Phase Retrieval from Gabor Measurements

Bojarovska, Irena; Flinth, Axel

doi:10.1007/s00041-015-9431-0

Cited by 50 publications

(45 citation statements)

References 27 publications

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“…In this case, the magnitude of the STFT determines the underlying signal uniquely under mild conditions. This conclusion was already derived in [11] based on different considerations. Nevertheless, the following proposition comes with an explicit recovery scheme as presented in Appendix A.…”

Section: Uniqueness and Basic Algorithmssupporting

confidence: 52%

Non-Convex Phase Retrieval From STFT Measurements

Bendory

Eldar

Boumal

2018

IEEE Trans. Inform. Theory

101

113

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Abstract-The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise.

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Section: Uniqueness and Basic Algorithmssupporting

confidence: 52%

Non-Convex Phase Retrieval From STFT Measurements

Bendory

Eldar

Boumal

2018

IEEE Trans. Inform. Theory

101

113

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show abstract

“…This case resembles the FROG setup, where a known reference window replaces the unknown shifted signal. Several works derived uniqueness results for this case under different conditions [15,17,18,19]. In [16] it was shown that, roughly speaking, it is sufficient to set L ≈ N/2 (namely, 2N measurements) to determine almost all non-vanishing signals.…”

Section: Frog Recoverymentioning

confidence: 99%

On signal reconstruction from FROG measurements

Bendory

Edidin

Eldar

2020

Applied and Computational Harmonic Analysis

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Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the illposedness, it is common to use a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaseless quartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities. In practice, one usually also has access to the signal's Fourier magnitude. We show that in this case only two translations suffice. Our results significantly improve upon earlier work.

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“…It is now known that by choosing windows with appropriate properties, only a small redundancy in the measurements (i.e. short windows) is needed to enforce uniqueness (for details, see [14], [15], [16], [17], [10]). The STFT of a 1D signal x ∈ C N with respect to a real sliding window g of length W is defined as where k = 0, .…”

Section: Introductionmentioning

confidence: 99%

Phase retrieval from STFT measurements via non-convex optimization

Bendory¹,

Eldar²

2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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The problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements arises in several applications, such as ultra-short pulse measurements and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases, the principle eigenvector of a designed matrix recovers the underlying signal. This matrix is constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest to use this principle eigenvector to initialize a gradient algorithm, minimizing a non-convex loss function. We prove that under appropriate conditions, this initialization results in a good estimate of the underlying signal. We further analyze the geometry of the loss function and show empirically that the gradient algorithm is robust to noise. Our method is both efficient and enjoys theoretical guarantees.

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Phase Retrieval from Gabor Measurements

Cited by 50 publications

References 27 publications

Non-Convex Phase Retrieval From STFT Measurements

Non-Convex Phase Retrieval From STFT Measurements

On signal reconstruction from FROG measurements

Phase retrieval from STFT measurements via non-convex optimization

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