Reconstruction of Thin Tubular Inclusions in Three-Dimensional Domains Using Electrical Impedance Tomography

We consider the inverse problem of reconstructing thin tubular inclusions inside a three-dimensional body from measurements of electrostatic currents and potentials on its boundary. By inclusions we mean objects with an electrical conductivity differing from that of the background material of the body. We apply an asymptotic expansion of the electrostatic potential on the boundary of the body as the thickness of the inclusions tends to zero to establish an asymptotic characterization of these inclusions in terms of the measurement data. This characterization is implemented in a noniterative reconstruction method similar to the factorization method for crack detection problems in two-dimensional domains. We present several numerical examples to illustrate our theoretical findings and to highlight the potentials and limitations of our method.