Numerical Reconstruction of Perfectly Conducting Inclusions from One Electrostatic Boundary Measurement

Electrical Impedance Tomography (EIT) aims to reconstruct inhomogeneities or inclusions in the interior of a medium based on current and voltage measurements at the boundary of the medium. In many practical applications of EIT, the e±ciency of the employed numerical inversion scheme is an essential parameter. The inversion methods typically evaluate an indicator function in order to estimate whether or not a given point is in the interior of the sought inclusion. The so-called sampling methods do
not assume any µa priori knowledge about the boundary condition valid at the inclusion boundary, but the evaluation of their indicator functions can be numerically expensive. On the other hand, decomposition methods express the solution of the Laplace equation in the medium in terms of layer potentials and estimate the inclusion using boundary value error minimisation. However,
matching the sources in the potentials with the measured current or voltage at the medium boundary can be numerically costly.
We describe a novel method for the reconstruction of perfectly electrically conducting inclusions in arbitrary homogeneous, simply connected media of ¯nite conductivity and with su±ciently smooth boundary. Similarly to the decomposition methods, our approach is based on a boundary layer representation of a solution of the Laplace equation in the medium, and it uses µa priori
knowledge of the boundary condition satisfied at the interface between the inclusion and the medium. However, both of these developments occur at the analytic stage only, and the actual numerical computation involves neither forward-model sources nor boundary-error minimisation. The method requires inclusions to be placed relatively close to the boundary of the medium, and
it is well-suited for detection of small inclusions and for detection and partial shape estimation of large inclusions. We give a mathematical justi¯cation for the indicator function used in the inversion method.Also, we illustrate the performance of the method using several numerical examples involving different medium geometries, as well as single and multiple inclusions of different shapes and positions within the medium. Finally, we compare the e±ciency and accuracy of the method to a decomposition scheme based on the Method of Auxiliary Sources.