Polarimetric phase retrieval: uniqueness and algorithms

TL;DR

This paper introduces a new polarimetric phase retrieval model that leverages polarization information in Fourier phase retrieval, providing theoretical uniqueness results and developing algebraic and algorithmic reconstruction methods, validated by numerical experiments.

Contribution

It presents the first comprehensive analysis of polarimetric phase retrieval, including uniqueness characterization and the development of novel algebraic and adapted algorithms for reconstruction.

Findings

Complete characterization of uniqueness properties.

Development of algebraic and adapted Fourier phase retrieval algorithms.

Numerical experiments demonstrating performance and scalability.

Abstract

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.