Compressive phase retrieval

Moravec, Matthew L.; Romberg, Justin; Baraniuk, Richard G.

doi:10.1117/12.736360

Cited by 119 publications

(107 citation statements)

References 34 publications

(38 reference statements)

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“…To the best of our knowledge, the first algorithm for compressive phase retrieval was proposed by Moravec et al in [12]. This approach requires knowledge of the 1 norm of the signal, making it impractical in most scenarios.…”

Section: Related Workmentioning

confidence: 99%

“…Therefore, we again end up with having a bipartite graph with left regular degree d. Assuming that f i = F + Θ(1) for all i and consequently F = Θ(K), the edge degree distribution of the right nodes does not change for large enough K and is given in (12). Therefore, the tree analysis and the density evolution equation stated in (14) remain the same, and one can essentially get all the previous results using this ensemble.…”

Section: A Ensemble Of Graphs Constructed By Chinese Remainder Theoremmentioning

confidence: 99%

“…Indeed, the problem of recovering a signal from only the magnitude of its Fourier transform has been a well-studied problem in the signal processing literature for several decades under the umbrella of phase retrieval [11]. It has recently received renewed interest in the "postcompressed-sensing" era [12]- [14], allowing for the insights from compressive sensing to be incorporated into the phase retrieval problem when the signal of interest is sparse, and the measurement matrix is unconstrained.…”

Section: Introduction a Phase Retrieval Problemmentioning

confidence: 99%

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PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

Pedarsani

Yin

Lee

et al. 2017

IEEE Trans. Inform. Theory

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Abstract-We consider the problem of recovering a complex signal x ∈ C n from m intensity measurements of the form |a H i x|, 1 ≤ i ≤ m, where a H i is the i-th row of measurement matrix A ∈ C m×n . Our main focus is on the case where the measurement vectors are unconstrained, and where x is exactly K-sparse, or the so-called general compressive phase retrieval problem. We introduce PhaseCode, a novel family of fast and efficient algorithms that are based on a sparsegraph coding framework. We show that in the noiseless case, the PhaseCode algorithm can recover an arbitrarily-close-toone fraction of the K non-zero signal components using only slightly more than 4K measurements when the support of the signal is uniformly random, with order-optimal time and memory complexity of Θ(K)1 . It is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is 4K − o(K) for the more difficult problem of recovering the signal exactly and with no assumptions on its support distribution [1], [2]. This shows that under mild relaxation of the conditions, our algorithm is the first constructive capacity-approaching compressive phase retrieval algorithm: in fact, our algorithm is also order-optimal in complexity and memory. Further, we show that for any signal x, PhaseCode can recover a random (1 − p)-fraction of the non-zero components of x with high probability, where p can be made arbitrarily close to zero, with sample complexity m = c(p)K, where c(p) is a small constant depending on p that can be precisely calculated, with optimal time and memory complexity. As a result, assuming that the non-zero components of x are lower bounded by Θ(1) and upper bounded by Θ(K γ ) for some positive constant γ < 1, we are able to provide a strong 1 guarantee for the estimated signalx as follows: x − x 1 ≤ p x 1(1 + o(1)), where p can be made arbitrarily close to zero. As one instance, the PhaseCode algorithm can provably recover, with high probability, a random 1 − 10 −7 fraction of the significant signal components, using at most m = 14K measurements.Next, motivated by some important practical classes of optical systems, we consider a "Fourier-friendly" constrained measurement setting, and show that its performance matches that of the unconstrained setting, when the signal is sparse in the Fourier domain with uniform support. In the Fourier-friendly setting that we consider, the measurement matrix is constrained to be a cascade of Fourier matrices (corresponding to optical lenses) and diagonal matrices (corresponding to diffraction mask patterns).Finally, we tackle the compressive phase retrieval problem in the presence of noise, where measurements are in the form Ramtin Pedarsani is with the ECE Department at UC Santa Barbara. email: ramtin@ece.ucsb.edu. Dong Yin and Kannan Ramchandran are with the EECS Department at UC Berkeley. email:{dongyin,kannanr}@eecs.berkeley.edu.Kangwook Lee is with the EE Department at KASIT. email: kw1jjang@kaist.ac.kr. This paper was presented in part in Allerton 2015...

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Section: Related Workmentioning

confidence: 99%

Section: A Ensemble Of Graphs Constructed By Chinese Remainder Theoremmentioning

confidence: 99%

Section: Introduction a Phase Retrieval Problemmentioning

confidence: 99%

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PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

Pedarsani

Yin

Lee

et al. 2017

IEEE Trans. Inform. Theory

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“…1 (see, e.g., [13,14,15,16,17,18,19]). In fact, the first paper about compressive phase retrieval [20] was addressing the very problem that we are trying to solve in the present paper, i.e., the recovery problem of a k-sparse complex signal x ∈ C N from Fourier intensity measurements | x[ω]| 2 . To our knowledge, the only work after [20] that directly addressed this problem is the recent paper by Pedarsani et al [7].…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the first paper about compressive phase retrieval [20] was addressing the very problem that we are trying to solve in the present paper, i.e., the recovery problem of a k-sparse complex signal x ∈ C N from Fourier intensity measurements | x[ω]| 2 . To our knowledge, the only work after [20] that directly addressed this problem is the recent paper by Pedarsani et al [7]. Based on a sparse graph codes framework, this paper provides a low complexity algorithm that achieves perfect reconstruction with very high probability using 14k measurements.…”

Section: Introductionmentioning

confidence: 99%

Fast compressive phase retrieval from Fourier measurements

Yapar

Pohl

Boche

2015

2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper considers the problem of recovering a k-sparse, N -dimensional complex signal from Fourier magnitude measurements. It proposes a Fourier optics setup such that signal recovery up to a global phase factor is possible with very high probability whenever M 4k log 2 (N/k) random Fourier intensity measurements are available. The proposed algorithm is comprised of two stages: An algebraic phase retrieval stage and a compressive sensing step subsequent to it. Simulation results are provided to demonstrate the applicability of the algorithm for noiseless and noisy scenarios.

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Coherent X‐Ray Diffraction Microscopy

Marchesini

Shapiro

2010

X‐Rays in Nanoscience

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Compressive phase retrieval

Cited by 119 publications

References 34 publications

PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

PhaseCode: Fast and Efficient Compressive Phase Retrieval Based on Sparse-Graph Codes

Fast compressive phase retrieval from Fourier measurements

Coherent X‐Ray Diffraction Microscopy

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