A weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging

TL;DR

This paper introduces a new weighted minimum gradient problem for conductivity imaging using interior current magnitude data from MRI, demonstrating how boundary measurements can determine the conductivity and current vector field despite non-uniqueness.

Contribution

It formulates a novel variational problem under the Complete Electrode Model and proves that boundary data suffices to recover the current vector field and conductivity with additional voltage measurements.

Findings

Interior current magnitude data determines level sets of solutions.

Boundary input current allows full current vector field recovery.

Additional voltage measurements ensure unique conductivity reconstruction.

Abstract

We consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one 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 of Somersalo, Cheney and Isaacson. This variational problem has non-unique solutions. The surprising discovery is that the physical data is still sufficient to determine the geometry of the level sets of the minimizers. In particular, we 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 the non-uniqueness in the variational problem. We also show that additional measurements of the voltage potential along one curve joining the electrodes yield unique determination of the conductivity. A nonlinear algorithm is proposed and implemented to illustrate the theoretical results.