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Generalized sampling and infinitedimensional compressed sensing
"... We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demo ..."
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Cited by 33 (20 self)
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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finitedimensional techniques are illsuited for solving a number of important problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.
Missing Samples Analysis in Signals for Applications to LEstimation and
 Compressive Sensing”, Signal Processing, Elsevier, Volume 94
, 2014
"... a b s t r a c t This paper provides statistical analysis for efficient detection of signal components when missing data samples are present. This analysis is important for both the areas of Lstatistics and compressive sensing. In both cases, few samples are available due to either noisy sample elim ..."
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Cited by 16 (2 self)
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a b s t r a c t This paper provides statistical analysis for efficient detection of signal components when missing data samples are present. This analysis is important for both the areas of Lstatistics and compressive sensing. In both cases, few samples are available due to either noisy sample elimination or random undersampling signal strategies. The analysis enables the determination of the sufficient number of observation and as such the minimum number of missing samples which still allow proper signal detection. Both single component and multicomponent signals are considered. The results are verified by computer simulations using different component frequencies and under various missingavailable samples scenarios.
Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing
, 2013
"... In this paper we bridge the substantial gap between existing compressed sensing theory and its current use in realworld applications. 1 We do so by introducing a new mathematical framework for overcoming the socalled coherence ..."
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Cited by 13 (4 self)
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In this paper we bridge the substantial gap between existing compressed sensing theory and its current use in realworld applications. 1 We do so by introducing a new mathematical framework for overcoming the socalled coherence
1 Cadzow Denoising Upgraded: A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements
, 2013
"... Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. T ..."
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Cited by 7 (4 self)
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Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed. Index Terms—Recovery of Dirac pulses, spike train, finite rate of innovation, superresolution, spectral estimation, maximum
A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements
"... We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve ..."
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We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed. Key words and phrases: Dirac pulses, spike train, finite rate of innovation, superresolution, sparse recovery, structured low rank approximation, alternating projections, Cadzow denoising
Robust Spike Train Recovery from Noisy Data by Structured Low Rank Approximation
"... Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to solve a difficult, even believed NPhard, matrix problem of structured low rank approxima ..."
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Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpassfiltered samples. We show that maximumlikelihood estimation of the unknown parameters amounts to solve a difficult, even believed NPhard, matrix problem of structured low rank approximation. We propose a new heuristic iterative optimization algorithm to solve it. Although it comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same implementation ease and speed. I. INTRODUCTION AND PROBLEM FORMULATION Reconstruction of signals lying in linear spaces, including bandlimited signals and splines, has long been the dominant