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153
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.
A Probabilistic and RIPless Theory of Compressed Sensing
, 2010
"... This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — discussed in the literature, ..."
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Cited by 95 (3 self)
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This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models — e.g. Gaussian, frequency measurements — discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) — they make use of a much weaker notion — or a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.
Stable image reconstruction using total variation minimization
 SIAM Journal on Imaging Sciences
, 2013
"... This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be ..."
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Cited by 50 (2 self)
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This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be reconstructed to within the best sterm approximation of its gradient, up to a logarithmic factor. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix. 1
Restricted isometries for partial random circulant matrices
 APPL. COMPUT. HARMON. ANAL
, 2010
"... In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampl ..."
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Cited by 47 (8 self)
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In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a dataacquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the sth order restricted isometry constant is small when the number m of samples satisfies m � (s log n) 3/2, where n is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.
Sparse Recovery from Combined Fusion Frame Measurements
 IEEE Trans. Inform. Theory
"... Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead ..."
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Cited by 43 (12 self)
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Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. This work combines these exciting fields to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it does not need to be sparse within each of the subspaces it occupies. This sparsity model is captured using a mixed ℓ1/ℓ2 norm for fusion frames. A signal sparse in a fusion frame can be sampled using very few random projections and exactly reconstructed using a convex optimization that minimizes this mixed ℓ1/ℓ2 norm. The provided sampling conditions generalize coherence and RIP conditions used in standard CS theory. It is demonstrated that they are sufficient to guarantee sparse recovery of any signal sparse in our model. Moreover, an average case analysis is provided using a probability model on the sparse signal that shows that under very mild conditions the probability of recovery failure decays exponentially with increasing dimension of the subspaces. Index Terms
Suprema of chaos processes and the restricted isometry property
 Comm. Pure Appl. Math
"... We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and timefrequency structured ..."
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Cited by 33 (6 self)
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We present a new bound for suprema of a special type of chaos processes indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and timefrequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m � s log 2 s log 2 n. Key words. Compressive sensing, restricted isometry property, structured random matrices, chaos processes, γ2functionals, generic chaining, partial random circulant matrices, random Gabor synthesis matrices.
IMPROVED BOUNDS ON RESTRICTED ISOMETRY CONSTANTS FOR GAUSSIAN MATRICES
"... Abstract. The Restricted Isometry Constants (RIC) of a matrix A measures how close to an isometry is the action of A on vectors with few nonzero entries, measured in the ℓ2 norm. Specifically, the upper and lower RIC of a matrix A of size n × N is the maximum and the minimum deviation from unity (on ..."
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Cited by 31 (4 self)
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Abstract. The Restricted Isometry Constants (RIC) of a matrix A measures how close to an isometry is the action of A on vectors with few nonzero entries, measured in the ℓ2 norm. Specifically, the upper and lower RIC of a matrix A of size n × N is the maximum and the minimum deviation from unity (one) of the largest and smallest, respectively, square of singular values of all `N ´ matrices formed by taking k columns from A. Calculation of the k RIC is intractable for most matrices due to its combinatorial nature; however, many random matrices typically have bounded RIC in some range of problem sizes (k, n, N). We provide the best known bound on the RIC for Gaussian matrices, which is also the smallest known bound on the RIC for any large rectangular matrix. Improvements over prior bounds are achieved by exploiting similarity of singular values for matrices which share a substantial number of columns. Key words. Wishart Matrices, Compressed sensing, sparse approximation, restricted isometry constant, phase transitions, Gaussian matrices, singular values of random matrices.
Sparse recovery for spherical harmonic expansions
 In Proceedings of the 9th International Conference on Sampling Theory and Applications
, 2011
"... We show that sparse spherical harmonic expansions can be recovered from a small number of randomly chosen samples on the sphere using ℓ1minimization. To this end, we establish the restricted isometry property of an associated preconditioned random measurement matrix. 1. ..."
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Cited by 26 (6 self)
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We show that sparse spherical harmonic expansions can be recovered from a small number of randomly chosen samples on the sphere using ℓ1minimization. To this end, we establish the restricted isometry property of an associated preconditioned random measurement matrix. 1.