This paper considers phase retrieval from the magnitude of one-dimensional 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 then show that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution). 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.
Bibliographical noteFunding Information:
The work of K. Huang and N. D. Sidiropoulos was supported by the National Science Foundation under Grants CIF-1525194 and IIS-1247632. The work of Y. C. Eldar was supported in part by the European Unions Horizon 2020 Research and Innovation Program through the ERC-BNYQ Project, and in part by the Israel Science Foundation under Grant 335/14.
- Phase retrieval
- alternating direction method of multipliers
- auto-correlation retrieval
- minimum phase
- over-sampled Fourier measurements
- semi-definite programming