Phase Retrieval Overview

Principles

Definition

Phase retrieval is a research problem and methodological approach focused on reconstructing a complex signal or image from measurements of its magnitude or intensity, without direct measurement of its corresponding phase information. This field is significant in diverse areas such as optical imaging, crystallography, and signal processing, where the phase component, crucial for full reconstruction, is often lost during data acquisition.

Ontological type

Core Algorithms

Measurement Models

Key Applications

Key developments

Key Figures

Iterative Holographic Recovery era

Gerchberg and Saxton crystallized the era of iterative phase retrieval in the early 1970s by introducing a practical two-plane projection algorithm to recover phase from limited intensity measurements in holographic and diffraction contexts. Their method alternates between the real-space image domain and the Fourier domain, yielding progressively accurate phase with simple, lab-friendly computations. This iterative holographic recovery approach sits on the holography foundation laid by Dennis Gabor, translating phase problems into experimentally demonstrable procedures. Across optics and electron microscopy laboratories in the 1970s, researchers implemented two-image reconstructions, holographic sampling strategies, and phase-based alignment to validate reconstructions through defocus, correlation, and holographic references.

Constraint and Convex Formalization era

George M. Sheldrick[1], associated with Washington University in St. Louis[3] and the University of Cambridge[4], emerges as a foundational figure in early phase retrieval-inspired methods during this era. In the 1990 paper Phase annealing in SHELX-90: direct methods for larger structures[7], Sheldrick[1] developed phase annealing techniques that enhanced direct methods for larger crystals, addressing key challenges in magnitude-only phase reconstruction and enabling more accurate structure solutions. Emmanuel J. Candès[2], with affiliations at Stanford University[5] and Université Paris-Sud[6], stands as a central figure in convex formulations of phase retrieval during this era. His 2012 paper PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming[8], by Candès[2], introduced convex lifting techniques that guarantee exact and stable recovery from magnitude-only data, anchoring robust, scalable phase retrieval in both theory and practice during this era.

Hybrid Physics–Data Methods era

During the Hybrid Physics–Data Methods era (2017–2023), foundational theory by Emmanuel Candès and Mahdi Soltanolkotabi on nonconvex phase retrieval and the Wirtinger Flow algorithm anchored robust, data-aware solvers. Yonina C. Eldar's work on structured illumination and coded diffraction patterns provided physics-informed priors that enable stable recovery under realistic optical measurements. Jianwei Miao's imaging-science program linked phase retrieval to practical microscopy and X-ray imaging, demonstrating how computational reconstructions translate into deployable optical systems. Akshay Bora and collaborators advanced the use of neural priors and generative models to constrain phase-retrieval solutions, illustrating the synthesis of data-driven priors with physical models.