swMATH ID: 31783
Software Authors: Candès, Emmanuel J.; Eldar, Yonina C.; Strohmer, Thomas; Voroninski, Vladislav
Description: Phase retrieval via matrix completion. This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging, and many other applications. Our approach, called PhaseLift, combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that a complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover.
Homepage: https://arxiv.org/abs/1109.0573
Dependencies: Matlab
Keywords: phase retrieval; matrix completion; diffraction; Fourier transform; convex optimization; trace-norm minimization
Related Software: Wirtinger Flow; GESPAR; PhaseMax; TFOCS; CVX; SparsePR; PhaseCut; SDPLR; BlockPR; SDPT3; PDCO; Mosek; UNLocBoX; CoSaMP; ADMiRA; YALMIP; Manopt; GitHub; RRR; SNLSDP
Referenced in: 228 Publications
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Referenced by 394 Authors

15 Wang, Yang
8 Cai, Jian-Feng
7 Candès, Emmanuel J.
7 Strohmer, Thomas
6 Huang, Meng
6 Iwen, Mark A.
6 Xu, Zhiqiang
5 Beinert, Robert
5 Krahmer, Felix
5 Rauhut, Holger
5 Voroninski, Vladislav
5 Wei, Ke
4 Alaifari, Rima
4 Bahmani, Sohail
4 Cheng, Cheng
4 Eldar, Yonina Chana
4 Grohs, Philipp
4 Han, Deguang
4 Hand, Paul E.
4 Kueng, Richard
4 Li, Ji
4 Li, Xiaodong
4 Lou, Yifei
4 Moscoso, Miguel
4 Novikov, Alexei
4 Osher, Stanley Joel
4 Papanicolaou, George C.
4 Sun, Qiyu
4 Wang, Hongxia
3 Ashraphijuo, Morteza
3 Balan, Radu V.
3 Bendory, Tamir
3 Chang, Huibin
3 Daubechies, Ingrid Chantal
3 Ding, Lijun
3 Drusvyatskiy, Dmitriy
3 Fadili, Jalal M.
3 Foucart, Simon
3 Friedlander, Michael P.
3 Fung, Samy Wu
3 Gribonval, Rémi
3 Guo, Yukun
3 Jaming, Philippe
3 Juste, Ted
3 Ling, Shuyang
3 Marchesini, Stefano
3 Mixon, Dustin G.
3 Pereyra, Marcelo
3 Romberg, Justin K.
3 Saab, Rayan
3 Terstiege, Ulrich
3 Thao, Nguyen Hieu
3 Viswanathan, Aditya
3 Waldspurger, Irène
3 Wang, Xiaodong
3 Zhang, Deyue
2 Aravkin, Aleksandr Y.
2 Bandeira, Afonso S.
2 Bardsley, Patrick
2 Blanchard, Gilles
2 Bodmann, Bernhard G.
2 Cahill, Jameson
2 Cai, Tony Tony
2 Cevher, Volkan
2 Chan, Tony Fan-Cheong
2 Chen, Pengwen
2 Chen, Yuxin
2 Chesneau, Christophe
2 Chi, Yuejie
2 d’Aspremont, Alexandre
2 Demanet, Laurent
2 Durmus, Alain
2 Dym, Nadav
2 Edidin, Dan
2 Ehler, Martin
2 Fannjiang, Albert C.
2 Gao, Bing
2 Guevara Vasquez, Fernando
2 Hammen, Nathaniel
2 Heaton, Howard
2 Kech, Michael
2 Keriven, Nicolas
2 Kılınç Karzan, Fatma
2 Kyrillidis, Anastasios
2 Lavaei, Javad
2 Leung, Shingyu
2 Li, Jingzhi
2 Li, Youfa
2 Liu, Haixia
2 Luo, Qi
2 Luu, Tung Duy
2 Ma, Cong
2 Mallat, Stephane Georges
2 Nelson, Aaron A.
2 Neykov, Matey
2 Plonka, Gerlind
2 Preskitt, Brian
2 Sanghavi, Sujay
2 Schaeffer, Hayden
2 Shen, Zuowei
...and 294 more Authors
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Referenced in 68 Serials

