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 219 articles , 2 standard articles )

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  1. Cai, Jian-Feng; Huang, Meng; Li, Dong; Wang, Yang: Solving phase retrieval with random initial guess is nearly as good as by spectral initialization (2022)
  2. Cai, Jian-Feng; Li, Jingzhi; Lu, Xiliang; You, Juntao: Sparse signal recovery from phaseless measurements via hard thresholding pursuit (2022)
  3. Alaifari, Rima; Wellershoff, Matthias: Stability estimates for phase retrieval from discrete Gabor measurements (2021)
  4. Aziznejad, Shayan; Unser, Michael: Duality mapping for Schatten matrix norms (2021)
  5. Bahmani, Sohail; Lee, Kiryung: Low-rank matrix estimation from rank-one projections by unlifted convex optimization (2021)
  6. Beinert, Robert; Bredies, Kristian: Tensor-free proximal methods for lifted bilinear/quadratic inverse problems with applications to phase retrieval (2021)
  7. Charisopoulos, Vasileios; Chen, Yudong; Davis, Damek; Díaz, Mateo; Ding, Lijun; Drusvyatskiy, Dmitriy: Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence (2021)
  8. Cheng, Cheng; Daubechies, Ingrid; Dym, Nadav; Lu, Jianfeng: Stable phase retrieval from locally stable and conditionally connected measurements (2021)
  9. Cheng, Cheng; Sun, Qiyu: Stable phaseless sampling and reconstruction of real-valued signals with finite rate of innovation (2021)
  10. Chen, Xuemei; Hardin, Douglas P.; Saff, Edward B.: On the search for tight frames of low coherence (2021)
  11. Cosse, Augustin; Demanet, Laurent: Stable rank-one matrix completion is solved by the level (2) Lasserre relaxation (2021)
  12. Ding, Lijun; Udell, Madeleine: On the simplicity and conditioning of low rank semidefinite programs (2021)
  13. Ding, Lijun; Yurtsever, Alp; Cevher, Volkan; Tropp, Joel A.; Udell, Madeleine: An optimal-storage approach to semidefinite programming using approximate complementarity (2021)
  14. Fawzi, Hamza: On polyhedral approximations of the positive semidefinite cone (2021)
  15. Gao, Bing; Liu, Haixia; Wang, Yang: Phase retrieval for sub-Gaussian measurements (2021)
  16. 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)
  17. Hieu Thao, Nguyen; Soloviev, Oleg; Luke, Russell; Verhaegen, Michel: Projection methods for high numerical aperture phase retrieval (2021)
  18. Huang, Meng; Rong, Yi; Wang, Yang; Xu, Zhiqiang: Almost everywhere generalized phase retrieval (2021)
  19. Huang, Meng; Xu, Zhiqiang: Phase retrieval from the norms of affine transformations (2021)
  20. Iwen, Mark A.; Krahmer, Felix; Krause-Solberg, Sara; Maly, Johannes: On recovery guarantees for one-bit compressed sensing on manifolds (2021)

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