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38
Compressed sensing with coherent and redundant dictionaries
, 2010
"... This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not ..."
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Cited by 165 (13 self)
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This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possible via an `1analysis optimization problem. We introduce a condition on the measurement/sensing matrix, which is a natural generalization of the now wellknown restricted isometry property, and which guarantees accurate recovery of signals that are nearly sparse in (possibly) highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary and our results may be the first of this kind. We discuss practical examples and the implications of our results on those applications, and complement our study by demonstrating the potential of `1analysis for such problems. 1
Two are better than one: Fundamental parameters of frame coherence
, 2011
"... This paper investigates two parameters that measure the coherence of a frame: worstcase and average coherence. We first use worstcase and average coherence to derive nearoptimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. Next, we provide ..."
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Cited by 16 (8 self)
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This paper investigates two parameters that measure the coherence of a frame: worstcase and average coherence. We first use worstcase and average coherence to derive nearoptimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. Next, we provide a catalog of nearly tight frames with small worstcase and average coherence. Later, we find a new lower bound on worstcase coherence; we compare it to the Welch bound and use it to interpret recently reported signal reconstruction results. Finally, we give an algorithm that transforms frames in a way that decreases average coherence without changing the spectral norm or worstcase coherence.
Coherencepattern guided compressive sensing with unresolved grids
 SIAM J. Imaging Sci
"... Abstract. Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such c ..."
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Cited by 9 (3 self)
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Abstract. Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such coherent sensing matrices. These techniques are embedded in the existing compressed sensing algorithms such as Orthogonal Matching Pursuit (OMP),
Robust support recovery using sparse compressive sensing matrices
 in Proc. 45th Annual Conf. on Information Sciences and Systems
, 2011
"... Abstract—This paper considers the task of recovering the support of a sparse, highdimensional vector from a small number of measurements. The procedure proposed here, which we call the SignSketch procedure, is shown to be a robust recovery method in settings where the measurements are corrupted by ..."
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Cited by 5 (1 self)
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Abstract—This paper considers the task of recovering the support of a sparse, highdimensional vector from a small number of measurements. The procedure proposed here, which we call the SignSketch procedure, is shown to be a robust recovery method in settings where the measurements are corrupted by various forms of uncertainty, including additive Gaussian noise and (possibly unbounded) outliers, and even subsequent quantization of the measurements to a single bit. The SignSketch procedure employs sparse random measurement matrices, and utilizes a computationally efficient support recovery procedure that is a variation of a technique from the sketching literature. We show here that O(max {k log(n − k), k log k}) nonadaptive linear measurements suffice to recover the support of any unknown ndimensional vector having no more than k nonzero entries, and that our proposed procedure requires at most O(n log n) total operations for both acquisition and inference. Index Terms—Support recovery, sparsity pattern recovery, model selection, feature selection, sparse recovery, robust inference, sketching, compressive sensing.
ReducedDimension Multiuser Detection
"... Abstract—We present a new framework for reduceddimension multiuser detection (RDMUD) that trades off complexity for biterrorrate (BER) performance. This approach can significantly reduce the number of matched filter branches required by classic multiuser detection designs. We show that the RDMUD ..."
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Cited by 4 (2 self)
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Abstract—We present a new framework for reduceddimension multiuser detection (RDMUD) that trades off complexity for biterrorrate (BER) performance. This approach can significantly reduce the number of matched filter branches required by classic multiuser detection designs. We show that the RDMUD can perform similarly to the linear MUD detector when M is sufficiently large relative to N and K, where N and K are the number of total and active users, respectively. We also study the inherent RDMUD tradeoff between complexity (the number of correlating signals) and BER performance. This leads to a new notion of approximate sufficient statistics, whereby sufficient statistics are approximated to reduce complexity at the expense of some BER performance loss. 1 I.
Multiuser Detection in Asynchronous On–Off Random Access Channels Using Lasso
"... Abstract—This paper considers on–off random access channels where users transmit either a one or a zero to a base station. Such channels represent an abstraction of control channels used for scheduling requests in thirdgeneration cellular systems and uplinks in wireless sensor networks deployed for ..."
