\gamma \circ {\Phi }^{-1}$ (see Anisotropic Conductivities). The 2D conductivity equation is conformally invariant. The boundary operators transform by composition with $\Phi$ and multiplication by the boundary stretching factor $|{\Phi }^{\prime }|$. This much larger (infinite-dimensional) symmetry is special to two dimensions and is used, for example, to map arbitrary simply connected domains to the unit disc.

Mesh permutations. Renumbering the nodes of a finite element mesh (the symmetric group $S_N$) changes nothing physically. Graph neural networks are permutation-equivariant by construction.

Complex EIT. A global phase rotation $e^{i\varphi }$ ($U(1)$) of the complex potentials leaves the admittivity problem unchanged (see Complex Conductivity).

References

1. R. V. Kohn, M. Vogelius (1984). Determining conductivity by boundary measurements. Comm. Pure Appl. Math. 37(3), 289–298. doi:10.1002/cpa.3160370302
2. J. Sylvester (1990). An anisotropic inverse boundary value problem. Comm. Pure Appl. Math. 43(2), 201–232. doi:10.1002/cpa.3160430203
3. M. M. Bronstein, J. Bruna, T. Cohen, P. Veličković (2021). Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges. arXiv:2104.13478