  This work introduces a novel Fourier phase retrieval model, called
polarimetric phase retrieval that enables a systematic use of polarization
information in Fourier phase retrieval problems. We provide a complete
characterization of uniqueness properties of this new model by unraveling
equivalencies with a peculiar polynomial factorization problem. We introduce
two different but complementary categories of reconstruction methods. The first
one is algebraic and relies on the use of approximate greatest common divisor
computations using Sylvester matrices. The second one carefully adapts existing
algorithms for Fourier phase retrieval, namely semidefinite positive relaxation
and Wirtinger-Flow, to solve the polarimetric phase retrieval problem. Finally,
a set of numerical experiments permits a detailed assessment of the numerical
behavior and relative performances of each proposed reconstruction strategy. We
further highlight a reconstruction strategy that combines both approaches for
scalable, computationally efficient and asymptotically MSE optimal performance.
