Reconstructing anisotropic conductivities on two-dimensional Riemannian manifolds from power densities

TL;DR

This paper demonstrates that on a genus-zero 2D Riemannian manifold with known metric, anisotropic conductivities can be uniquely reconstructed from a few power densities, with numerical validation on a catenoid surface.

Contribution

It provides a constructive method for reconstructing anisotropic conductivities on genus-zero manifolds from limited power density data, extending previous theoretical results.

Findings

Unique reconstruction of anisotropic conductivities demonstrated

Numerical reconstruction performed on a catenoid surface

Method applicable to manifolds with known metric and genus zero

Abstract

We consider an electrically conductive compact two-dimensional Riemannian manifold with smooth boundary. This setting defines a natural conductive Laplacian on the manifold and hence also voltage potentials, current fields and corresponding power densities arising from suitable boundary conditions. Motivated by Acousto-Electric Tomography we show that if the manifold has genus zero and the metric is known, then the anisotropic conductivity can be recovered uniquely and constructively from knowledge of a few power densities. We illustrate the procedure numerically by reconstructing an anisotropic conductivity on the catenoid, i.e. the classical genus zero minimal surface in three-space.