Exact reconstruction using Beurling minimal extrapolation

Castro, Yohann de; Gamboa, Fabrice

doi:10.1016/j.jmaa.2012.05.011

Cited by 155 publications

(264 citation statements)

References 19 publications

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“…Similar results are given for support detection from low Fourier coefficients [4], [23], recovery of non-uniform splines from their projection onto spaces of algebraic polynomials [8], [18] and recovery of streams of pulses [6], [9]. (see also [17]). …”

Section: Introductionsupporting

confidence: 62%

Super-Resolution on the Sphere Using Convex Optimization

Bendory

Dekel²,

Feuer

2015

IEEE Trans. Signal Process.

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This paper considers the problem of recovering an ensemble of Diracs on a sphere from its low resolution measurements. The Diracs can be located at any location on the sphere, not necessarily on a grid. We show that under a separation condition, one can recover the ensemble with high precision by a three-stage algorithm, which consists of solving a semi-definite program, root finding and least-square fitting. The algorithm's computation time depends solely on the number of measurements, and not on the required solution accuracy. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for recovery. Furthermore, in the discrete setting, we estimate the recovery error in the presence of noise as a function of the noise level and the super-resolution factor.

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Section: Introductionsupporting

confidence: 62%

Super-Resolution on the Sphere Using Convex Optimization

Bendory

Dekel²,

Feuer

2015

IEEE Trans. Signal Process.

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“…This result in its general form is given in [6] (see also [8,11]). For completeness, we provide here the proof for the case of real coefficients …”

Section: The Dual Problem Of Polynomial Interpolationmentioning

confidence: 87%

Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

2014

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In this work we consider the problem of recovering an ensemble of Diracs on the sphere from its projection onto spaces of spherical harmonics. We show that under an appropriate separation condition on the unknown locations of the Diracs, the ensemble can be recovered through Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of 'dual' interpolating polynomials and is based on [8], where the theory was developed for trigonometric polynomials. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for exact recovery.

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“…Another article about off-grid spikes was [1] by Aubel et al, who also arrived at MSF > 1 if windowed Fourier (or short-time Fourier transform) measurements are available. Furthermore, there are two works that do not require MS. De Castro and Gamboa [10] showed that K spikes can be resolved from 2K +1 Fourier samples; and with additional positive assumption of point sources, Donoho et al [14] showed that 2K noiseless measurements are sufficient to yield exact recovery of K positive spikes. In addition to these exact recovery results, errors in spike detection and noise robustness are of great interest as well.…”

Section: Definition 1 (Minimum Separation)mentioning

confidence: 99%

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

2016

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We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1, 2) of the Rayleigh length (physical resolution limit), L 1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 2 norms (L 1−2 ) and capped L 1 (CL 1 ), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global L 1 solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in finding a sparse solution satisfying the constraints more accurately.

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Exact reconstruction using Beurling minimal extrapolation

Cited by 155 publications

References 19 publications

Super-Resolution on the Sphere Using Convex Optimization

Super-Resolution on the Sphere Using Convex Optimization

Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

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