The factorization method in impedance tomography

Abstract

This chapter examines the classical problem of impedance tomography. The corresponding direct problem is a boundary value problem for a simple elliptic differential equation of second order in divergence form in a bounded domain G. The conductivity can vary in space but is assumed to be constant one near the boundary of the domain G. The inverse problem is to determine the support D of the contrast (i.e., the region where the conductivity differs from one) from the knowledge of all possible Cauchy data on the boundary of G. As in the previous chapters, the chapter reviews results on the direct problem, introduces the Neuman-to-Dirichlet operator, and formulates the inverse problem of impedance tomography to determine the support D of the contrast from the knowledge of the Neumann-to-Dirichlet operator. It then proves a factorization of the difference of the Neumann-to-Dirichlet operators which correspond to the presence and the absence of the contrast, respectively. Using this factorization and the general results from Chapters 1 and 2, the support D is again characterized by the convergence of a Picard series.