Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms
This addresses a fundamental challenge in signal processing and imaging for scientists and engineers, offering a novel convex approach with uniqueness guarantees, though it builds incrementally on existing semidefinite relaxation methods.
The paper tackles the classical problem of phase retrieval from 1D over-sampled Fourier measurements by showing that an optimal solution can be found via a convex problem, revealing hidden convexity, and proposes a new measuring technique that guarantees uniqueness and an efficient algorithm for large-scale problems with optimality guarantees.
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. We show 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. We also show 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.