SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 632-671, March 2024.
Abstract. 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 measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of 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 toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.

中文翻译：

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偏振傅立叶相位检索

SIAM 影像科学杂志，第 17 卷，第 1 期，第 632-671 页，2024 年 3 月。
摘要。这项工作介绍了偏振傅里叶相位检索 (PPR)，这是一种物理启发模型，可在傅里叶相位检索问题中利用光信息的偏振。我们通过阐明与两个相关问题（即二元相位检索和多项式自相关因式分解问题）的等价性来提供其唯一性属性的完整表征。特别是，我们表明该问题存在唯一的解决方案，可以将其表示为测量多项式的最大公约数（GCD）。因此，我们利用西尔维斯特矩阵的零空间特性，基于近似 GCD 计算提出了 PPR 的代数解。或者，仔细调整现有的相位检索、半定正松弛和 Wirtinger 流迭代算法来解决 PPR 问题。最后，一组数值实验允许对每个提出的重建策略的数值行为和相对性能进行详细评估。他们进一步展示了代数和迭代方法的富有成效的组合，以实现可扩展、计算高效且对噪声鲁棒的 PPR 重建策略。