Vectorial Phase Retrieval of 1-D Signals

Raz, Oren; Dudovich, Nirit; Nadler, Boaz

doi:10.1109/tsp.2013.2239994

Cited by 32 publications

(47 citation statements)

References 39 publications

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“…The case of the windowed Fourier transform was considered in the '80s [29]; see also [5] for a frame based approach to this case. A different approach is taken in [33]. The authors propose a novel algorithm adapted to compactly supported signals and FFT that uses individual signal spectral powers and two additional interferences between signals.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Bălan

2015

Found Comput Math

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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional 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 as an iterative algorithm that finds the least-square solution which is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer-Rao lower bound.

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Section: Introductionmentioning

confidence: 99%

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Bălan

2015

Found Comput Math

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“…(ii) In [21], [22], the authors propose a framework called vectorial phase retrieval (VPR). Mathematically, the framework proposed in this section is equivalent to VPR.…”

Section: A Phase Retrievalmentioning

confidence: 99%

Reconstruction of Signals From Their Autocorrelation and Cross-Correlation Vectors, With Applications to Phase Retrieval and Blind Channel Estimation

Hassibi

2019

IEEE Trans. Signal Process.

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We consider the problem of reconstructing two signals from the autocorrelation and cross-correlation measurements. This inverse problem is a fundamental one in signal processing, and arises in many applications, including phase retrieval and blind channel estimation. In a typical phase retrieval setup, only the autocorrelation measurements are obtainable. We show that, when the measurements are obtained using three simple "masks", phase retrieval reduces to the aforementioned reconstruction problem.The classic solution to this problem is based on finding common factors between the z-transforms of the autocorrelation and cross-correlation vectors. This solution has enjoyed limited practical success, mainly due to the fact that it is not sufficiently stable in the noisy setting. In this work, inspired by the success of convex programming in provably and stably solving various quadratic constrained problems, we develop a semidefinite programming-based algorithm and provide theoretical guarantees. In particular, we show that almost all signals can be uniquely recovered by this algorithm (up to a global phase). Comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting.

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“…in [14] it knows the information about the spectrum of the original signal |x| 2 , inference signal y and |x +ŷ| 2 . In [15] [16], |x| 2 , |x +ŷ| 2 and |ŷ| 2 are given. Phase retrieval with background information can be regarded as one of the special type of utilizing the inference signal y.…”

Section: A)mentioning

confidence: 99%

“…Utilizing the redundant property of STFT frames, the solution of (1) can be unique with proper constraints about the quantities or the bandwidth of the frames. Except for the multiple measurements, the information of the inference signal can also be used to guarantee the uniqueness of phase retrieval in [14][15] [16]. Interested readers can refer to [17] which has a comprehensive review about the method to guarantee the uniqueness solution of phase retrieval.…”

Section: Introductionmentioning

confidence: 99%

Phase retrieval with background information

Yuan

Wang

2019

Inverse Problems

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Phase retrieval problem has been studied in various applications. It is an inverse problem without the standard uniqueness guarantee. To make complete theoretical analyses and devise efficient algorithms to recover the signal is sophisticated. In this paper, we come up with a model called phase retrieval with background information which recovers the signal with the known background information from the intensity of their combinational Fourier transform spectrum. We prove that the uniqueness of phase retrieval can be guaranteed even considering those trivial solutions when the background information is sufficient. Under this condition, we construct a loss function and utilize the projected gradient descent method to search for the ground truth. We prove that the stationary point is the global optimum with probability 1. Numerical simulations demonstrate the projected gradient descent method performs well both for 1-D and 2-D signals. Furthermore, this method is quite robust to the Gaussian noise and the bias of the background information.

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Vectorial Phase Retrieval of 1-D Signals

Cited by 32 publications

References 39 publications

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

Reconstruction of Signals From Their Autocorrelation and Cross-Correlation Vectors, With Applications to Phase Retrieval and Blind Channel Estimation

Phase retrieval with background information

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