Conformal Transplantation of Networks

In two dimensions EIT on any simply connected domain $\Omega$ is equivalent to EIT on the unit disk $D$. With a conformal map $\Phi :D\to \Omega$, a conductivity $\sigma$ on $\Omega$ corresponds to the isotropic conductivity $\tilde{\sigma }=\sigma \circ \Phi$ on $D$, and the boundary data transform by transporting voltages and currents (see Conformal Invariance of the Conductivity Equation). A network designed for the disk, or for a square via a second map, therefore applies to every simply connected domain:

1. compute the map $\Phi$ for the domain (see Numerical Conformal Mapping);
2. pull back the conductivity (or the reconstruction to be denoised), $\tilde{\sigma }=\sigma \circ \Phi$, onto a fixed grid on $D$;
3. apply the network on $D$;
4. push the result forward, $\sigma =\tilde{\sigma }\circ {\Phi }^{-1}$.

This is the natural shape-agnostic U-Net for 2D EIT. The architecture never sees the shape, because the physics does not depend on it after the pull-back.

What changes, and what does not

• The equation does not change. Unlike a general diffeomorphism, a conformal map keeps the conductivity isotropic and its values unchanged. A reconstruction method on $D$ remains a reconstruction method.
• The resolution does. A uniform grid on $D$ corresponds to a grid on $\Omega$ with local spacing $|{\Phi }^{\prime }|$ times the reference spacing. Where the boundary bulges out, $|{\Phi }^{\prime }|$ is large and the resolution is coarse. At re-entrant parts of a non-convex domain it is fine, and the conformal “crowding” of elongated domains can make it extremely uneven.
• The image statistics do too. A prior learned from images on $\Omega$ is not the same as one learned on $D$. Shapes are stretched by $|{\Phi }^{\prime }|$ and rotated by $\arg {\Phi }^{\prime }$. A prior trained on the reference domain therefore expects reference-domain statistics. It is exact for priors that are themselves conformally invariant, and approximately right when $|{\Phi }^{\prime }|$ varies little.
• Electrodes move. The electrodes of $\Omega$ land at other positions on $\partial D$, with lengths scaled by $1/|{\Phi }^{\prime }|$ and a varying contact impedance. The forward model on $D$ must use them.

For convex, roughly round domains (a thorax, a head cross-section) the distortion is mild and transplantation works well. For elongated or strongly non-convex domains a rectangle as reference domain, or networks on the mesh itself (see Graph Convolutions on Finite Element Meshes), avoid the crowding.

In three dimensions nothing comparable exists: by Liouville’s theorem the only conformal maps are Möbius transformations.

References

1. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
2. S. J. Hamilton, A. Hauptmann (2018). Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks. IEEE Trans. Med. Imaging 37(10), 2367–2377. doi:10.1109/TMI.2018.2828303