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Compressive Phase Retrieval via Generalized Approximate Message Passing
"... Abstract—In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PRGAMP algorithm has excellent phasetransition behavior ..."
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Cited by 30 (8 self)
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Abstract—In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PRGAMP algorithm has excellent phasetransition behavior, noise robustness, and runtime. In particular, for successful recovery of synthetic BernoullicircularGaussian signals, PRGAMP requires ≈ 4 times the number of measurements as a phaseoracle version of GAMP and, at moderate to large SNR, the NMSE of PRGAMP is only ≈ 3 dB worse than that of phaseoracle GAMP. A comparison to the recently proposed convexrelation approach known as “CPRL ” reveals PRGAMP’s superior phase transition and ordersofmagnitude faster runtimes, especially as the problem dimensions increase. When applied to the recovery of a 65kpixel grayscale image from 32k randomly masked magnitude measurements, numerical results show a median PRGAMP runtime of only 13.4 seconds. A. Phase retrieval I.
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.
An algebraic characterization of injectivity in phase retrieval,” arXiv:1312.0158
, 2013
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Determination of all pure quantum states from a minimal number of observables
, 2014
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Reconstruction of signals from magnitudes of redundant representations
"... Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm ..."
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Cited by 12 (6 self)
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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm that finds the leastsquare solution and is robust in the presence of noise. We analyze its numerical performance by comparing it to two versions of the CramerRao lower bound. 1.
Phase retrieval from very few measurements
, 2013
"... In many applications, signals are measured according to a linear process, but the phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as phase retrieval. This paper focuses on completely determining signals with as few inte ..."
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Cited by 4 (1 self)
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In many applications, signals are measured according to a linear process, but the phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as phase retrieval. This paper focuses on completely determining signals with as few intensity measurements as possible, and on efficient phase retrieval algorithms from such measurements. For the case of complex Mdimensional signals, we construct a measurement ensemble of size 4M − 4 which yields injective intensity measurements; this is conjectured to be the smallest such ensemble. For the case of real signals, we devise a theory of “almost” injective intensity measurements, and we characterize such ensembles. Later, we show that phase retrieval from M + 1 almost injective intensity measurements is NPhard, indicating that computationally efficient phase retrieval must come at the price of measurement redundancy.
Phasecode: Fast and efficient compressive phase retrieval based on sparsegraphcodes,” arXiv preprint arXiv:1408.0034
, 2014
"... We consider the problem of recovering a complex signal x ∈ Cn from m intensity measurements of the form aix, 1 ≤ i ≤ m, where ai is a measurement row vector. We address multiple settings corresponding to whether the measurement vectors are unconstrained choices or not, and to whether the signal to ..."
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Cited by 2 (0 self)
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We consider the problem of recovering a complex signal x ∈ Cn from m intensity measurements of the form aix, 1 ≤ i ≤ m, where ai is a measurement row vector. We address multiple settings corresponding to whether the measurement vectors are unconstrained choices or not, and to whether the signal to be recovered is sparse or not. However, our main focus is on the case where the measurement vectors are unconstrained, and where x is exactly Ksparse, or the socalled general compressive phaseretrieval problem. We introduce PhaseCode, a novel family of fast and efficient mergeandcolor algorithms (that includes Unicolor PhaseCode and Multicolor PhaseCode) that are based on a sparsegraphcodes framework. As one instance, our Unicolor PhaseCode algorithm can provably recover, with high probability, all but a tiny 10−7 fraction of the significant signal components, using at most m = 14K measurements, which is a small constant factor from the fundamental limit, with an optimal O(K) decoding time and an optimal O(K) memory complexity. Next, motivated by some important practical classes of optical systems, we consider a “Fourierfriendly ” constrained measurement setting, and show that its performance matches that of the unconstrained setting. In the Fourierfriendly setting that we consider, the
Stability of Phase Retrievable Frames
 proceedings of SPIE 2013
"... In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then th ..."
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In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound ρ so that any frame set within ρ from F has the same property. In particular this proves the recent construction in15 is stable under perturbations. By the same token we reduce the critical cardinality conjectured in11 to proving a stability result for non phaseretrievable frames.