There are many practical problems in which it is required to detect and characterize hidden structures or remote objects by virtue of the scattered acoustic or electromagnetic fields they generate. It remains an open question, however, as to which reconstruction algorithms offer the most informative images for a given set of field measurements. Commonly used time-domain beamforming techniques, and their equivalent frequency-domain implementations, are conceptually simple and stable in the presence of noise, however, large proportions of missing measurements can quickly degrade the image quality. We apply a new algorithm based on the maximum entropy method (MEM) to the reconstruction of images from sparsely sampled coherent field data. The general principles and limitations of the new method are discussed in the framework of regularization theory, and the results of monostatic imaging experiments confirm that superior resolution and artifact suppression are obtained relative to a commonly used linear inverse filtering approach.