On uniqueness and non-uniqueness for current reconstruction from magnetic fields

The goal of this paper is to provide a basis for the analysis of the limits of the reconstructability of current densities from their magnetic fields as used for non-destructive testing and monitoring of fuel cells. For the reconstruction of a current density from its magnetic field, we study the properties of the Biot-Savart operator W. In particular, the nullspace N(W) of the Biot–Savart operator and its orthogonal space N(W)⊥ with respect to the L2 scalar product are characterized. The characterization of these spaces is a basic step for the evaluation of the principal limits of magnetic tomography for fuel cells and for the development of efficient reconstruction algorithms. Further, practically realizable examples for elements in the nullspace N(W) are provided. Finally, for a discrete wire network we show uniqueness for current reconstructions, i.e. the result N(W) = {0}.