GESPAR:Efficient Sparse Phase Retrieval with Application to Optics

The problem of phase retrieval, namely, recovery of a signal from the magnitude of its Fourier transform is illposed since the Fourier phase information is lost. Therefore, prior information on the signal is needed in order to recover it. In this work we consider the case in which the prior information on the signal is that it is sparse, i.e., it consists of a small number of nonzero elements. We propose GESPAR: A fast local search method for recovering a sparse signal from measurements of its Fourier transform magnitude. Our algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images. Simulation results indicate that the proposed algorithm is fast and more accurate than existing techniques. We demonstrate applications in optics where GESPAR is generalized and used for finding sparse solutions to sets of quadratic measurements.