32 Applied and Computational Harmonic Analysis
22 SIAM Journal on Imaging Sciences
19 The Journal of Fourier Analysis and Applications
15 Inverse Problems
10 Foundations of Computational Mathematics
8 Mathematical Programming. Series A. Series B
7 Linear Algebra and its Applications
7 SIAM Journal on Optimization
5 Journal of Machine Learning Research (JMLR)
4 Journal of Computational and Applied Mathematics
4 Advances in Computational Mathematics
3 The Annals of Statistics
3 Mathematics of Operations Research
3 Journal of Global Optimization
3 SIAM Journal on Applied Mathematics
3 SIAM Journal on Scientific Computing
3 Mathematical Problems in Engineering
3 Optimization Letters
3 Information and Inference
3 SIAM Journal on Mathematics of Data Science
2 Annals of the Institute of Statistical Mathematics
2 Numerical Functional Analysis and Optimization
2 Results in Mathematics
2 Advances in Applied Mathematics
2 Discrete & Computational Geometry
2 SIAM Journal on Matrix Analysis and Applications
2 Journal of Scientific Computing
2 Numerical Algorithms
2 SIAM Journal on Mathematical Analysis
2 SIAM Review
2 Journal of Mathematical Imaging and Vision
2 Annals of Mathematics and Artificial Intelligence
2 Multiscale Modeling & Simulation
2 Inverse Problems and Imaging
2 Science China. Mathematics
2 Journal of the Operations Research Society of China
2 Annals of Applied Mathematics
2 Mathematical Statistics and Learning
1 Analysis Mathematica
1 Communications on Pure and Applied Mathematics
1 IEEE Transactions on Information Theory
1 Journal of Computational Physics
1 Journal of the Franklin Institute
1 Information Sciences
1 Mathematics and Computers in Simulation
1 Systems & Control Letters
1 Acta Applicandae Mathematicae
1 Statistical Science
1 Applied Mathematics Letters
1 Multidimensional Systems and Signal Processing
1 Machine Learning
1 Computational Optimization and Applications
1 Bernoulli
1 Methodology and Computing in Applied Probability
1 Communications in Mathematical Sciences
1 Analysis and Applications (Singapore)
1 International Journal of Wavelets, Multiresolution and Information Processing
1 Journal of Statistical Mechanics: Theory and Experiment
1 Inverse Problems in Science and Engineering
1 Electronic Journal of Statistics
1 Mathematical Programming Computation
1 The Journal of Mathematical Neuroscience
1 SIAM/ASA Journal on Uncertainty Quantification
1 Waves, Wavelets and Fractals. Advanced Analysis
1 Advances in Operator Theory
1 SIAM Journal on Applied Algebra and Geometry
1 Electronic Research Archive
1 Fixed Point Theory and Algorithms for Sciences and Engineering
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Referenced in 38 Fields

98 Information and communication theory, circuits (94-XX)
87 Operations research, mathematical programming (90-XX)
71 Numerical analysis (65-XX)
50 Harmonic analysis on Euclidean spaces (42-XX)
34 Calculus of variations and optimal control; optimization (49-XX)
32 Statistics (62-XX)
29 Computer science (68-XX)
26 Linear and multilinear algebra; matrix theory (15-XX)
18 Optics, electromagnetic theory (78-XX)
14 Functional analysis (46-XX)
14 Probability theory and stochastic processes (60-XX)
13 Partial differential equations (35-XX)
8 Operator theory (47-XX)
5 Convex and discrete geometry (52-XX)
4 Quantum theory (81-XX)
3 Combinatorics (05-XX)
3 Functions of a complex variable (30-XX)
3 Integral equations (45-XX)
3 Biology and other natural sciences (92-XX)
2 Commutative algebra (13-XX)
2 Algebraic geometry (14-XX)
2 Measure and integration (28-XX)
2 Potential theory (31-XX)
2 Approximations and expansions (41-XX)
2 Geometry (51-XX)
2 Differential geometry (53-XX)
2 Mechanics of deformable solids (74-XX)
2 Systems theory; control (93-XX)
1 Group theory and generalizations (20-XX)
1 Real functions (26-XX)
1 Dynamical systems and ergodic theory (37-XX)
1 Abstract harmonic analysis (43-XX)
1 General topology (54-XX)
1 Global analysis, analysis on manifolds (58-XX)
1 Fluid mechanics (76-XX)
1 Statistical mechanics, structure of matter (82-XX)
1 Astronomy and astrophysics (85-XX)
1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

Referencing Publications by Year