Inversion formulas for the linearized impedance tomography problem

Abstract

We consider the linearized electrical impedance tomography problem in two dimensions on the unit disk. By a linearization around constant coefficients and using a trigonometric basis, we calculate the linearized Dirichlet-to-Neumann operator in terms of moments of the conduction coefficient of the problem. By expanding this coefficient into angular trigonometric functions and Legendre-Müntz polynomials in radial coordinates, we can find a lower-triangular representation of the parameter to data mapping. As a consequence, we find an explicit solution formula for the corresponding inverse problem. Furthermore, we also consider the problem with boundary data given only on parts of the boundary while setting homogeneous Dirichlet values on the rest. We show that the conduction coefficient is uniquely determined from incomplete data of the linearized Dirichlet-to-Neumann operator with an explicit solution formula provided.