Numerical conductivity reconstruction from partial interior current density information in three dimensions

This paper focuses on conductivity imaging of a three dimensional object using interior current density information (ICDI). Applications include the emerging hybrid tomographic methods known as magnetic resonance electrical impedance tomography and current density impedance imaging, which potentially have both high contrast and high resolution. For all possible forms of ICDI data, the Fréchet derivative of the map between conductivity and ICDI is derived. Then, an iterative reconstruction method is formulated based on the Newton scheme. The method is implemented numerically and its properties are investigated on simulated data obtained from two different phantoms. The method is also benchmarked against the J-substitution method. We systematically study the possibilities, challenges, shortcomings, and artifacts due the different forms of full and partial ICDI data and one or several boundary conditions. The results establish that at least two components of two non-parallel interior current densities are required to obtain good reconstructions; this is an important outcome for conductivity imaging methods which use only one component of the magnetic field. The results hold promise for the near real-time and high resolution conductivity reconstruction in practical applications.