Phase Retrieval Using Feasible Point Pursuit: Algorithms and Cramér-Rao Bound

Reconstructing a signal from squared linear (rank-1 quadratic) measurements is a challenging problem with important applications in optics and imaging, where it is known as phase retrieval. This paper proposes two new phase retrieval algorithms based on nonconvex quadratically constrained quadratic programming) formulations, and a recently proposed approximation technique dubbed feasible point pursuit (FPP). The first is designed for uniformly distributed bounded measurement errors, such as those arising from high-rate quantization (B-FPP). The second is designed for Gaussian measurement errors, using a least-squares criterion (LS-FPP). Their performance is measured against state-of-the-art algorithms and the Cramér-Rao bound (CRB), which is also derived here. Simulations show that LS-FPP outperforms the existing schemes and operates close to the CRB. Compact CRB expressions, properties, and insights are obtained by explicitly computing the CRB in various special cases - including when the signal of interest admits a sparse parametrization, using harmonic retrieval as an example.