Reconstruction of a current distribution from its magnetic field

We consider the inverse problem of reconstructing a current distribution from measurements of its magnetic field. Uniqueness issues and simulations for the reconstruction are studied. Given the magnetic field on a surface surrounding the current distribution we show that a projection of the current distribution can be reconstructed uniquely. In addition, we derive some properties of directed current distributions that reflect the properties and difficulties of the reconstruction. A Tikhonov-projection scheme complemented by an artefact-correction algorithm is employed to reconstruct the current distribution within a cuboid. By numerical examples in three dimensions we show that for measurement errors up to 1% we can detect areas of low-current density within the cuboid.