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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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Cited by 98 (15 self)
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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.
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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Cited by 80 (8 self)
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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.
Sparsitycognizant total leastsquares for perturbed compressive sampling
 Signal Processing, IEEE Transactions on
, 2011
"... Abstract—Solving linear regression problems based on the total leastsquares (TLS) criterion has welldocumented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However, existing TLS approaches do not account for sparsity possib ..."
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Cited by 27 (4 self)
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Abstract—Solving linear regression problems based on the total leastsquares (TLS) criterion has welldocumented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However, existing TLS approaches do not account for sparsity possibly present in the unknown vector of regression coefficients. On the other hand, sparsity is the key attribute exploited by modern compressive sampling and variable selection approaches to linear regression, which include noise in the data, but do not account for perturbations in the regression matrix. The present paper fills this gap by formulating and solving (regularized) TLS optimization problems under sparsity constraints. Nearoptimum and reducedcomplexity suboptimum sparse (S) TLS algorithms are developed to address the perturbed compressive sampling (and the related dictionary learning) challenge, when there is a mismatch between the true and adopted bases over which the unknown vector is sparse. The novel STLS schemes also allow for perturbations in the regression matrix of the leastabsolute selection and shrinkage selection operator (Lasso), and endow TLS approaches with ability to cope with sparse, underdetermined “errorsinvariables ” models. Interesting generalizations can further exploit prior knowledge on the perturbations to obtain novel weighted and structured STLS solvers. Analysis and simulations demonstrate the practical impact of STLS in calibrating the mismatch effects of contemporary gridbased approaches to cognitive radio sensing, and robust directionofarrival estimation using antenna arrays. Index Terms—Directionofarrival estimation, errorsinvariables models, sparsity, spectrum sensing, total leastsquares.
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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Cited by 26 (8 self)
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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.
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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Cited by 23 (4 self)
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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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Cited by 19 (4 self)
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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.
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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Cited by 15 (1 self)
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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.
Blind calibration for compressed sensing by convex optimization
 in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process
"... We consider the problem of calibrating a compressed sensing measurement system under the assumption that the decalibration consists in unknown gains on each measure. We focus on blind calibration, using measures performed on a few unknown (but sparse) signals. A naive formulation of this blind cal ..."
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Cited by 12 (6 self)
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We consider the problem of calibrating a compressed sensing measurement system under the assumption that the decalibration consists in unknown gains on each measure. We focus on blind calibration, using measures performed on a few unknown (but sparse) signals. A naive formulation of this blind calibration problem, using `1 minimization, is reminiscent of blind source separation and dictionary learning, which are known to be highly nonconvex and riddled with local minima. In the considered context, we show that in fact this formulation can be exactly expressed as a convex optimization problem, and can be solved using offtheshelf algorithms. Numerical simulations demonstrate the effectiveness of the approach even for highly uncalibrated measures, when a sufficient number of (unknown, but sparse) calibrating signals is provided. We observe that the success/failure of the approach seems to obey sharp phase transitions. 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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Cited by 8 (2 self)
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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.
PERTURBATIONS OF MEASUREMENT MATRICES AND DICTIONARIES IN COMPRESSED SENSING
"... Abstract. The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ℓ q min ..."
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Cited by 7 (2 self)
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Abstract. The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on ℓ q minimization are known to provide an effective signal recovery tool in this setting. This note explores conditions under which ℓ q minimization is robust to measurement noise, and stable with respect to perturbations of the sensing matrix A and the dictionary D. We propose a new condition, the D null space property, which guarantees that ℓ q minimization produces solutions that are robust and stable against perturbations of A and D. We also show that ℓ q minimization is jointly stable with respect to imprecise knowledge of the measurement matrix A and the dictionary D when A satisfies the restricted isometry property. 1.