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372
Exploiting structure in waveletbased Bayesian compressive sensing
, 2009
"... Bayesian compressive sensing (CS) is considered for signals and images that are sparse in a wavelet basis. The statistical structure of the wavelet coefficients is exploited explicitly in the proposed model, and therefore this framework goes beyond simply assuming that the data are compressible in a ..."
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Cited by 85 (14 self)
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Bayesian compressive sensing (CS) is considered for signals and images that are sparse in a wavelet basis. The statistical structure of the wavelet coefficients is exploited explicitly in the proposed model, and therefore this framework goes beyond simply assuming that the data are compressible in a wavelet basis. The structure exploited within the wavelet coefficients is consistent with that used in waveletbased compression algorithms. A hierarchical Bayesian model is constituted, with efficient inference via Markov chain Monte Carlo (MCMC) sampling. The algorithm is fully developed and demonstrated using several natural images, with performance comparisons to many stateoftheart compressivesensing inversion algorithms.
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 83 (16 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.
Kalman filtered compressed sensing
 in Proc. IEEE Int. Conf. Image (ICIP), 2008
"... We consider the problem of reconstructing time sequences of spatially sparse signals (with unknown and timevarying sparsity patterns) from a limited number of linear “incoherent ” measurements, in realtime. The signals are sparse in some transform domain referred to as the sparsity basis. For a si ..."
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Cited by 82 (19 self)
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We consider the problem of reconstructing time sequences of spatially sparse signals (with unknown and timevarying sparsity patterns) from a limited number of linear “incoherent ” measurements, in realtime. The signals are sparse in some transform domain referred to as the sparsity basis. For a single spatial signal, the solution is provided by Compressed Sensing (CS). The question that we address is, for a sequence of sparse signals, can we do better than CS, if (a) the sparsity pattern of the signal’s transform coefficients’ vector changes slowly over time, and (b) a simple prior model on the temporal dynamics of its current nonzero elements is available. The overall idea of our solution is to use CS to estimate the support set of the initial signal’s transform vector. At future times, run a reduced order Kalman filter with the currently estimated support and estimate new additions to the support set by applying CS to the Kalman innovations or filtering error (whenever it is “large”). Index Terms/Keywords: compressed sensing, Kalman filtering, compressive sampling, sequential MMSE estimation
Highly undersampled magnetic resonance image reconstruction via homotopic ℓ0minimization
 IEEE Trans. Med. Imaging
, 2009
"... any reduction in scan time offers a number of potential benefits ranging from hightemporalrate observation of physiological processes to improvements in patient comfort. Following recent developments in Compressive Sensing (CS) theory, several authors have demonstrated that certain classes of MR i ..."
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Cited by 68 (1 self)
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any reduction in scan time offers a number of potential benefits ranging from hightemporalrate observation of physiological processes to improvements in patient comfort. Following recent developments in Compressive Sensing (CS) theory, several authors have demonstrated that certain classes of MR images which possess sparse representations in some transform domain can be accurately reconstructed from very highly undersampled Kspace data by solving a convex ℓ1minimization problem. Although ℓ1based techniques are extremely powerful, they inherently require a degree of oversampling above the theoretical minimum sampling rate to guarantee that exact reconstruction can be achieved. In this paper, we propose a generalization of the Compressive Sensing paradigm based on homotopic approximation of the ℓ0 quasinorm and show how MR image reconstruction can be pushed even further below the Nyquist limit and significantly closer to the theoretical bound. Following a brief review of standard Compressive Sensing methods and the developed theoretical extensions, several example MRI reconstructions from highly undersampled Kspace data are presented.
FIXEDPOINT CONTINUATION FOR ℓ1MINIMIZATION: METHODOLOGY AND CONVERGENCE
"... We present a framework for solving largescale ℓ1regularized convex minimization problem: min �x�1 + µf(x). Our approach is based on two powerful algorithmic ideas: operatorsplitting and continuation. Operatorsplitting results in a fixedpoint algorithm for any given scalar µ; continuation refers ..."
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Cited by 65 (9 self)
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We present a framework for solving largescale ℓ1regularized convex minimization problem: min �x�1 + µf(x). Our approach is based on two powerful algorithmic ideas: operatorsplitting and continuation. Operatorsplitting results in a fixedpoint algorithm for any given scalar µ; continuation refers to approximately following the path traced by the optimal value of x as µ increases. In this paper, we study the structure of optimal solution sets; prove finite convergence for important quantities; and establish qlinear convergence rates for the fixedpoint algorithm applied to problems with f(x) convex, but not necessarily strictly convex. The continuation framework, motivated by our convergence results, is demonstrated to facilitate the construction of practical algorithms.
