Absolute Uniqueness of Phase Retrieval with Random Illumination

Abstract: Random illumination is proposed to enforce absolute uniqueness and resolve all types of ambiguity, trivial or nontrivial, from phase retrieval. Almost sure irreducibility is proved for any complex-valued object of a full rank support. While the new irreducibility result can be viewed as a probabilistic version of the classical result by Bruck, Sodin and Hayes, it provides a novel perspective and an effective method for phase retrieval. In particular, almost sure uniqueness, up to a global phase, is proved for complex-valued objects under general two-point conditions. Under a tight sector constraint absolute uniqueness is proved to hold with probability exponentially close to unity as the object sparsity increases. Under a magnitude constraint with random amplitude illumination, uniqueness modulo global phase is proved to hold with probability exponentially close to unity as object sparsity increases. For general complex-valued objects without any constraint, almost sure uniqueness up to global phase is established with two sets of Fourier magnitude data under two independent illuminations. Numerical experiments suggest that random illumination essentially alleviates most, if not all, numerical problems commonly associated with the standard phasing algorithms.