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49
Phase Retrieval via Matrix Completion
, 2011
"... This paper develops a novel framework for phase retrieval, a problem which arises in Xray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach, called PhaseLift, combines multiple structured illuminations together with ideas from convex programming to ..."
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Cited by 75 (11 self)
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This paper develops a novel framework for phase retrieval, a problem which arises in Xray 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 any complexvalued 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 noiseaware algorithms are stable in the sense that the reconstruction degrades gracefully as the signaltonoise 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.
Stable optimizationless recovery from phaseless linear measurements. arXiv:1208.1803
, 2012
"... We address the problem of recovering an nvector from m linear measurements lacking sign or phase information. We show that lifting and semidefinite relaxation suffice by themselves for stable recovery in the setting of m = O(n logn) random sensing vectors, with high probability. The recovery metho ..."
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Cited by 37 (2 self)
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We address the problem of recovering an nvector from m linear measurements lacking sign or phase information. We show that lifting and semidefinite relaxation suffice by themselves for stable recovery in the setting of m = O(n logn) random sensing vectors, with high probability. The recovery method is optimizationless in the sense that trace minimization in the PhaseLift procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem. The optimizationless perspective allows for a DouglasRachford numerical algorithm that is unavailable for PhaseLift. This method exhibits linear convergence with a favorable convergence rate and without any parameter tuning. Acknowledgements. The authors acknowledge generous funding from the National Science
Phase retrieval with polarization
 SIAM J. ON IMAGING SCI
, 2013
"... In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and ..."
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Cited by 22 (5 self)
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In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [14]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternatingprojectionstype procedure.
Phase Retrieval from Coded Diffraction Patterns
, 2013
"... This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the inten ..."
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Cited by 21 (5 self)
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This paper considers the question of recovering the phase of an object from intensityonly measurements, a problem which naturally appears in Xray crystallography and related disciplines. We study a physically realistic setup where one can modulate the signal of interest and then collect the intensity of its diffraction pattern, each modulation thereby producing a sort of coded diffraction pattern. We show that PhaseLift, a recent convex programming technique, recovers the phase information exactly from a number of random modulations, which is polylogarithmic in the number of unknowns. Numerical experiments with noiseless and noisy data complement our theoretical analysis and illustrate our approach.
Convex recovery of a structured signal from independent random linear measurements,” 2014, preprint: http://arxiv.org/abs/1405.1102
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A Partial Derandomization of PhaseLift using Spherical Designs
, 2013
"... ABSTRACT. The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally efficient, numerically stable, and comes with rigorous pe ..."
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Cited by 12 (4 self)
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ABSTRACT. The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally efficient, numerically stable, and comes with rigorous performance guarantees. PhaseLift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. However, it specifically demands Gaussian random measurement vectors — a limitation that restricts practical utility and obscures the specific properties of measurement ensembles that enable phase retrieval. Here we present a partial derandomization of PhaseLift that only requires sampling from certain polynomial size vector configurations, called tdesigns. Such configurations have been studied in algebraic combinatorics, coding theory, and quantum information. We prove reconstruction guarantees for a number of measurements that depends on the degree t of the design. If the degree is allowed to to grow logarithmically with the dimension, the bounds become tight up to polylogfactors. Beyond the specific case of PhaseLift, this work highlights the utility of spherical designs for the derandomization of data recovery schemes. 1.
Fast Algorithms for Signal Reconstruction without Phase
"... We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in signal processing, especially speech recognition technology, and has relevance for state tomography in quantum theory. We show that a generic frame gives reconstruction from the absolute val ..."
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Cited by 12 (1 self)
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We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in signal processing, especially speech recognition technology, and has relevance for state tomography in quantum theory. We show that a generic frame gives reconstruction from the absolute value of the frame coefficients in polynomial time. An improved efficiency of reconstruction is obtained with a family of sparse frames or frames associated with complex projective 2designs.