Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms

This paper considers phase retrieval from the magnitude of 1D over-sampled Fourier measurements, a classical problem that has challenged researchers in various fields of science and engineering. It is shown that an optimal vector in a least-squares sense can be found by solving a convex problem, thus establishing a hidden convexity in Fourier phase retrieval. It is also shown that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution) in this case. A method is then derived to retrieve an optimal minimum phase solution in polynomial time. Using these results, a new measuring technique is proposed which guarantees uniqueness of the solution, along with an efficient algorithm that can solve large-scale Fourier phase retrieval problems with uniqueness and optimality guarantees.