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.

References in zbMATH (referenced in 171 articles , 2 standard articles )

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  1. Aldroubi, Akram; Krishtal, I.; Tang, S.: Phaseless reconstruction from space-time samples (2020)
  2. Ashraphijuo, Morteza; Wang, Xiaodong: Characterization of sampling patterns for low-tt-rank tensor retrieval (2020)
  3. Barnett, Alexander H.; Epstein, Charles L.; Greengard, Leslie F.; Magland, Jeremy F.: Geometry of the phase retrieval problem (2020)
  4. Bendory, Tamir; Edidin, Dan; Eldar, Yonina C.: On signal reconstruction from FROG measurements (2020)
  5. Burer, Samuel; Ye, Yinyu: Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs (2020)
  6. Carlsson, Marcus; Gerosa, Daniele: On phase retrieval via matrix completion and the estimation of low rank PSD matrices (2020)
  7. Chen, Yang; Cheng, Cheng; Sun, Qiyu; Wang, Haichao: Phase retrieval of real-valued signals in a shift-invariant space (2020)
  8. Chowdhury, Mujibur Rahman; Qin, Jing; Lou, Yifei: Non-blind and blind deconvolution under Poisson noise using fractional-order total variation (2020)
  9. Foucart, Simon; Gribonval, Rémi; Jacques, Laurent; Rauhut, Holger: Jointly low-rank and bisparse recovery: questions and partial answers (2020)
  10. Fung, Samy Wu; Di, Zichao: Multigrid optimization for large-scale ptychographic phase retrieval (2020)
  11. Grohs, Philipp; Koppensteiner, Sarah; Rathmair, Martin: Phase retrieval: uniqueness and stability (2020)
  12. Iwen, Mark A.; Preskitt, Brian; Saab, Rayan; Viswanathan, Aditya: Phase retrieval from local measurements: improved robustness via eigenvector-based angular synchronization (2020)
  13. Jorgensen, Palle; Song, Myung-Sin; Tian, James: A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS (L^2) spaces (2020)
  14. Luu, Tung Duy; Fadili, Jalal; Chesneau, Christophe: Sharp oracle inequalities for low-complexity priors (2020)
  15. Ma, Cong; Wang, Kaizheng; Chi, Yuejie; Chen, Yuxin: Implicit regularization in nonconvex statistical estimation: gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution (2020)
  16. Neykov, Matey; Wang, Zhaoran; Liu, Han: Agnostic estimation for phase retrieval (2020)
  17. Pang, Tongyao; Li, Qingna; Wen, Zaiwen; Shen, Zuowei: Phase retrieval: a data-driven wavelet frame based approach (2020)
  18. Shojaei, Shayan; Haddadi, Farzan: Blind three dimensional deconvolution via convex optimization (2020)
  19. Wei, Ke; Cai, Jian-Feng; Chan, Tony F.; Leung, Shingyu: Guarantees of Riemannian optimization for low rank matrix completion (2020)
  20. Zhang, Deyue; Guo, Yukun; Sun, Fenglin; Liu, Hongyu: Unique determinations in inverse scattering problems with phaseless near-field measurements (2020)

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