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86
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.
Xampling: Signal acquisition and processing in union of subspaces
, 2011
"... We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two: Analog compression that narrows down the input bandwidth prior to sampling with commercial devices followed by a nonlinear algorithm that ..."
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Cited by 43 (21 self)
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We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two: Analog compression that narrows down the input bandwidth prior to sampling with commercial devices followed by a nonlinear algorithm that detects the input subspace prior to conventional signal processing. A representative union model of spectrally sparse signals serves as a testcase to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy, and software complexities. We conduct a comprehensive comparison between two subNyquist acquisition strategies for spectrally sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address low rate signal processing and develop an algorithm for that purpose that enables convenient signal processing at subNyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.
Spectral Compressive Sensing
, 2010
"... Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency do ..."
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Cited by 39 (5 self)
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Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency domain. In practical applications, the standard frequency domain signal representation is the discrete Fourier transform (DFT). Unfortunately, the DFT coefficients of a frequencysparse signal are themselves sparse only in the contrived case where the sinusoid frequencies are integer multiples of the DFT’s fundamental frequency. As a result, practical DFTbased CS acquisition and recovery of smooth signals does not perform nearly as well as one might expect. In this paper, we develop a new spectral compressive sensing (SCS) theory for general frequencysparse signals. The key ingredients are an oversampled DFT frame, a signal model that inhibits closely spaced sinusoids, and classical sinusoid parameter estimation algorithms from the field of spectrum estimation. Using peridogram and eigenanalysis based spectrum estimates (e.g., MUSIC), our new SCS algorithms significantly outperform the current stateoftheart CS algorithms while providing provable bounds on the number of measurements required for stable recovery.
Generalized sampling and infinitedimensional compressed sensing
"... We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demo ..."
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Cited by 33 (20 self)
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We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finitedimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finitedimensional techniques are illsuited for solving a number of important problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.
Atomic norm denoising with applications to line spectral estimation
, 2012
"... The subNyquist estimation of line spectra is a classical problem in signal processing, but currently popular subspacebased techniques have few guarantees in the presence of noise and rely on a priori knowledge about system model order. Motivated by recent work on atomic norms in inverse problems, ..."
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Cited by 29 (4 self)
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The subNyquist estimation of line spectra is a classical problem in signal processing, but currently popular subspacebased techniques have few guarantees in the presence of noise and rely on a priori knowledge about system model order. Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the meansquarederror performance in the presence of noise and without advance knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials with guaranteed bounds on the meansquared error. We show that the associated convex optimization problem, called Atomic norm Soft Thresholding (AST), can be solved in polynomial time via semidefinite programming. For very large scale problems we provide an alternative, efficient algorithm, called Discretized Atomic norm Soft Thresholding (DAST), based on the Fast Fourier Transform that achieves nearly the same error rate as that guaranteed by the semidefinite programming approach. We compare both AST and DAST with Cadzow’s canonical alternating projection algorithm and demonstrate that AST outperforms DAST which outperforms Cadzow in terms of meansquare reconstruction error over a wide range of signaltonoise ratios. For very large problems DAST is considerably faster than both AST and Cadzow.
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 29 (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.
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 20 (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.
Spectral Compressed Sensing via Structured Matrix Completion
"... The paper studies the problem of recovering a spectrally sparse object from a small number of time domain samples. Specifically, the object of interest with ambient dimension n is assumed to be a mixture of r complex multidimensional sinusoids, while the underlying frequencies can assume any value ..."
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Cited by 17 (6 self)
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The paper studies the problem of recovering a spectrally sparse object from a small number of time domain samples. Specifically, the object of interest with ambient dimension n is assumed to be a mixture of r complex multidimensional sinusoids, while the underlying frequencies can assume any value in the unit disk. Conventional compressed sensing paradigms suffer from the basis mismatch issue when imposing a discrete dictionary on the Fourier representation. To address this problem, we develop a novel nonparametric algorithm, called enhanced matrix completion (EMaC), based on structured matrix completion. The algorithm starts by arranging the data into a lowrank enhanced form with multifold Hankel structure, then attempts recovery via nuclear norm minimization. Under mild incoherence conditions, EMaC allows perfect recovery as soon as the number of samples exceeds the order of O(rlog 2 n). We also show that, in many instances, accurate completion of a lowrank multifold Hankel matrix is possible when the number of observed entries is proportional to the information theoretical limits (except for a logarithmic gap). The robustness of EMaC against bounded noise and its applicability to super resolution are further demonstrated by numerical experiments. 1.
Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing
, 2013
"... In this paper we bridge the substantial gap between existing compressed sensing theory and its current use in realworld applications. 1 We do so by introducing a new mathematical framework for overcoming the socalled coherence ..."
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Cited by 13 (4 self)
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In this paper we bridge the substantial gap between existing compressed sensing theory and its current use in realworld applications. 1 We do so by introducing a new mathematical framework for overcoming the socalled coherence