Conformal Invariance of the Conductivity Equation

In two dimensions the Conductivity Equation is invariant under conformal maps. Every simply connected domain is the conformal image of the unit disk, so EIT on any such domain can be computed on the disk, with an isotropic conductivity, and mapped back.

Invariance of the Dirichlet energy

Let $\Phi :D\to \Omega$ be conformal (holomorphic with ${\Phi }^{\prime }\ne 0$) and $v=u\circ \Phi$. By the Cauchy–Riemann equations $D\Phi$ is a rotation times $|{\Phi }^{\prime }|$, so $|\nabla v{|}^2=|\nabla u\circ \Phi {|}^2|{\Phi }^{\prime }{|}^2$ and $\mathrm{d}{x}_{\Omega }=|{\Phi }^{\prime }{|}^2\mathrm{d}{x}_D$. Hence

$${\int }_{\Omega }\sigma |\nabla u{|}^2\mathrm{d}x={\int }_D(\sigma \circ \Phi )|\nabla v{|}^2\mathrm{d}x,$$

and by the Dirichlet principle $u$ solves $\nabla \cdot \sigma \nabla u=0$ in $\Omega$ exactly when $v$ solves $\nabla \cdot \tilde{\sigma }\nabla v=0$ in $D$, with

$$\tilde{\sigma }=\sigma \circ \Phi .$$

The pulled-back conductivity stays isotropic, and its values are just transported. In higher dimensions nothing comparable holds: by Liouville’s theorem the only conformal maps in $d\ge 3$ are Möbius transformations.

General diffeomorphisms

For a diffeomorphism that is not conformal, the same computation gives the anisotropic conductivity

$$\tilde{\sigma }=|\det D\Phi |(D\Phi {)}^{-1}\sigma (D\Phi {)}^{-\top }\circ \Phi .$$

Its anisotropy is measured by the distortion $K=\max |D\Phi {|}^2/|\det D\Phi |$, which is $K\ge 1$ with equality exactly for conformal maps. This is the mechanism behind the non-uniqueness of Anisotropic Conductivities: diffeomorphisms that fix the boundary leave all boundary data unchanged.

Boundary data and electrodes

On the boundary, arc length transforms as $\mathrm{d}{s}_{\Omega }=|{\Phi }^{\prime }|\mathrm{d}{s}_D$. Therefore:

• Voltages are transported: $v=u\circ \Phi$ on $\partial D$.
• Current densities scale with the length factor: $\tilde{\sigma }{\partial }_{\nu }v=(g\circ \Phi )|{\Phi }^{\prime }|$. The current through any boundary piece, in particular every electrode current $I_{\ell }={\int }_{e_{\ell }}g\mathrm{d}s$, is invariant.
• In the Complete Electrode Model, the contact term ${\int }_{e_{\ell }}(u-U_{\ell }{)}^2/z_{\ell }\mathrm{d}s$ becomes ${\int }_{{\tilde{e}}_{\ell }}(v-U_{\ell }{)}^2|{\Phi }^{\prime }|/z_{\ell }\mathrm{d}s$. The pulled-back contact impedance $z_{\ell }/|{\Phi }^{\prime }|$ varies along the electrode.
• In the Gap Model, the uniform current density $I_{\ell }/|e_{\ell }|$ becomes $I_{\ell }|{\Phi }^{\prime }|/|e_{\ell }|$, and mean voltages are weighted by $|{\Phi }^{\prime }|$.

The Neumann-to-Dirichlet Map of $(\Omega ,\sigma )$ and the one of $(D,\tilde{\sigma })$ are therefore related by these weights. Electrode data, meaning currents in and electrode voltages out, are unchanged.

Consequences

• Model domains. Forward and inverse problems on any simply connected domain can be solved on the disk: reconstruct $\tilde{\sigma }$ and push it forward by $\sigma =\tilde{\sigma }\circ {\Phi }^{-1}$. If the true boundary is only approximately known, reconstructing on a wrong domain produces an anisotropic error, and only its conformal part can be corrected.
• Fast solvers. A mesh of $\Omega$ obtained by mapping a disk mesh with $\Phi$ has a stiffness matrix that is spectrally equivalent to the one of the disk mesh, with constants close to $1$. Each element is mapped by an almost-similarity. Disk solvers therefore precondition the mapped problem (see Fast Solvers on Disk Domains, Numerical Conformal Mapping).

In ModularEIT.jl: ConformalMap, conformal_grid.

References

1. V. Kolehmainen, M. Lassas, P. Ola (2005). The Inverse Conductivity Problem with an Imperfectly Known Boundary. SIAM J. Appl. Math. 66(2), 365–383. doi:10.1137/040612737
2. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
3. K. Astala, L. Päivärinta, M. Lassas (2005). Calderón’s inverse problem for anisotropic conductivity in the plane. Comm. Partial Differential Equations 30(1–2), 207–224. doi:10.1081/PDE-200044485