DIRECT REGULARIZED RECONSTRUCTION FOR THE THREE-DIMENSIONAL CALDERON PROBLEM

Electrical Impedance Tomography gives rise to the severely ill-posed Calderon problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderon problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data il-lustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.