Image reconstruction from Fourier transform magnitude with applications to synthetic aperture radar imaging

The problem of reconstructing an unknown signal starting from the knowledge of only magnitude information about its Fourier transform is addressed. To this end a phase retrieval (PR) method, based on the inversion of a quadratic operator, is proposed, presented, and discussed. The main feature of the approach is the use of a square amplitude distribution rather than an amplitude distribution. In the lack of any a priori information about the phase to be recovered, the success of the method is related to the availability of a sufficiently large ratio between the dimension of data and the dimension of unknowns; the retrieval procedure could converge to a meaningless trap when this ratio is not large enough. It should be noted, however, that the proposed method has a range of global effectiveness wider than that of previous approaches. Moreover, when available, the use of a priori information, such as the knowledge of the support of the scene to be imaged or the knowledge of a part of the scene, allows one to achieve accurate final images starting from a completely random initial guess of the image values. We apply our PR-based method to a synthetic aperture radar (SAR) case. As is well known, random motion of the platform and/or propagation effects in turbulent random media can affect the correct phase synchronization of the received SAR raw data needed for well-focused SAR images. As a matter of fact, even relatively small phase errors greatly impair the quality of the image. Under common conditions phase errors on the received signal affect just the phase of the Fourier transform of the intensity image without impairing its amplitude. The proposed technique is able to compensate these phase errors by retrieving the phase of the Fourier transform of the image intensity. We also present several experiments performed either on numerically simulated data or on actual data relative to an airborne SAR mission that show the effectiveness of the proposed PR method.