An anisotropic inverse boundary value problem

We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ? Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ_*γ which will produce the same voltage and current measurements on ? Ω. We prove the converse in two dimensions (i.e., if γ₁ and γ₂ produce the same boundary measurements, then γ₁, = Ψ_*γ₂ for an appropriate Ψ) for conductivities which are near a constant.