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

摘要

This paper considers phase retrieval from the magnitude of 1D over-sampledFourier measurements, a classical problem that has challenged researchers invarious fields of science and engineering. We show that an optimal vector in aleast-squares sense can be found by solving a convex problem, thus establishinga hidden convexity in Fourier phase retrieval. We also show that the standardsemidefinite relaxation approach yields the optimal cost function value (albeitnot necessarily an optimal solution) in this case. A method is then derived toretrieve an optimal minimum phase solution in polynomial time. Using theseresults, a new measuring technique is proposed which guarantees uniqueness ofthe solution, along with an efficient algorithm that can solve large-scaleFourier phase retrieval problems with uniqueness and optimality guarantees.