An Alternating Optimization Approach for Phase Retrieval

In this paper, we address the problem of phase retrieval to recover a signal from the magnitude of its Fourier transform. In many applications of phase retrieval, the signals encountered are naturally sparse. In this work, we consider the case where the signal is sparse under the assumption that few components are nonzero. We exploit further the sparse nature of the signals and propose a two stage sparse phase retrieval algorithm. A simple iterative minimization algorithm recovers a sparse signal from measurements of its Fourier transform (or other linear transform) magnitude based on the minimization of a block l(1) norm. We show in the experiments that the proposed algorithm achieves a competitive performance. It is robust to noise and scalable in practical implementation. The proposed method converges to a more accurate and stable solution than other existing techniques for synthetic signals. For speech signals, experiments show that the voice quality of reconstructed speech signals is almost as good as the original signals.