Dequantizing compressed sensing: When oversampling and nongaussian constraints combine
, 2009
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Optimally tuned iterative reconstruction algorithms for compressed sensing
 Selected Topics in Signal Processing
"... Abstract — We conducted an extensive computational experiment, lasting multiple CPUyears, to optimally select parameters for two important classes of algorithms for finding sparse solutions of underdetermined systems of linear equations. We make the optimally tuned implementations available at spar ..."
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Cited by 57 (4 self)
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Abstract — We conducted an extensive computational experiment, lasting multiple CPUyears, to optimally select parameters for two important classes of algorithms for finding sparse solutions of underdetermined systems of linear equations. We make the optimally tuned implementations available at sparselab.stanford.edu; they run ‘out of the box ’ with no user tuning: it is not necessary to select thresholds or know the likely degree of sparsity. Our class of algorithms includes iterative hard and soft thresholding with or without relaxation, as well as CoSaMP, subspace pursuit and some natural extensions. As a result, our optimally tuned algorithms dominate such proposals. Our notion of optimality is defined in terms of phase transitions, i.e. we maximize the number of nonzeros at which the algorithm can successfully operate. We show that the phase transition is a welldefined quantity with our suite of random underdetermined linear systems. Our tuning gives the highest transition possible within each class of algorithms. We verify by extensive computation the robustness of our recommendations to the amplitude distribution of the nonzero coefficients as well as the matrix ensemble defining the underdetermined system. Our findings include: (a) For all algorithms, the worst amplitude distribution for nonzeros is generally the constantamplitude randomsign distribution, where all nonzeros are the same amplitude. (b) Various random matrix ensembles give the same phase transitions; random partial isometries may give different transitions and require different tuning; (c) Optimally tuned subspace pursuit dominates optimally tuned CoSaMP, particularly so when the system is almost square. I.
Fast Bayesian compressive sensing using Laplace priors
 in IEEE Int. Conf. on Acoustics, Speech, and Sig. Proc. (ICASSP09
, 2009
"... In this paper we model the components of the compressive sensing (CS) problem using the Bayesian framework by utilizing a hierarchical form of the Laplace prior to model sparsity of the unknown signal. This signal prior includes some of the existing models as special cases and achieves a high degree ..."
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Cited by 57 (11 self)
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In this paper we model the components of the compressive sensing (CS) problem using the Bayesian framework by utilizing a hierarchical form of the Laplace prior to model sparsity of the unknown signal. This signal prior includes some of the existing models as special cases and achieves a high degree of sparsity. We develop a constructive (greedy) algorithm resulting from this formulation where necessary parameters are estimated solely from the observation and therefore no userintervention is needed. We provide experimental results with synthetic 1D signals and images, and compare with the stateoftheart CS reconstruction algorithms demonstrating the superior performance of the proposed approach. Index Terms — Bayesian methods, compressive sensing, inverse problems, sparse Bayesian learning, relevance vector machine
Restoration of Poissonian images using alternating direction optimization
 IEEE Trans. Image Process
, 2010
"... Abstract—Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using stateoftheart regularizers (such as those based upon multiscale representations or total variation) is still an a ..."
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Cited by 53 (5 self)
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Abstract—Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using stateoftheart regularizers (such as those based upon multiscale representations or total variation) is still an active research area, since the associated optimization problems are quite challenging. In this paper, we propose an approach to deconvolving Poissonian images, which is based upon an alternating direction optimization method. The standard regularization [or maximum a posteriori (MAP)] restoration criterion, which combines the Poisson loglikelihood with a (nonsmooth) convex regularizer (logprior), leads to hard optimization problems: the loglikelihood is nonquadratic and nonseparable, the regularizer is nonsmooth, and there is a nonnegativity constraint. Using standard convex analysis tools, we present sufficient conditions for existence and uniqueness of solutions of these optimization problems, for several types of regularizers: totalvariation, framebased analysis, and framebased synthesis. We attack these problems with an instance of the alternating direction method of multipliers (ADMM), which belongs to the family of augmented Lagrangian algorithms. We study sufficient conditions for convergence and show that these are satisfied, either under totalvariation or framebased (analysis and synthesis) regularization. The resulting algorithms are shown to outperform alternative stateoftheart methods, both in terms of speed and restoration accuracy. Index Terms—Alternating direction methods, augmented Lagrangian, convex optimization, image deconvolution, image restoration, Poisson images. I.