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Cited by 4 (0 self)
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Abstract—This paper considers on–off random access channels where users transmit either a one or a zero to a base station. Such channels represent an abstraction of control channels used for scheduling requests in thirdgeneration cellular systems and uplinks in wireless sensor networks deployed for target detection. This paper introduces a novel convexoptimizationbased scheme for multiuser detection (MUD) in asynchronous on–off random access channels that does not require knowledge of the delays or the instantaneous received signaltonoise ratios of the individual users at the base station. For any fixed number of temporal signal space dimensions N and maximum delay τ in the system, the proposed scheme can accommodate M � exp(O(N 1/3)) total users and k � N/logM active users in the system—a significant improvement over thek ≤ M � N scaling suggested by the use of classical matchedfilteringbased approaches to MUD employing orthogonal signaling. Furthermore, the computational complexity of the proposed scheme differs from that of a similar oraclebased scheme with perfect knowledge of the user delays by at most a factor oflog(N+τ). Finally, the results presented in here are nonasymptotic, in contrast to related previous work for synchronous channels that only guarantees that the probability of MUD error at the base station goes to zero asymptotically in M. I.
Compressive Demodulation of Mutually Interfering Signals
 IEEE Trans. on Information Theory
, 2013
"... The challenge of Multiuser Detection (MUD) is that of demodulating mutually interfering signals given that at any time instant the number of active users is typically small. The promise of compressed sensing is the demodulation of sparse superpositions of signature waveforms from very few measure ..."
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Cited by 4 (3 self)
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The challenge of Multiuser Detection (MUD) is that of demodulating mutually interfering signals given that at any time instant the number of active users is typically small. The promise of compressed sensing is the demodulation of sparse superpositions of signature waveforms from very few measurements. This paper considers signature waveforms that are are drawn from a Gabor frame. It describes a MUD architecture that uses subsampling to convert analog input to a digital signal, and then uses iterative matching pursuit to recover the active users. Compressive demodulation requires K logN samples to recover K active users whereas standard MUD requires N samples. The paper provides theoretical performance guarantees and consistent numerical simulations.
Revisiting Model Selection and Recovery of Sparse Signals Using OneStep Thresholding
"... Abstract—This paper studies nonasymptotic model selection and recovery of sparse signals in highdimensional, linear inference problems. In contrast to the existing literature, the focus here is on the general case of arbitrary design matrices and arbitrary nonzero entries of the signal. In this re ..."
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Cited by 3 (2 self)
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Abstract—This paper studies nonasymptotic model selection and recovery of sparse signals in highdimensional, linear inference problems. In contrast to the existing literature, the focus here is on the general case of arbitrary design matrices and arbitrary nonzero entries of the signal. In this regard, it utilizes two easily computable measures of coherence—termed as the worstcase coherence and the average coherence—among the columns of a design matrix to analyze a simple, modelorder agnostic onestep thresholding (OST) algorithm. In particular, the paper establishes that if the design matrix has reasonably small worstcase and average coherence then OST performs nearoptimal model selection when either (i) the energy of any nonzero entry of the signal is close to the average signal energy per nonzero entry or (ii) the signaltonoise ratio (SNR) in the measurement system is not too high. Further, the paper shows that if the design matrix in addition has sufficiently small spectral norm then OST also exactly recovers most sparse signals whose nonzero entries have approximately the same magnitude even if the number of nonzero entries scales almost linearly with the number of rows of the design matrix. Finally, the paper also presents various classes of random and deterministic design matrices that can be used together with OST to successfully carry out nearoptimal model selection and recovery of sparse signals under certain SNR regimes or for certain classes of signals. I.
FAST LEVEL SET ESTIMATION FROM PROJECTION MEASUREMENTS
"... Estimation of the level set of a function (i.e., regions where the function exceeds some value) is an important problem with applications in digital elevation maps, medical imaging, and astronomy. In many applications, however, the function of interest is acquired through indirect measurements, such ..."
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Cited by 2 (0 self)
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Estimation of the level set of a function (i.e., regions where the function exceeds some value) is an important problem with applications in digital elevation maps, medical imaging, and astronomy. In many applications, however, the function of interest is acquired through indirect measurements, such as tomographic projections, codedaperture measurements, or pseudorandom projections associated with compressed sensing. This paper describes a new methodology and associated theoretical analysis for rapid and accurate estimation of the level set from such projection measurements. The proposed method estimates the level set from projection measurements without an intermediate function reconstruction step, thereby leading to significantly faster computation. In addition, the coherence of the projection operator and McDiarmid’s inequality are used to characterize the estimator’s performance. Index Terms — Compressed sensing, coherence, level sets, performance bounds, segmentation, thresholding