The Inverse Conductivity Problem with an Imperfectly Known Boundary in Three Dimensions

We consider the inverse conductivity problem in a strictly convex domain whose boundary is not known. Usually the numerical reconstruction from the measured current and voltage data is done assuming the domain has a known fixed geometry. However, in practical applications the geometry of the domain is usually not known. This introduces an error, and effectively changes the problem into an anisotropic one. The main result of this paper is a uniqueness result characterizing the isotropic conductivities on convex domains in terms of measurements done on a different domain, which we call the model domain, up to an affine isometry. As data for the inverse problem, we assume the Robin-to-Neumann map and the contact impedance function on the boundary of the model domain to be given. Also, we present a minimization algorithm based on the use of Cotton–York tensor, that finds the pushforward of the isotropic conductivity to our model domain, and also finds the boundary of the original domain up to an affine isometry. This algorithm works also in dimensions higher than three, but then the Cotton–York tensor has to replaced with the Weyl–tensor. AMS classification: 35J25, 35R30, 58J32.