Results 1  10
of
58
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinato ..."
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Cited by 202 (31 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
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Cited by 59 (13 self)
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These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
Highly sparse representations from dictionaries are unique and independent of the sparseness measure
, 2003
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Blocksparse signals: Uncertainty relations and efficient recovery
 IEEE Trans. Signal Process
, 2010
"... Abstract—We consider efficient methods for the recovery of blocksparse signals—i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, wh ..."
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Cited by 51 (13 self)
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Abstract—We consider efficient methods for the recovery of blocksparse signals—i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, which we introduce. We then show that a blockversion of the orthogonal matching pursuit algorithm recovers block ksparse signals in no more than k steps if the blockcoherence is sufficiently small. The same condition on blockcoherence is shown to guarantee successful recovery through a mixed `2=`1optimization approach. This complements previous recovery results for the blocksparse case which relied on small blockrestricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of blocksparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem. Index Terms—Basis pursuit, blocksparsity, compressed sensing, matching pursuit. I.
Stability results for random sampling of sparse trigonometric polynomials
, 2006
"... Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that ..."
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Cited by 49 (17 self)
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Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that recovery both by a BP variant and by OMP is stable under perturbation of the samples values by noise. For BP in addition, the stability result is extended to (nonsparse) trigonometric polynomials that can be wellapproximated by sparse ones. The theoretical findings are illustrated by numerical experiments. Key Words: random sampling, trigonometric polynomials, Orthogonal Matching Pursuit, Basis Pursuit, compressed sensing, stability under noise, fast Fourier transform, nonequispaced
Random sampling of sparse trigonometric polynomials
 Appl. Comput. Harm. Anal
, 2006
"... We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, ..."
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Cited by 41 (18 self)
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We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
A simple test to check the optimality of sparse signal approximations
 Tech. Rep., IRISA, 2004, in preparation. V  720
"... Approximating a signal or an image with a sparse linear expansion from an overcomplete dictionary of atoms is an extremely useful tool to solve many signal processing problems. Finding the sparsest approximation of a signal from an arbitrary dictionary is an NPhard problem. Despite of this, several ..."
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Cited by 30 (2 self)
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Approximating a signal or an image with a sparse linear expansion from an overcomplete dictionary of atoms is an extremely useful tool to solve many signal processing problems. Finding the sparsest approximation of a signal from an arbitrary dictionary is an NPhard problem. Despite of this, several algorithms have been proposed that provide suboptimal solutions. However, it is generally difficult to know how close the computed solution is to being “optimal”, and whether another algorithm could provide a better result. In this paper we provide a simple test to check whether the output of a sparse approximation algorithm is nearly optimal, in the sense that no significantly different linear expansion from the dictionary can provide both a smaller approximation error and a better sparsity. As a byproduct of our theorems, we obtain results on the identifiability of sparse overcomplete models in the presence of noise, for a fairly large class of sparse priors. 1.
Denoising by Sparse Approximation: Error Bounds Based on RateDistortion Theory
, 2006
"... If a signal x is known to have a sparse representation with respect to a frame, it can be estimated from a noisecorrupted observation y by finding the best sparse approximation to y. Removing noise in this manner depends on the frame efficiently representing the signal while it inefficiently repres ..."
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Cited by 30 (6 self)
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If a signal x is known to have a sparse representation with respect to a frame, it can be estimated from a noisecorrupted observation y by finding the best sparse approximation to y. Removing noise in this manner depends on the frame efficiently representing the signal while it inefficiently represents the noise. The meansquared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal are analyzed. First an MSE bound that depends on a new bound on approximating a Gaussian signal as a linear combination of elements of an overcomplete dictionary is given. Further analyses are for dictionaries generated randomly according to a sphericallysymmetric distribution and signals expressible with single dictionary elements. Easilycomputed approximations for the probability of selecting the correct dictionary element and the MSE are given. Asymptotic expressions reveal a critical input signaltonoise ratio for signal recovery.
Compressed Sensing of BlockSparse Signals: Uncertainty Relations and Efficient Recovery
, 2009
"... We consider compressed sensing of blocksparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, which we introduce. We then show that a blockversion of the orthogonal ..."
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Cited by 29 (10 self)
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We consider compressed sensing of blocksparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, which we introduce. We then show that a blockversion of the orthogonal matching pursuit algorithm recovers block ksparse signals in no more than k steps if the blockcoherence is sufficiently small. The same condition on blockcoherence is shown to guarantee successful recovery through a mixed ℓ2/ℓ1optimization approach. This complements previous recovery results for the blocksparse case which relied on small blockrestricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of blocksparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.