Imaging of conducting materials via the Kernel Method

In this work, we present a new non-iterative imaging method for Electrical Resistance Tomography (ERT). The problem in ERT is retrieving the spatial behaviour of the electrical conductivity by means of boundary measurements in steady-state conditions. Specifically, the interest is focused on the inverse obstacle problem, that consists in reconstructing the shape, position and dimension of one or more anomalies embedded in a known background. The proposed method, called Kernel Method, is based on the idea that if there exists a current density Jn that applied at the boundary ∂𝛺 of the domain under investigation 𝛺 produces the same scalar potential (on ∂𝛺), with and without anomalies, then the power density corresponding to Jn, evaluated on a configuration without anomalies, is vanishing in the region occupied by the latter. The proposed method has a very low computational cost. Indeed, the evaluation of the desired current density Jn on ∂𝛺 requires a negligible computational effort, and the reconstructions require only one forward problem.