Polarimetric Fourier phase retrieval
Résumé
This work introduces polarimetric Fourier phase retrieval (PPR), a physically-inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurements polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger-Flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches towards a scalable, computationally efficient and robust to noise reconstruction strategy for PPR.
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