The Numerics of Phase Retrieval

Abstract

Phase retrieval, i.e., the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications such as X-ray crystallography, diffraction imaging, optics, quantum mechanics, and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it becomes increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the last decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.

References in corpus (44)

• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
• Phase recovery and holographic image reconstruction using deep learning in neural networks
• Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
• Recovering low-rank matrices from few coefficients in any basis
• A unified evaluation of iterative projection algorithms for phase retrieval
• Relaxed Averaged Alternating Reflections for Diffraction Imaging
• An overview of low-rank matrix recovery from incomplete observations
• Guaranteed Matrix Completion via Non-convex Factorization
• Experimental robustness of Fourier Ptychography phase retrieval algorithms
• GESPAR: Efficient Phase Retrieval of Sparse Signals
• SIC-POVMs and the Extended Clifford Group
• Compressive Phase Retrieval via Generalized Approximate Message Passing
• SIC-POVMs: A new computer study
• The SIC Question: History and State of Play
• Practical algorithms for simulation and reconstruction of digital in-line holograms
• Self-Calibration and Biconvex Compressive Sensing
• Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution
• An algebraic characterization of injectivity in phase retrieval
• A Geometric Analysis of Phase Retrieval
• Quantum Tomography under Prior Information
• SHARP: a distributed, GPU-based ptychographic solver
• Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming
• Iterative phase retrieval for digital holography
• Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns
• Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval
• Imaging proteins at the truly single-molecule level
• A Partial Derandomization of PhaseLift using Spherical Designs
• Sparse Phase Retrieval from Short-Time Fourier Measurements
• Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns
• Optimal Spectral Initialization for Signal Recovery with Applications to Phase Retrieval
• When holography meets coherent diffraction imaging
• DOLPHIn - Dictionary Learning for Phase Retrieval
• Absolute Uniqueness of Phase Retrieval with Random Illumination
• Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers
• Holographic Phase Retrieval and Reference Design
• Phase recovery from a Bayesian point of view: the variational approach
• Coherent Diffractive Imaging Using Randomly Coded Masks
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Shaping Coherent X-rays with Binary Optics
• Sparse phase retrieval of one-dimensional signals by Prony's method
• Coded Aperture Ptychography: Uniqueness and Reconstruction
• On Relaxed Averaged Alternating Reflections (RAAR) Algorithm for Phase Retrieval from Structured Illuminations
• Solving large-scale general phase retrieval problems via a sequence of convex relaxations
• Fourier Phase Retrieval with Extended Support Estimation via Deep Neural Network

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• Phase retrieval with Bregman divergences and application to audio signal recovery
• Distortion-Aware Phase Retrieval Receiver for High-Order QAM Transmission with Carrierless Intensity-Only Measurements
• SIC-POVMs from Stark units: Prime dimensions n^2+3
• Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
• Extended Successive Convex Approximation for Phase Retrieval with Dictionary Learning
• Bayes-Optimal Estimation in Generalized Linear Models via Spatial Coupling
• H-FISTA: A hierarchical algorithm for phase retrieval with application to pulsar dynamic spectra
• Uniqueness Theorems for Tomographic Phase Retrieval with Few Diffraction Patterns
• Provable Phase Retrieval with Mirror Descent
• SPRING: an effective and reliable framework for image reconstruction in single-particle Coherent Diffraction Imaging
• Approximation of Functions: Optimal Sampling and Complexity
• Solving non-linear integral equations for laser pulse retrieval with Newton's method
• 3D Tomographic Phase Retrieval and Unwrapping
• Total variation-based reconstruction and phase retrieval for diffraction tomography