GESPAR: Efficient Phase Retrieval of Sparse Signals
The authors consider the problem of One Dimensional (1D) phase retrieval, namely, recovery of a 1D signal from the magnitude of its Fourier transform. This problem is ill-posed since the Fourier phase information is lost. Therefore, prior information on the signal is needed in order to recover it. In this paper, they 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. They propose a fast local search method for recovering a sparse 1D signal from measurements of its Fourier transform magnitude. Their algorithm does not require matrix lifting, unlike previous approaches, and therefore is potentially suitable for large scale problems such as images.