The information present in the phase of gradient-echo images has opened a new window to look at fine brain anatomy. To obtain high quality phase images depicting small field perturbations produced by tissue susceptibility differences, large slowly spatially varying phase shifts due to air-tissue interfaces have to be removed. Various techniques have been proposed to fit this background field with the field generated by magnetic dipoles that either demand extra information regarding the object, or user input, or have require high number of iterations. In this work, a fast converging method based on an approximate solution of the Matching Pursuit iterative procedure is presented, which regularizes this ill-posed deconvolution problem by exploiting the spatial sparsity of the sources of the background field.