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An inverse transformation based on the fast Fourier transform can convert a two-dimensional image of the normal component of magnetic field into a corresponding image of the two-dimensional source currents that generated the field. Applying such a transformation to a magnetic image from a scanning Superconducting Quantum Interference Device (SQUID) microscope reveals that the spatial resolution s in the current image can be over 20 times better than that found in the raw magnetic field image, and up to about 5 times smaller than the SQUID sample separation z. We describe a quantitative theory for the noise and spatial resolution found in such current density images. We find that s is proportional to z and logarithmically related to the magnetic field noise in the image, the current applied to the sample, and the pixel size. We discuss the unusual functional dependence of these parameters and compare our theory to experimental data obtained from a scanning SQUID microscope. Finally, we describe how selective filtering in Fourier space can reduce noise and other artifacts in the current density images. © 2002 American Institute of Physics.