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Sensitivity to basis mismatch of compressed sensing,” preprint
, 2009
"... Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the c ..."
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Abstract—The theory of compressed sensing suggests that successful inversion of an image of the physical world (e.g., a radar/sonar return or a sensor array snapshot vector) for the source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in passive and active radar and sonar, this basis is usually taken to be a DFT basis. The compressed sensing measurements are then inverted using an ℓ1minimization principle (basis pursuit) for the nonzero source amplitudes. This seems to make compressed sensing an ideal image inversion principle for high resolution modal analysis. However, in reality no physical field is sparse in the DFT basis or in an apriori known basis. In fact the main goal in image inversion is to identify the modal structure. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and there is always mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of basis pursuit to mismatch between the assumed and the actual sparsity bases and compare the performance of basis pursuit with that of classical image inversion. Our mathematical analysis and numerical examples show that the performance of basis pursuit degrades considerably in the presence of mismatch, and they suggest that the use of compressed sensing as a modal analysis principle requires more consideration and refinement, at least for the problem sizes common to radar/sonar. I.
Structured compressed sensing: From theory to applications
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard ..."
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Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuoustime signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
COMPRESSED REMOTE SENSING OF SPARSE OBJECTS
"... Abstract. The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a logsq ..."
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Abstract. The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a logsquare factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The prediction of theorems are confirmed by numerical simulations. 1.
Group Testing with Probabilistic Tests: Theory, Design and Application
"... Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a class ..."
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Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design nonadaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items.
Y Eldar, Noise folding in compressed sensing
 IEEE Signal Process. Lett
, 2011
"... Abstract—The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show th ..."
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Abstract—The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show that, for the vast majority of measurement schemes employed in compressed sensing, the two models are equivalent with the important difference that the signaltonoise ratio (SNR) is divided by a factor proportional to,whereis the dimension of the signal and is the number of observations. Since is often large, this leads to noise folding which can have a severe impact on the SNR. Index Terms—Analog noise versus digital noise, compressed sensing, matching pursuit, noise folding, sparse signals. I.
Robustly Stable Signal Recovery in Compressed Sensing with Structured Matrix Perturbation
, 2011
"... The sparse signal recovery in standard compressed sensing (CS) requires that the sensing matrix be known a priori. The CS problem subject to perturbation in the sensing matrix is often encountered in practice and results in the literature have shown that the signal recovery error grows linearly with ..."
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The sparse signal recovery in standard compressed sensing (CS) requires that the sensing matrix be known a priori. The CS problem subject to perturbation in the sensing matrix is often encountered in practice and results in the literature have shown that the signal recovery error grows linearly with the perturbation level. This paper assumes a structure for the perturbation. Under mild conditions on the perturbed sensing matrix, it is shown that a sparse signal can be recovered by 1 minimization with the recovery error being at most proportional to the measurement noise level, similar to that in the standard CS. The recovery is exact in the special noise free case provided that the signal is sufficiently sparse with respect to the perturbation level. A similar result holds for compressible signals under an additional assumption of small perturbation. Algorithms are proposed for implementing the 1 minimization problem and numerical simulations are carried out that verify our analysis.
Mixed operators in compressed sensing
 in Proc. 44th Ann. Conf. Inf. Sci. Syst. (CISS
, 2010
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A Bayesian perspective on sparse regularization for STAP postprocessing
 Proc. IEEE Radar Conf., 1471–1475
, 2010
"... formulations cast the problem as a detection task which results in an optimal decision statistic for a single target in colored Gaussian noise. In the present work, inspired by recent theoretical and algorithmic advances in the field known as compressed sensing, we impose a Laplacian prior on the t ..."
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formulations cast the problem as a detection task which results in an optimal decision statistic for a single target in colored Gaussian noise. In the present work, inspired by recent theoretical and algorithmic advances in the field known as compressed sensing, we impose a Laplacian prior on the targets themselves which encourages sparsity in the resulting reconstruction of the angle/Doppler plane. By casting the problem in a Bayesian framework, it becomes readily apparent that sparse regularization can be applied as a postprocessing step after the use of a traditional STAP algorithm for clutter estimation. Simulation results demonstrate that this approach allows closely spaced targets to be more easily distinguished. I.
COMPRESSIVE RADAR WITH OFFGRID TARGETS: A PERTURBATION APPROACH
"... Abstract. Compressed sensing (CS) schemes are proposed for monostatic as well as synthetic aperture radar (SAR) imaging with chirped signals and UltraNarrowband (UNB) continuous waveforms. In particular, a simple, perturbation method is developed to reduce the gridding error for offgrid targets. A ..."
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Abstract. Compressed sensing (CS) schemes are proposed for monostatic as well as synthetic aperture radar (SAR) imaging with chirped signals and UltraNarrowband (UNB) continuous waveforms. In particular, a simple, perturbation method is developed to reduce the gridding error for offgrid targets. A coherence bound is obtained for the resulting measurement matrix. A greedy pursuit algorithm, SupportConstrained Orthogonal Matching Pursuit (SCOMP), is proposed to take advantage of the support constraint in the perturbation formulation and proved to have the capacity of determining the offgrid targets to the grid accuracy under favorable conditions. Alternatively, the Locally Optimized Thresholding (LOT) is proposed to enhance the performance of the CS method, Basis Pursuit (BP). For the advantages of higher signaltonoise ratio and signaltointerference ratio, it is proposed that Spotlight SAR imaging be implemented with CS techniques and multifrequency UNB waveforms. Numerical simulations show promising results of the proposed approach and algorithms. 1.
Compressive Sensing under Matrix Uncertainties: An Approximate Message Passing Approach
"... Abstract—In this work, we consider a general form of noisy compressive sensing (CS) when there is uncertainty in the measurement matrix as well as in the measurements. Matrix uncertainty is motivated by practical cases in which there are imperfections or unknown calibration parameters in the signal ..."
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Abstract—In this work, we consider a general form of noisy compressive sensing (CS) when there is uncertainty in the measurement matrix as well as in the measurements. Matrix uncertainty is motivated by practical cases in which there are imperfections or unknown calibration parameters in the signal acquisition hardware. While previous work has focused on analyzing and extending classical CS algorithms like the LASSO and Dantzig selector for this problem setting, we propose a new algorithm whose goal is either minimization of meansquared error or maximization of posterior probability in the presence of these uncertainties. In particular, we extend the Approximate Message Passing (AMP) approach originally proposed by Donoho, Maleki, and Montanari, and recently generalized by Rangan, to the case of probabilistic uncertainties in the elements of the measurement matrix. Empirically, we show that our approach performs near oracle bounds. We then show that our matrixuncertain AMP can be applied in an alternating fashion to learn both the unknown measurement matrix and signal vector. We also present a simple analysis showing that, for suitably large systems, it suffices to treat uniform matrix uncertainty as additive white Gaussian noise. I.