Magnetic diagnostics are used to reconstruct current density profiles in noninductively driven plasmas in the CDX-U tokamak. The boundary magnetic fields are measured by 32 pickup coils and three flux loops attached at different poloidal positions inside the CDX-U vessel; in addition, internal magnetic probes are also used. Significant eddy currents are measured on the thick aluminum vessel and accounted for in the analysis. Two-dimensional current density distributions and magnetic flux contours are reconstructed from these measurements using a least-square-error technique. A finite element description of plasma currents is useful for general magnetic geometries which evolve from open field lines to closed flux surfaces. By increasing the number of elements, a converged current density distribution is obtained.