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 196 articles , 2 standard articles )

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  1. Alaifari, Rima; Wellershoff, Matthias: Stability estimates for phase retrieval from discrete Gabor measurements (2021)
  2. Aziznejad, Shayan; Unser, Michael: Duality mapping for Schatten matrix norms (2021)
  3. Cheng, Cheng; Sun, Qiyu: Stable phaseless sampling and reconstruction of real-valued signals with finite rate of innovation (2021)
  4. Chen, Xuemei; Hardin, Douglas P.; Saff, Edward B.: On the search for tight frames of low coherence (2021)
  5. Gao, Bing; Liu, Haixia; Wang, Yang: Phase retrieval for sub-Gaussian measurements (2021)
  6. Gong, Yuxuan; Li, Peijun; Wang, Xu; Xu, Xiang: Numerical solution of an inverse random source problem for the time fractional diffusion equation via phaselift (2021)
  7. Huang, Meng; Rong, Yi; Wang, Yang; Xu, Zhiqiang: Almost everywhere generalized phase retrieval (2021)
  8. Huang, Meng; Xu, Zhiqiang: Phase retrieval from the norms of affine transformations (2021)
  9. Iwen, Mark A.; Krahmer, Felix; Krause-Solberg, Sara; Maly, Johannes: On recovery guarantees for one-bit compressed sensing on manifolds (2021)
  10. Li, Ji; Cai, Jian-Feng; Zhao, Hongkai: Scalable incremental nonconvex optimization approach for phase retrieval (2021)
  11. Na, Sen; Kolar, Mladen: High-dimensional index volatility models via Stein’s identity (2021)
  12. Needham, Tom; Shonkwiler, Clayton: Symplectic geometry and connectivity of spaces of frames (2021)
  13. Preskitt, Brian; Saab, Rayan: Admissible measurements and robust algorithms for ptychography (2021)
  14. Tasissa, Abiy; Lai, Rongjie: Low-rank matrix completion in a general non-orthogonal basis (2021)
  15. Thao, Nguyen Hieu; Soloviev, Oleg; Verhaegen, Michel: Convex combination of alternating projection and Douglas-Rachford operators for phase retrieval (2021)
  16. Zhang, Deyue; Guo, Yukun: Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory (2021)
  17. Aldroubi, Akram; Krishtal, I.; Tang, S.: Phaseless reconstruction from space-time samples (2020)
  18. Ashraphijuo, Morteza; Wang, Xiaodong: Characterization of sampling patterns for low-tt-rank tensor retrieval (2020)
  19. Barnett, Alexander H.; Epstein, Charles L.; Greengard, Leslie F.; Magland, Jeremy F.: Geometry of the phase retrieval problem (2020)
  20. Bendory, Tamir; Edidin, Dan; Eldar, Yonina C.: On signal reconstruction from FROG measurements (2020)

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