Recovering a signal from its Fourier intensity underlies many vital science and engi- neering applications, including but not limited to astronomy, X-ray crystallography, lensless imaging, imaging through scattering media, etc. To guarantee the uniqueness of solutions, constraints from prior information are needed. Object support is the most widely used constraint for recovery from the phaseless Fourier measurement, which is known as the support-constraint Fourier phase retrieval. Despite the problem being notoriously ill-posed, the iterative algorithms can solve it efficiently when the measurement and support are ideal; however, their performance is greatly hindered when (i) noise is present in the measurement, and/or (ii) the support is inaccurate. Furthermore, the reasoning of these algorithms is often not informative and hard to extend, making it difficult to develop better and more reliable algorithms. In this dissertation, we extend the iterative algorithms to overcome their issues. Divided into three parts, our major contributions include:
i. Regularized iterative phase retrieval, which is able to combine iterative methods with image statistics from deep denoisers. Adopting the framework of regularization- by-denoising, we develop hybrid methods that inherit the advantages of each approach, which not only achieve greater noise robustness, but also relax the initialization requirement;
ii. A systematic study of loose support constraints. Based on empirical observa- tions of algorithm stagnation that causes the degraded performance, we introduce a novel family of spatial regularization and propose a support-robust variant, which significantly outperform baseline methods;
iii. Automated algorithm search for Fourier phase retrieval. From the perspective of solving the feasibility problem, we propose parametric proximal splitting algorithms for phase retrieval and use data-driven approaches to search for the parameters. We discover multiple new algorithms with higher success rates and lower reconstruction errors, for both clean and noisy measurements.
Besides the significant contributions to the three major existing problems in phase retrieval, the methodologies presented in this thesis pave a way for improving algorithms even when their inner logic is opaque.