Super-Resolution from Noisy Data

Candès, Emmanuel J.; Fernandez‐Granda, Carlos

doi:10.1007/s00041-013-9292-3

Cited by 384 publications

(455 citation statements)

References 41 publications

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“…Another line of research that relates to this example is that of superresolution (see, e.g., [CFG14,CFG13] and many followup works). These works study the recovery of frequency sparse signals from equispaced samples.…”

Section: Our Main Results and Its Applicationsmentioning

confidence: 99%

Compressive Sensing with Redundant Dictionaries and Structured Measurements

Krahmer¹,

Needell²,

Ward³

2015

SIAM J. Math. Anal.

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ABSTRACT. Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary D. This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary, if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the D-restricted isometry property (D-RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in D. While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the D-RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary D. We also extend this analysis to allow for weighted sparse expansions. Consequently, we arrive at compressive sensing recovery guarantees for structured measurements and redundant dictionaries, opening the door to a wide array of practical applications.

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Section: Our Main Results and Its Applicationsmentioning

confidence: 99%

Compressive Sensing with Redundant Dictionaries and Structured Measurements

Krahmer¹,

Needell²,

Ward³

2015

SIAM J. Math. Anal.

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“…CS is limited by basis mismatch 17 which occurs when the DOAs do not coincide with the look directions of the angular spectrum, and by basis coherence. Solutions to basis mismatch involve for example using atomic norm and solving the dual problem 5,18 that are not addressed here. Grid refinement alleviates basis mismatch for high signal to noise ratio (SNR) at the expense of increased computational complexity.…”

Section: Introductionmentioning

confidence: 99%

Multiple and single snapshot compressive beamforming

Gerstoft

Xenaki

Mecklenbräuker

2015

The Journal of the Acoustical Society of America

206

110

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For a sound field observed on a sensor array, compressive sensing (CS) reconstructs the directionof-arrival (DOA) of multiple sources using a sparsity constraint. The DOA estimation is posed as an underdetermined problem by expressing the acoustic pressure at each sensor as a phaselagged superposition of source amplitudes at all hypothetical DOAs. Regularizing with an 1-norm constraint renders the problem solvable with convex optimization, and promoting sparsity gives highresolution DOA maps. Here, the sparse source distribution is derived using maximum a posteriori (MAP) estimates for both single and multiple snapshots. CS does not require inversion of the data covariance matrix and thus works well even for a single snapshot where it gives higher resolution than conventional beamforming. For multiple snapshots, CS outperforms conventional high-resolution methods, even with coherent arrivals and at low signal-to-noise ratio. The superior resolution of CS is demonstrated with vertical array data from the SWellEx96 experiment for coherent multi-paths.

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“…This can be modeled as a CS problem where the sensing matrix is highly coherent and the signal has a spread out support. Recent work on this problem has shown that several optimization based or greedy methods are successful in accurately recovering these types of signals [13,28,29]. Although later we will also consider spread signals, these works are fundamentally different than ours since their goal is exact reconstruction that overcomes the coherent sensing, whereas we are promoting the advantages of coherence sampling when tolerant detection is the goal.…”

Section: Related Workmentioning

confidence: 93%

Tolerant compressed sensing with partially coherent sensing matrices

Birnbaum

Eldar

Needell

2017

Wavelets and Sparsity XVII

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Most of compressed sensing (CS) theory to date is focused on incoherent sensing, that is, columns from the sensing matrix are highly uncorrelated. However, sensing systems with naturally occurring correlations arise in many applications, such as signal detection, motion detection and radar. Moreover, in these applications it is often not necessary to know the support of the signal exactly, but instead small errors in the support and signal are tolerable. Despite the abundance of work utilizing incoherent sensing matrices, for this type of tolerant recovery we suggest that coherence is actually beneficial. We promote the use of coherent sampling when tolerant support recovery is acceptable, and demonstrate its advantages empirically. In addition, we provide a first step towards theoretical analysis by considering a specific reconstruction method for selected signal classes.

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Super-Resolution from Noisy Data

Cited by 384 publications

References 41 publications

Compressive Sensing with Redundant Dictionaries and Structured Measurements

Compressive Sensing with Redundant Dictionaries and Structured Measurements

Multiple and single snapshot compressive beamforming

Tolerant compressed sensing with partially coherent sensing matrices

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