Phase Retrieval via Matrix Completion

被引:340
作者
Candes, Emmanuel J. [1 ,2 ]
Eldar, Yonina C. [3 ]
Strohmer, Thomas [4 ]
Voroninski, Vladislav [5 ]
机构
[1] Stanford Univ, Dept Math, Stanford, CA 94305 USA
[2] Stanford Univ, Dept Stat, Stanford, CA 94305 USA
[3] Technion Israel Inst Technol, Dept Elect Engn, IL-32000 Haifa, Israel
[4] Univ Calif Davis, Dept Math, Davis, CA 95616 USA
[5] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA
来源
SIAM JOURNAL ON IMAGING SCIENCES | 2013年 / 6卷 / 01期
基金
美国国家科学基金会;
关键词
diffraction; Fourier transform; convex optimization; trace-norm minimization; X-RAY CRYSTALLOGRAPHY; FOURIER-TRANSFORM; RANK MINIMIZATION; RECONSTRUCTION; CONVEX; MAGNITUDE; ALGORITHM; OPTIMIZATION; NANOSCALE; IMAGE;
D O I
10.1137/110848074
中图分类号
TP18 [人工智能理论];
学科分类号
081104 ; 0812 ; 0835 ; 1405 ;
摘要
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.
引用
收藏
页码:199 / 225
页数:27
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