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133
Compressive sensing
 IEEE Signal Processing Mag
, 2007
"... The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too m ..."
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Cited by 299 (40 self)
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The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too many samples and must compress in order to store or transmit them. In other applications, including imaging systems (medical scanners, radars) and highspeed analogtodigital converters, increasing the sampling rate or density beyond the current stateoftheart is very expensive. In this lecture, we will learn about a new technique that tackles these issues using compressive sensing [1, 2]. We will replace the conventional sampling and reconstruction operations with a more general linear measurement scheme coupled with an optimization in order to acquire certain kinds of signals at a rate significantly below Nyquist. 2
Algorithms for simultaneous sparse approximation. Part II: Convex relaxation
, 2004
"... Abstract. A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals th ..."
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Cited by 205 (3 self)
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Abstract. A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals that participate. These elementary signals typically model coherent structures in the input signals, and they are chosen from a large, linearly dependent collection. The first part of this paper proposes a greedy pursuit algorithm, called Simultaneous Orthogonal Matching Pursuit, for simultaneous sparse approximation. Then it presents some numerical experiments that demonstrate how a sparse model for the input signals can be identified more reliably given several input signals. Afterward, the paper proves that the SOMP algorithm can compute provably good solutions to several simultaneous sparse approximation problems. The second part of the paper develops another algorithmic approach called convex relaxation, and it provides theoretical results on the performance of convex relaxation for simultaneous sparse approximation. Date: Typeset on March 17, 2005. Key words and phrases. Greedy algorithms, Orthogonal Matching Pursuit, multiple measurement vectors, simultaneous
Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
, 2006
"... Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our pr ..."
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Cited by 168 (20 self)
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Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the sth stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a speciallychosen threshold, solves a leastsquares problem using the selected coordinates, and subtracts the leastsquares fit, producing a new residual. After a fixed number of stages (e.g. 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving largescale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a ksparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)
Compressed sensing and best kterm approximation
 J. Amer. Math. Soc
, 2009
"... Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements to be taken from signals while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and app ..."
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Cited by 146 (11 self)
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Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements to be taken from signals while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and approximation theory by Kashin [23] but were recently brought into the forefront by the work of Candès, Romberg and Tao [7, 5, 6] and Donoho [9] who constructed concrete algorithms and showed their promise in application. There remain several fundamental questions on both the theoretical and practical side of compressed sensing. This paper is primarily concerned about one of these theoretical issues revolving around just how well compressed sensing can approximate a given signal from a given budget of fixed linear measurements, as compared to adaptive linear measurements. More precisely, we consider discrete signals x ∈ IR N, allocate n < N linear measurements of x, and we describe the range of k for which these measurements encode enough information to recover x in the sense of ℓp to the accuracy of best kterm approximation. We also consider the problem of having such accuracy only with high probability.
Iterative hard thresholding for compressed sensing
 Appl. Comp. Harm. Anal
"... Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery probl ..."
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Cited by 136 (13 self)
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Compressed sensing is a technique to sample compressible signals below the Nyquist rate, whilst still allowing near optimal reconstruction of the signal. In this paper we present a theoretical analysis of the iterative hard thresholding algorithm when applied to the compressed sensing recovery problem. We show that the algorithm has the following properties (made more precise in the main text of the paper) • It gives nearoptimal error guarantees. • It is robust to observation noise. • It succeeds with a minimum number of observations. • It can be used with any sampling operator for which the operator and its adjoint can be computed. • The memory requirement is linear in the problem size. Preprint submitted to Elsevier 28 January 2009 • Its computational complexity per iteration is of the same order as the application of the measurement operator or its adjoint. • It requires a fixed number of iterations depending only on the logarithm of a form of signal to noise ratio of the signal. • Its performance guarantees are uniform in that they only depend on properties of the sampling operator and signal sparsity.
Bayesian Compressive Sensing
, 2007
"... The data of interest are assumed to be represented as Ndimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number M ≪ N of basisfunction coefficients associated with B. Compressive sensing ..."
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Cited by 132 (15 self)
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The data of interest are assumed to be represented as Ndimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number M ≪ N of basisfunction coefficients associated with B. Compressive sensing is a framework whereby one does not measure one of the aforementioned Ndimensional signals directly, but rather a set of related measurements, with the new measurements a linear combination of the original underlying Ndimensional signal. The number of required compressivesensing measurements is typically much smaller than N, offering the potential to simplify the sensing system. Let f denote the unknown underlying Ndimensional signal, and g a vector of compressivesensing measurements, then one may approximate f accurately by utilizing knowledge of the (underdetermined) linear relationship between f and g, in addition to knowledge of the fact that f is compressible in B. In this paper we employ a Bayesian formalism for estimating the underlying signal f based on compressivesensing measurements g. The proposed framework has the following properties: (i) in addition to estimating the underlying signal f, “error bars ” are also estimated, these giving a measure of confidence in the inverted signal; (ii) using knowledge of the error bars, a principled means is provided for determining when a sufficient
Sparsity and Incoherence in Compressive Sampling
, 2006
"... We consider the problem of reconstructing a sparse signal x 0 ∈ R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0, where U is an orthonormal matrix, we show that ℓ1 minimization recovers x 0 exactly when the number of measurements exceeds m ≥ Const · µ 2 (U) ..."
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Cited by 123 (9 self)
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We consider the problem of reconstructing a sparse signal x 0 ∈ R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0, where U is an orthonormal matrix, we show that ℓ1 minimization recovers x 0 exactly when the number of measurements exceeds m ≥ Const · µ 2 (U) · S · log n, where S is the number of nonzero components in x 0, and µ is the largest entry in U properly normalized: µ(U) = √ n · maxk,j Uk,j. The smaller µ, the fewer samples needed. The result holds for “most ” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0 for each nonzero entry on T and the observed values of Ux 0 are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples.
Distributed compressed sensing
, 2005
"... Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algori ..."
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Cited by 84 (21 self)
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Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algorithms for multisignal ensembles that exploit both intra and intersignal correlation structures. The DCS theory rests on a new concept that we term the joint sparsity of a signal ensemble. We study in detail three simple models for jointly sparse signals, propose algorithms for joint recovery of multiple signals from incoherent projections, and characterize theoretically and empirically the number of measurements per sensor required for accurate reconstruction. We establish a parallel with the SlepianWolf theorem from information theory and establish upper and lower bounds on the measurement rates required for encoding jointly sparse signals. In two of our three models, the results are asymptotically bestpossible, meaning that both the upper and lower bounds match the performance of our practical algorithms. Moreover, simulations indicate that the asymptotics take effect with just a moderate number of signals. In some sense DCS is a framework for distributed compression of sources with memory, which has remained a challenging problem for some time. DCS is immediately applicable to a range of problems in sensor networks and arrays.
Counting faces of randomlyprojected polytopes when the projection radically lowers dimension
 J. of the AMS
, 2009
"... 1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random highdimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications in statistics, probability, information theory, and signal processing with ..."
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Cited by 76 (6 self)
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1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random highdimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications in statistics, probability, information theory, and signal processing with potential impacts in
Compressed Sensing and Redundant Dictionaries
"... This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry con ..."
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Cited by 74 (12 self)
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This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.