The phase retrieval problem

Klibanov, Michael V.; Sacks, Paul; Tikhonravov, Alexander V.

doi:10.1088/0266-5611/11/1/001

Cited by 147 publications

(119 citation statements)

References 21 publications

(11 reference statements)

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“…We note that phase retrieval has seen a resurgence in activity in recent years, fueled on the one hand by the desire to image individual molecules and other nanoparticles, and on the other hand by new imaging capabilities: one such recent modality is the availability of new X-ray synchrotron sources that provide extraordinary X-ray fluxes, see for example [9,21,49,53,62]. References and various instances of the phase retrieval problem as well as some theoretical and numerical solutions can be found in [35,40,44].…”

Section: The Phase Retrieval Problemmentioning

confidence: 99%

Phase Retrieval via Matrix Completion

Candès

Eldar²,

Strohmer

et al. 2013

SIAM J. Imaging Sci.

659

850

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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 any 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.

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Section: The Phase Retrieval Problemmentioning

confidence: 99%

Phase Retrieval via Matrix Completion

Candès

Eldar²,

Strohmer

et al. 2013

SIAM J. Imaging Sci.

659

850

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“…its bandwidth is limited to B sampled at N frequency points. This leads us to the conclusion that 2 N valid solutions exist with N sampling points for one symbol (Klibanov et al, 1995). Therefore, Eq.…”

Section: Consequences For the Signal Recoverymentioning

confidence: 91%

Orthogonal frequency division multiplexing with amplitude shift keying subcarrier modulation as a reliable and efficient transmission scheme for self-mixing receivers

2017

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Abstract. A new receiving scheme for self-mixing receivers is presented that overcomes the disadvantages of the selfheterodyne concept. Generally speaking, the self-mixing receiver offers immunity to phase noise and frequency offsets, especially at very high frequencies, since it does not require radio frequency local oscillators. Our proposed technique eliminates the drawbacks of the self-heterodyne transmission scheme, which are the poor power efficiency and the strong dependence on the continously transmitted carrier.A nonlinear system of equations is constructed that describes a phase retrieval problem for the reconstruction of the original transmit signal before self-mixing. Two different solution strategies, with restrictions in time and frequency domain, are presented. As a consequence, the self-mixing equation system is shown to be solvable with some a-priori information about the transmit signal. With this novel approach, the transmitted information is distributed over the full available bandwidth, and there is no special dependence on a certain subcarrier for the down-conversion.The general performance, regarding bit error ratio over signal to noise ratio, is improved by at least 2 dB as compared to the self-heterodyne transmission scheme. In the case of frequency selective channels, e.g. multi-path propagation, this improvement is shown to be much larger, because the presented approach is able to reconstruct the received subcarriers without the necessity of receiving all subcarriers.

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“…In the classical compressed sensing, we are interested in finding the sparsest solution to the system of equations from its linear measurements . Nowadays, nonlinear inverse problems have attracted much attention [2], one kind of which is phase retrieval problem [3]. In the sampling of this problem, the magnitude of its measurement is known, while the symbol of its measurement is unknown.…”

Section: Introductionmentioning

confidence: 99%

Phase Retrieval via Iterative Hard Thresholding Backtracking Algorithm

Zhu¹,

Wang²

2016

dtetr

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Abstract. In this paper, we study the problem of phase retrieval via an improved iterative hard thresholding (IHT) algorithm, which is called the backtracking-based iterative hard thresholding (BIHT) algorithm. By adding the idea of backtracking, BIHT optimizes the selection of the support in the iterative process of the algorithm. The IHT algorithm should start with a careful initialization, but the BIHT does not need to meet this requirement. In addition, BIHT overcomes the shortcoming of the instability of IHT, and improves the calculation time and accuracy. Under the reasonable assumption, we illustrate the effectiveness of the BIHT algorithm by several different experiments.

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The phase retrieval problem

Cited by 147 publications

References 21 publications

Phase Retrieval via Matrix Completion

Phase Retrieval via Matrix Completion

Orthogonal frequency division multiplexing with amplitude shift keying subcarrier modulation as a reliable and efficient transmission scheme for self-mixing receivers

Phase Retrieval via Iterative Hard Thresholding Backtracking Algorithm

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