# Phase Retrieval and Applications

Recover phaseless complex signals from measured magnitudes, or how to revamp Fourier optics and invent algorithmic detectors. Emphasis will be put on recent development. (**Update: Jul 03 2017**)

[**S**] indicates my contribution.

### Provable Results with Algorithms

- Convergence of the randomized Kaczmarz method for phase retrieval (2017)
- Phase Retrieval via Randomized Kaczmarz: Theoretical Guarantees (2017)
- Phase retrieval using alternating minimization in a batch setting (2017)
- Solving Almost all Systems of Random Quadratic Equations (2017)
- Phase Retrieval Using Structured Sparsity: A Sample Efficient Algorithmic Framework (2017)
- Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval (2017)
- Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption (2017)
- Phase Retrieval via Sparse Wirtinger Flow (2017)
- Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization (2017)
- Multiple Illumination Phaseless Super-Resolution (MIPS) with Applications To Phaseless DOA Estimation and Diffraction Imaging (2017)
- Sparse phase retrieval of one-dimensional signals by Prony’s method (2017)
- Corruption Robust Phase Retrieval via Linear Programming (2016)
- Phase Retrieval from Local Measurements: Improved Robustness via Eigenvector-Based Angular Synchronization (2016)
- Sparse Phase Retrieval via Truncated Amplitude Flow (2016)
- Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Spac (2016)
- An Elementary Proof of Convex Phase Retrieval in the Natural Parameter Space via the Linear Program PhaseMax (2016)
- Solving Large-scale Systems of Random Quadratic Equations via Stochastic Truncated Amplitude Flow (2016)
- Low rank matrix recovery from Clifford orbits (2016)
- PhaseMax: Convex Phase Retrieval via Basis Pursuit (2016)
- Phase Retrieval From the Magnitudes of Affine Linear Measurements (2016)
- Phaseless super-resolution in the continuous domain (2016)
- Reconstruction of signals from their autocorrelation and cross-correlation vectors, with applications to phase retrieval and blind channel estimation (2016)
- Low Rank Phase Retrieval (2016)
- Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation (2016)
- Phase retrieval with random Gaussian sensing vectors by alternating projections (2016)
- Non-Convex Phase Retrieval from STFT Measurements (2016)
- Gauss-Newton Method for Phase Retrieval (2016)
- Phase Retrieval via Incremental Truncated Wirtinger Flow (2016)
- Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow (2016)
- Reshaped Wirtinger Flow for Solving Quadratic Systems of Equations (2016)
- Provable Non-convex Phase Retrieval with Outliers: Median Truncated Wirtinger Flow (2016)
- Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms (2016)
- A Geometric Analysis of Phase Retrieval ([
**S**], 2016) - Phaseless super-resolution using masks (2016)
- A Least Squares Approach for Stable Phase Retrieval from Short-Time Fourier Transform Magnitude (2015)
- STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms (2015)
- The Local Convexity of Solving Quadratic Equations (2015)
- Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems (2015)

- Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study (2015)
- Phase Retrieval via Wirtinger Flow: Theory and Algorithms (2014)
- Phase Retrieval using Alternating Minimization (2013)
- Phase Retrieval from Coded Diffraction Patterns (2013)
- Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns (2012)
- PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming (2011)

### Recovery Limits

- On Signal Reconstruction from FROG Measurements (2017)
- On the Uniqueness of FROG Methods (2017)
- On The 2D Phase Retrieval Problem (2016)
- Generalized phase retrieval : measurement number, matrix recovery and beyond (2016)
- Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes (2015)
- Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case (2015)
- Phase retrieval for the Cauchy wavelet transform (2014)
- Phase Transitions in Phase Retrieval (2014)
- Phase retrieval from very few measurements (2013)
- Invertibility and Robustness of Phaseless Reconstruction (2013)
- Phase Retrieval: Stability and Recovery Guarantees (2012)
- Stability of unique Fourier-transform phase reconstruction (1983)
- Reducible polynomials in more than one variable (1982)
- The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform (1982)

### Experimental Results

- Coordinate Descent Algorithms for Phase Retrieval (2017)

- Benchmark problems for phase retrieval (2017)
- A general framework for denoising phaseless diffraction measurements (2016)
- Phase retrieval for wavelet transforms (2015)
- Fourier Phase Retrieval with a Single Mask by Douglas-Rachford Algorithm (2015)
- Experimental robustness of Fourier Ptychographic phase retrieval algorithms (2015)
- Phase retrieval for wavelet transforms (2015)
- Sparse Phase Retrieval from Short-Time Fourier Measurements (2014)
- Phase retrieval for imaging problems (2013)
- Phase Recovery, MaxCut and Complex Semidefinite Programming (2012)
- Alternating direction methods for classical and ptychographic phase retrieval (2012)
- Phase Retrieval via Matrix Completion (2011)
- Phase retrieval and saddle-point optimization (2006)
- A unified evaluation of iterative projection algorithms for phase retrieval (2006)
- Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization (2002)

### Survey/Introduction Articles

- Fourier Phase Retrieval: Uniqueness and Algorithms (2017)
- Phase Retrieval: An Overview of Recent Developments (2015)
- Phase Retrieval with Application to Optical Imaging (2014)
- The phase retrieval problem (1995)

Disclaimer- This page is meant to serve a hub for references on this problem, and does not represent in any way personal endorsement of papers listed here. So I do not hold any responsibility for quality and technical correctness of each paper listed here. The reader is advised to use this resource with discretion.

If you’d like your paper to be listed here- Just drop me a few lines via email (which can be found on “Welcome” page). If you don’t bother to spend a word, just deposit your paper on arXiv. I get email alert about new animals there every morning, and will be happy to hunt one for this zoo if it seemsfit.