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Compressive sensing

by Richard Baraniuk - 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 ..."
Abstract - Cited by 696 (62 self) - Add to MetaCart
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

Compressed sensing

by Yaakov Tsaig, David L. Donoho , 2004
"... We study the notion of Compressed Sensing (CS) as put forward in [14] and related work [20, 3, 4]. The basic idea behind CS is that a signal or image, unknown but supposed to be compressible by a known transform, (eg. wavelet or Fourier), can be subjected to fewer measurements than the nominal numbe ..."
Abstract - Cited by 3625 (22 self) - Add to MetaCart
We study the notion of Compressed Sensing (CS) as put forward in [14] and related work [20, 3, 4]. The basic idea behind CS is that a signal or image, unknown but supposed to be compressible by a known transform, (eg. wavelet or Fourier), can be subjected to fewer measurements than the nominal

Bayesian Compressive Sensing

by Shihao Ji, Ya Xue, Lawrence Carin , 2007
"... The data of interest are assumed to be represented as N-dimensional 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 basis-function coefficients associated with B. Compressive sensing ..."
Abstract - Cited by 330 (24 self) - Add to MetaCart
The data of interest are assumed to be represented as N-dimensional 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 basis-function coefficients associated with B. Compressive sensing

Iterative hard thresholding for compressed sensing

by Thomas Blumensath, Mike E. Davies - 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 ..."
Abstract - Cited by 329 (18 self) - Add to MetaCart
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

Compressive Sensing

by Massimo Fornasier, Holger Rauhut , 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1-minimization can be used for recovery. The theory has many poten ..."
Abstract - Cited by 50 (12 self) - Add to MetaCart
Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1-minimization can be used for recovery. The theory has many

Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems

by Mário A. T. Figueiredo, Robert D. Nowak, Stephen J. Wright - IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING , 2007
"... Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a spa ..."
Abstract - Cited by 539 (17 self) - Add to MetaCart
sparseness-inducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution, and compressed sensing are a few well-known examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound

Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging

by Michael Lustig, David Donoho, John M. Pauly - MAGNETIC RESONANCE IN MEDICINE 58:1182–1195 , 2007
"... The sparsity which is implicit in MR images is exploited to significantly undersample k-space. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finit ..."
Abstract - Cited by 538 (11 self) - Add to MetaCart
The sparsity which is implicit in MR images is exploited to significantly undersample k-space. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finite-differences or their wavelet coefficients. According to the recently developed mathematical theory of compressedsensing, images with a sparse representation can be recovered from randomly undersampled k-space data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noise-like interference. In the sparse transform domain the significant coefficients stand out above the interference. A nonlinear thresholding scheme can recover the sparse coefficients, effectively recovering the image itself. In this article, practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference. Incoherence is introduced by pseudo-random variable-density undersampling of phase-encodes. The reconstruction is performed by minimizing the ℓ1 norm of a transformed image, subject to data fidelity constraints. Examples demonstrate improved spatial resolution and accelerated acquisition for multislice fast spinecho brain imaging and 3D contrast enhanced angiography.

Compressed Sensing and Redundant Dictionaries

by Holger Rauhut, Karin Schnass, Pierre Vandergheynst
"... 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 ..."
Abstract - Cited by 137 (13 self) - Add to MetaCart
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

compressed sensing

by Hiroshi Inoue, Hiroshi Inoue , 2013
"... This paper discusses new bounds for restricted isometry property in compressed sensing. In the literature, E.J. Candès has proved that δ2s < √ 2 − 1 is a sufficient condition via l1 optimization having s-sparse vector solution. Later, many researchers have improved the sufficient conditions on δ2 ..."
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This paper discusses new bounds for restricted isometry property in compressed sensing. In the literature, E.J. Candès has proved that δ2s < √ 2 − 1 is a sufficient condition via l1 optimization having s-sparse vector solution. Later, many researchers have improved the sufficient conditions on δ

Compressive Sensing

by Xuejun Liao, Hui Li, Lawrence Carin
"... •Compressive sensing (CS) is a sampling strategy for signals that are sparse in an arbitrary orthonormal basis Ψ. • It is a generalization of classic Nyquist-Shannon sampling for sig-nals that have a continuous finite support in Fourier basis.•Given a signal x ∈ RN = Ψw satisfying ‖w‖`0: = |T | = | ..."
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•Compressive sensing (CS) is a sampling strategy for signals that are sparse in an arbitrary orthonormal basis Ψ. • It is a generalization of classic Nyquist-Shannon sampling for sig-nals that have a continuous finite support in Fourier basis.•Given a signal x ∈ RN = Ψw satisfying ‖w‖`0: = |T
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