Image reconstruction in Electrical Impedance Tomography using an integral equation of the Lippmann-Schwinger type

We consider the inverse problem of reconstructing the conductivity distribution in the interior of an object using only electrical data measured on its boundary. In this work we assume the knowledge of the boundary values of and its normal derivative @ =@n, the electic potential and the current density @ =@n. We use a Green's function formulation, similar to that appearing in the derivation of the Lippmann{Schwinger equation, to transform this problem into a linear Fredholm integral equation of the rst kind. This formulation con rms the ill-posedness of the Inverse Problem and isolates it in the form of the well-known instability of the solution of such equations. It also suggests a way of regularizing the reconstruction process leading to a Fredholm integral equation of the second kind which can be solved explicitly. We illustrate the use of our technique in the case of a simply connected two dimensional body.