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Iterative hard thresholding for compressed sensing
 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 ..."
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Cited by 142 (13 self)
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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 problem. We show that the algorithm has the following properties (made more precise in the main text of the paper) • It gives nearoptimal error guarantees. • It is robust to observation noise. • It succeeds with a minimum number of observations. • It can be used with any sampling operator for which the operator and its adjoint can be computed. • The memory requirement is linear in the problem size. Preprint submitted to Elsevier 28 January 2009 • Its computational complexity per iteration is of the same order as the application of the measurement operator or its adjoint. • It requires a fixed number of iterations depending only on the logarithm of a form of signal to noise ratio of the signal. • Its performance guarantees are uniform in that they only depend on properties of the sampling operator and signal sparsity.
Bregman iterative algorithms for ℓ1minimization with applications to compressed sensing
 SIAM J. Imaging Sci
, 2008
"... Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number o ..."
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Cited by 63 (14 self)
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Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 instances of the unconstrained problem minu∈Rn μ‖u‖1 + 2 ‖Au−fk ‖ 2 2 for given matrix A and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrixvector operations involving A and A ⊤ can be computed by fast transforms. Utilizing a fast fixedpoint continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.
HARD THRESHOLDING PURSUIT: AN ALGORITHM FOR COMPRESSIVE SENSING
"... We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and of the Compressive Sampling Matching Pursuit or Subspace Pursuit algorithms, is called Hard Thresholding Pursuit. ..."
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Cited by 23 (0 self)
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We introduce a new iterative algorithm to find sparse solutions of underdetermined linear systems. The algorithm, a simple combination of the Iterative Hard Thresholding algorithm and of the Compressive Sampling Matching Pursuit or Subspace Pursuit algorithms, is called Hard Thresholding Pursuit. We study its general convergence, and notice in particular that only a finite number of iterations are required. We then show that, under a certain condition on the restricted isometry constant of the matrix of the system, the Hard Thresholding Pursuit algorithm indeed finds all ssparse solutions. This condition, which reads δ3s < 1 / √ 3, is heuristically better than the sufficient conditions currently available for other Compressive Sensing algorithms. It applies to fast versions of the algorithm, too, including the Iterative Hard Thresholding algorithm. Stability with respect to sparsity defect and robustness with respect to measurement error are also guaranteed under the condition δ3s < 1 / √ 3. We conclude with some numerical experiments to demonstrate the good empirical performance and the low complexity of the Hard Thresholding Pursuit algorithm.
Rank Awareness in Joint Sparse Recovery
, 2010
"... In this paper we revisit the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem has received increasing interest as an extension of single channel sparse recovery, which lies at the hea ..."
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Cited by 16 (4 self)
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In this paper we revisit the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem has received increasing interest as an extension of single channel sparse recovery, which lies at the heart of the emerging field of compressed sensing. However, MMV approximation has origins in the field of array signal processing as we discuss in this paper. Inspired by these links, we introduce a new family of MMV algorithms based on the wellknow MUSIC method in array processing. We particularly highlight the role of the rank of the unknown signal matrix X in determining the difficulty of the recovery problem. We begin by deriving necessary and sufficient conditions for the uniqueness of the sparse MMV solution, which indicates that the larger the rank of X the less sparse X needs to be to ensure uniqueness. We also show that as the rank of X increases, the computational effort required to solve the MMV problem through a combinatorial search is reduced. In the second part of the paper we consider practical suboptimal algorithms for MMV recovery. We examine the rank awareness of popular methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques and show them to be rank blind in terms of worst case analysis. We then consider a family of greedy algorithms that are rank aware. The simplest such method is a discrete version of the MUSIC algorithm popular in array signal processing. This approach is guaranteed to recover the sparse vectors in the full rank MMV setting under mild conditions. We then extend this idea to develop a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case. This approach also provides guaranteed recovery in the full rank setting. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple measurements.
From Bernoulli–Gaussian Deconvolution to Sparse Signal Restoration
"... © 2011 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other w ..."
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Cited by 9 (4 self)
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© 2011 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Abstract—Formulated as a least square problem under an 0 constraint, sparse signal restoration is a discrete optimization problem, known to be NP complete. Classical algorithms include, by increasing cost and efficiency, matching pursuit (MP), orthogonal matching pursuit (OMP), orthogonal least squares (OLS), stepwise regression algorithms and the exhaustive search. We revisit the single most likely replacement (SMLR) algorithm, developed in the mid1980s for Bernoulli–Gaussian signal restoration. We show that the formulation of sparse signal restoration as a limit case of Bernoulli–Gaussian signal restoration leads to an 0penalized least square minimization problem, to which SMLR can be straightforwardly adapted. The resulting algorithm, called single best replacement (SBR), can be interpreted as a forward–backward extension of OLS sharing similarities with stepwise regression algorithms. Some structural properties of SBR are put forward. A fast and stable implementation is proposed. The approach is illustrated on two inverse problems involving highly correlated dictionaries. We show that SBR is very competitive with popular sparse algorithms in terms of tradeoff between accuracy and computation time. Index Terms—BernoulliGaussian (BG) signal restoration, inverse problems, mixed 2 0 criterion minimization, orthogonal least squares, SMLR algorithm, sparse signal estimation, stepwise regression algorithms. I.
On approximation of orientation distributions by means of spherical ridgelets
 In Proc. IEEE International Symposium on Biomedical Imaging: from Nano to Macro. IEEE
, 2008
"... Visualization and analysis of the microarchitecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, ..."
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Cited by 9 (0 self)
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Visualization and analysis of the microarchitecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, the conventional diffusion tensor imaging (DTI) used for estimating the local orientations of neural fibers, is incapable of performing reliably in the situations when a voxel of interest accommodates multiple fiber tracts. In this case, a much more accurate analysis is possible using the high angular resolution diffusion imaging (HARDI) that represents local diffusion by its apparent coefficients measured as a discrete function of spatial orientations. In this note, a novel approach to enhancing and modeling the HARDI signals using multiresolution bases of spherical ridgelets is presented. In addition to its desirable properties of being adaptive, sparsifying, and efficiently computable, the proposed modeling leads to analytical computation of the orientation distribution functions associated with the measured diffusion, thereby providing a fast and robust analytical solution for qball imaging.
CurveletWavelet Regularized Split Bregman Iteration for Compressed Sensing
"... Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images ..."
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Cited by 9 (3 self)
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Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent pointlike features while curvelets represent linelike features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Φu + ɛ with the measurement matrix Φ ∈ R K×N, K<<N, we consider a jointly sparsityconstrained optimization problem of the form argmin{‖ΛcΨcu‖1 + ‖ΛwΨwu‖1 + u 1 2‖f − Φu‖22}. Here Ψcand Ψw are the transform matrices corresponding to the two frames, and the diagonal matrices Λc, Λw contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the DouglasRachford Split algorithm. Numerical experiments for compressed sensing based Fourierdomain random imaging show good performances of the proposed curveletwavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
Recovery of sparse translationinvariant signals with continuous basis pursuit
 IEEE Trans Signal Processing
, 2011
"... Abstract—We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a �ni ..."
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Cited by 7 (0 self)
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Abstract—We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a �nite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of basic waveforms), and apply sparse optimization methods to select and solve for the relevant coef�cients. Here, we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coef�cients. We formulate a constrained convex optimization problem, in which the full set of dictionary coef�cients represents a linear approximation of the signal, the auxiliary coef�cients are constrained so as to only represent translated features, and sparsity is imposed on the primary coef�cients using an L1 penalty. The basis pursuit denoising