An alternating optimization approach for phase retrieval

In this paper, we address the problem of phase retrieval to re- cover 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 signal- s and propose a two stage sparse phase retrieval algorithm. A simple iterative minimization algorithm recovers a sparse sig- nal from measurements of its Fourier transform (or other lin- ear transform) magnitude based on the minimization of a block l₁ 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 ex- isting techniques for synthetic signals. For speech signals, ex- periments show that the voice quality of reconstructed speech signals is almost as good as the original signals.