We study a simple version of the general EIT problem, in which the medium has constant conductivity but might contain some buried object of unknown shape and different, but also constant, conductivity. Specifically, we present a procedure to reconstruct the object boundary in the case that we know the conductivities and the center and radius of a good circular approximation of the object boundary, using analytic solutions to the forward problem for circular objects with known conductivity. Assuming that the unknown object boundary is star-shaped with respect to the origin and a small perturbation of the approximating circle, we develop a linearized relation between the output voltage data that result from fixed input currents, entering as parameters, and the interface, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the interface. At least two measurements are needed to determine all coefficients, and more can be used for a least squares fit.
Numerical evidence supporting the validity and usefulness of the linearization is included.
The quality of the recovered perturbation depends on the input frequency as well as the frequencies of the perturbation. Low frequencies work best as input, and are most easily recovered, in that case even if the amplitude of the perturbation is not very small. On the other hand, several dipoles, corresponding to very close pairs of electrodes to induce the current, can be used successfully as input as well.