Conformal uniqueness results in anisotropic electrical impedance imaging

Abstract

The anisotropic conductivity inverse boundary value problem (or reconstruction problem for anisotropic electrical impedance tomography) is presented in a geometric formulation and a uniqueness result is proved, under two different hypotheses, for the case where the conductivity is known up to a multiplicative scalar field. The first of these results relies on the conductivity being determined by boundary measurements up to a diffeomorphism fixing points on the boundary, which has been shown for analytic conductivities in three and higher dimensions by Lee and Uhlmann and for conductivities close to constant by Sylvester. The apparatus of G-structures is then used to show that a conformal mapping of a Riemannian manifold which fixes all points on the boundary must be the identity. A second approach, which proves the result in the piecewise analytic category, is a straightforward extension of the work of Kohn and Vogelius on the isotropic problem.