Stable Optimizationless Recovery from Phaseless Linear Measurements

Demanet, Laurent; Hand, Paul

doi:10.1007/s00041-013-9305-2

Cited by 114 publications

(164 citation statements)

References 24 publications

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“…The original requirement of m = O(n log(n)) vectors has been relaxed to m = O(n) in [15]. Similar convex optimization solutions have been proposed by the authors of [20] and [34]. Additionally, [23] studied the duality gap in this approach and obtained a necessary and sufficient condition for the existence of a dual certificate.…”

Section: Introductionmentioning

confidence: 98%

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: 98%

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

Bălan

2015

Found Comput Math

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“…Matching oscillations pointwise generically runs into the cycle-skipping problem. Lack of convexity is also found in phase-retrieval problems (Demanet and Hand, 2014;Gholami, 2014), although in a less severe form.…”

Section: Methods Cost Function and Its Gradientmentioning

confidence: 88%

Phase and amplitude tracking for seismic event separation

Demanet

2015

GEOPHYSICS

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We have developed a method to decompose seismic records into atomic events, each defined by a smooth phase function and a smooth amplitude function. This decomposition is intrinsically nonlinear and calls for a nonconvex least-squares optimization formulation, along the lines of full-waveform inversion. To overcome the lack of convexity, we have developed an iterative refinement-expansion scheme to initialize and track the phase and amplitude for each atomic event. For short, we called the method phase tracking. The initialization is carried out by applying multiple signal classification to a few seed traces in which events can be separated and identified by their arrival times and amplitudes. We then construct the initial solution at the seed traces using linear phase functions from the arrival times and constant amplitude functions, assuming the medium is mostly dispersion free. We refine this initial solution to account for dispersion and imperfect knowledge of the wavelet at the seed traces by fitting the observed data using a gradient descent method. The resulting phase and amplitude functions are then carefully expanded across the traces in an adequately smooth way to match the whole data record. We have evaluated the proposed method on two synthetic records and a field record. Because the parametrization of the seismic events is physically meaningful, it also enables a simple form of bandwidth extension of the observed shot record to unobserved low and high frequencies. We tested this procedure on the same shot records. Bandwidth extension is in principle helpful to initialize full-waveform inversion with frequency sweeps and enhanced its resolution.

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“…It also implies that we must deal with a highly underdetermined problem since for the PL procedure, there are N 2 unknowns (instead of only N ) for M = O(N ) magnitude measurements. As explained in [21], this apparent lack of data is compensated by the fact that we are only looking for rank one matrices. It is also worth noting that the computational complexity of the PC algorithm (11) is higher than the one of PL (6) since we are looking in the space of Hermitian matrices of dimension M for PC instead of only N for PL.…”

Section: Discussionmentioning

confidence: 99%

Excitation Retrieval of Microwave Linear Arrays From Phaseless Far-Field Data

Fuchs

Coq

2015

IEEE Trans. Antennas Propagat.

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Abstract-A methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed. The approach combines tools from convex optimization (to solve the phase retrieval problem) and a simple measurement procedure (to mitigate the non uniqueness of the solution). Numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed. They show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem. In addition, the proposed approach, that only calls for readily available routines, is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases.

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Stable Optimizationless Recovery from Phaseless Linear Measurements

Cited by 114 publications

References 24 publications

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

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

Phase and amplitude tracking for seismic event separation

Excitation Retrieval of Microwave Linear Arrays From Phaseless Far-Field Data

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