  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.
