  Accurate determination of microscopic transport and magnetization currents is
of central importance for the study of the electric properties of low
dimensional materials and interfaces, of superconducting thin films and of
electronic devices. Current distribution is usually derived from the
measurement of the perpendicular component of the magnetic field above the
surface of the sample, followed by numerical inversion of the Biot-Savart law.
The inversion is commonly obtained by deriving the current stream function $g$,
which is then differentiated in order to obtain the current distribution.
However, this two-step procedure requires filtering at each step and, as a
result, oversmoothes the solution. To avoid this oversmoothing we develop a
direct procedure for inversion of the magnetic field that avoids use of the
stream function. This approach provides enhanced accuracy of current
reconstruction over a wide range of noise levels. We further introduce a
reflection procedure that allows for the reconstruction of currents that cross
the boundaries of the measurement window. The effectiveness of our approach is
demonstrated by several numerical examples.
