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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 137 (26 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.
Greedy solution of illposed problems: Error bounds and exact inversion
, 2009
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Learning with Structured Sparsity
"... This paper investigates a new learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature set, this concept generalizes the group sparsity idea. A gener ..."
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Cited by 127 (15 self)
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This paper investigates a new learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature set, this concept generalizes the group sparsity idea. A general theory is developed for learning with structured sparsity, based on the notion of coding complexity associated with the structure. Moreover, a structured greedy algorithm is proposed to efficiently solve the structured sparsity problem. Experiments demonstrate the advantage of structured sparsity over standard sparsity. 1.
ModifiedCS: Modifying compressive sensing for problems with partially known support
 in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2009
"... Abstract—We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known ” part of the support, denoted, may be available from prior knowledge. Alternatively, in a ..."
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Cited by 127 (33 self)
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Abstract—We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known ” part of the support, denoted, may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known ” part. The idea of our proposed solution (modifiedCS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of. We obtain sufficient conditions for exact reconstruction using modifiedCS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called regularized modifiedCS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation comparisons for both sparse and compressible signals are shown. Index Terms—Compressive sensing, modifiedCS, partially known support, prior knowledge, sparse reconstruction.
Bayesian compressive sensing via belief propagation
 IEEE Trans. Signal Processing
, 2010
"... Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can comple ..."
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Cited by 125 (19 self)
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Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform approximate Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast encoding and decoding is provided using sparse encoding matrices, which also improve BP convergence by reducing the presence of loops in the graph. To decode a lengthN signal containing K large coefficients, our CSBP decoding algorithm uses O(K log(N)) measurements and O(N log 2 (N)) computation. Finally, sparse encoding matrices and the CSBP decoding algorithm can be modified to support a variety of signal models and measurement noise. 1
Compressed Sensing: Theory and Applications
, 2012
"... Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that highdimensional signals, which allow a sparse representati ..."
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Cited by 120 (30 self)
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Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that highdimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements by using efficient algorithms. This article shall serve as an introduction to and a survey about compressed sensing. Key Words. Dimension reduction. Frames. Greedy algorithms. Illposed inverse problems. `1 minimization. Random matrices. Sparse approximation. Sparse recovery.
Signal Recovery From Incomplete and Inaccurate Measurements via Regularized Orthogonal Matching Pursuit
, 2007
"... We demonstrate a simple greedy algorithm that can reliably recover a vector v ∈ R d from incomplete and inaccurate measurements x = Φv + e. Here Φ is a N × d measurement matrix with N ≪ d, and e is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the ga ..."
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Cited by 115 (4 self)
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We demonstrate a simple greedy algorithm that can reliably recover a vector v ∈ R d from incomplete and inaccurate measurements x = Φv + e. Here Φ is a N × d measurement matrix with N ≪ d, and e is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the gap between two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix Φ that satisfies a Uniform Uncertainty Principle, ROMP recovers a signal v with O(n) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a Least Squares Problem. The noise level of the recovery is proportional to √ log n�e�2. In particular, if the error term e vanishes the reconstruction is exact. This stability result extends naturally to the very accurate recovery of approximately sparse signals.
Sensing by Random Convolution
 IEEE Int. Work. on Comp. Adv. MultiSensor Adaptive Proc., CAMPSAP
, 2007
"... Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in a ..."
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Cited by 112 (7 self)
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Abstract. This paper outlines a new framework for compressive sensing: convolution with a random waveform followed by random time domain subsampling. We show that sensing by random convolution is a universally efficient data acquisition strategy in that an ndimensional signal which is S sparse in any fixed representation can be recovered from m � S log n measurements. We discuss two imaging scenarios — radar and Fourier optics — where convolution with a random pulse allows us to seemingly superresolve finescale features, allowing us to recover highresolution signals from lowresolution measurements. 1. Introduction. The new field of compressive sensing (CS) has given us a fresh look at data acquisition, one of the fundamental tasks in signal processing. The message of this theory can be summarized succinctly [7, 8, 10, 15, 32]: the number of measurements we need to reconstruct a signal depends on its sparsity rather than its bandwidth. These measurements, however, are different than the samples that
Dictionaries for Sparse Representation Modeling
"... Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a prespecified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a p ..."
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Cited by 108 (4 self)
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Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a prespecified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a proper dictionary can be done using one of two ways: (i) building a sparsifying dictionary based on a mathematical model of the data, or (ii) learning a dictionary to perform best on a training set. In this paper we describe the evolution of these two paradigms. As manifestations of the first approach, we cover topics such as wavelets, wavelet packets, contourlets, and curvelets, all aiming to exploit 1D and 2D mathematical models for constructing effective dictionaries for signals and images. Dictionary learning takes a different route, attaching the dictionary to a set of examples it is supposed to serve. From the seminal work of Field and Olshausen, through the MOD, the KSVD, the Generalized PCA and others, this paper surveys the various options such training has to offer, up to the most recent contributions and structures.
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 104 (16 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.