Fast Compressive Phase Retrieval from Fourier Measurements

This paper considers the problem of recovering a $k$-sparse, $N$-dimensional complex signal from Fourier magnitude measurements. It proposes a Fourier optics setup such that signal recovery up to a global phase factor is possible with very high probability whenever $M\gtrsim 4k{\log }_2(N\mathrm{/}k)$ random Fourier intensity measurements are available. The proposed algorithm is comprised of two stages: An algebraic phase retrieval stage and a compressive sensing step subsequent to it. Simulation results are provided to demonstrate the applicability of the algorithm for noiseless and noisy scenarios.