Fast Phase Retrieval: A Robust and Efficient Multidimensional Phase Retrieval Algorithm

Abstract

We present the first phase retrieval algorithm with a set of deterministic recovery guarantees. We show that for a class of objects known as "Schwarz Objects", the algorithm is guaranteed to reconstruct the object given only the magnitudes of its discrete Fourier transform. We present numerical evidence that the algorithm additionally succeeds quite often for non-Schwarz objects. We also present a set of measurement matrices for which the algorithm is guaranteed to recover any object. We derive the algorithm by converting instances of the phase-retrieval problem to the Schwarz problem and refine the solution with local optimization. The result is an algorithm that is fast, universal and robust against noise.