The importance of random phase for image reconstruction from frequency offset Fourier data

Motivated by the ability of synthetic aperture radar and related imaging systems to produce images of suprisingly high quality, we consider the problem of reconstructing the magnitude of a complex signal from samples of the Fourier transform of located in a small region offset from the origin. It is shown that high quality speckle reconstructions are possible so long as the phase of is highly random. In this case the quality of the reconstruction is insensitive to the location of the known Fourier data, and edges at all orientations are reproduced equally well. Computer examples are presented demonstrating these attributes.