Convex backscattering support in electric impedance tomography

This paper reinvestigates a recently introduced notion of backscattering for the inverse obstacle problem in impedance tomography. Under mild restrictions on the topological properties of the obstacles, it is shown that the corresponding backscatter data are the boundary values of a function that is holomorphic in the exterior of the obstacle(s), which allows to reformulate the obstacle problem as an inverse source problem for the Laplace equation. For general obstacles, the convex backscattering support is then defined to be the smallest convex set that carries an admissible source, i.e., a source that yields the given (backscatter) data as the trace of the associated potential. The convex backscattering support can be computed numerically; numerical reconstructions are included to illustrate the viability of the method.