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We consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude ofone current density field generated by applying current on a set of electrodes. The required interior data can be obtained by means of MRI measurements. On the boundary we only require knowledge of the electrodes, their impedances, and the corresponding average input currents. From the mathematical point of view, this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the complete electrode model (CEM) of Somersalo, Cheney, and Isaacson [SIAM J. Appl. Math., 52 (1992), pp. 1023--1040]. We show that this variational problem has nonunique solutions. The surprising discovery is that the physical data is still sufficient to determine the geometry of (the connected components of) the level sets of the minimizers. We thus obtain an interesting phase retrieval result: knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude. We characterize locally the nonuniqueness in the variational problem. In two and three dimensions we also show that additional measurements of the voltage potential along a curve joining the electrodes yield unique determination of the conductivity. The proofs involve a maximum principle and a new regularity up to the boundary result for the CEM boundary conditions. A nonlinear algorithm is proposed and implemented to illustrate the theoretical results.

A Weighted Minimum Gradient Problem with Complete Electrode Model Boundary Conditions for Conductivity Imaging