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68
Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging
 MAGNETIC RESONANCE IN MEDICINE 58:1182–1195
, 2007
"... The sparsity which is implicit in MR images is exploited to significantly undersample kspace. 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 ..."
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Cited by 538 (11 self)
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The sparsity which is implicit in MR images is exploited to significantly undersample kspace. 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 finitedifferences or their wavelet coefficients. According to the recently developed mathematical theory of compressedsensing, images with a sparse representation can be recovered from randomly undersampled kspace data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noiselike 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 pseudorandom variabledensity undersampling of phaseencodes. 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.
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 78 (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.
An efficient algorithm for compressed MR imaging using total variation and wavelets
 in IEEE Conference on Computer Vision and Pattern Recognition
, 2008
"... Compressed sensing, an emerging multidisciplinary field involving mathematics, probability, optimization, and signal processing, focuses on reconstructing an unknown signal from a very limited number of samples. Because information such as boundaries of organs is very sparse in most MR images, compr ..."
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Cited by 49 (3 self)
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Compressed sensing, an emerging multidisciplinary field involving mathematics, probability, optimization, and signal processing, focuses on reconstructing an unknown signal from a very limited number of samples. Because information such as boundaries of organs is very sparse in most MR images, compressed sensing makes it possible to reconstruct the same MR image from a very limited set of measurements significantly reducing the MRI scan duration. In order to do that however, one has to solve the difficult problem of minimizing nonsmooth functions on large data sets. To handle this, we propose an efficient algorithm that jointly minimizes the ℓ1 norm, total variation, and a least squares measure, one of the most powerful models for compressive MR imaging. Our algorithm is based upon an iterative operatorsplitting framework. The calculations are accelerated by continuation and takes advantage of fast wavelet and Fourier transforms enabling our code to process MR images from actual real life applications. We show that faithful MR images can be reconstructed from a subset that represents a mere 20 percent of the complete set of measurements. 1.
Accelerating SENSE using compressed sensing
"... Both parallel magnetic resonance imaging (pMRI) and compressed sensing (CS) are emerging techniques to accelerate conventional MRI by reducing the number of acquired data. The combination of pMRI and CS for further acceleration is of great interests. In this paper, we propose two methods to combine ..."
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Cited by 21 (3 self)
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Both parallel magnetic resonance imaging (pMRI) and compressed sensing (CS) are emerging techniques to accelerate conventional MRI by reducing the number of acquired data. The combination of pMRI and CS for further acceleration is of great interests. In this paper, we propose two methods to combine SENSE, one of the standard methods for pMRI, and SparseMRI, a recently proposed method for CSMRI with Cartesian trajectories. The first method, named SparseSENSE, directly formulates the reconstruction from multichannel reduced kspace data as the same nonlinear convex optimization problem as SparseMRI, except that the encoding matrix is the Fourier transform of the channelspecific sensitivity modulation. The second method, named CSSENSE, first employs SparseMRI to reconstruct a set of aliased reducedfieldofview images in each channel, and then applies Cartesian SENSE to reconstruct the final image. The results from simulations, phantom and in vivo experiments demonstrate that both SparseSENSE and CSSENSE can achieve a reduction factor higher than those achieved by SparseMRI and SENSE individually, and CSSENSE outperforms SparseSENSE in most cases. MR imaging speed is usually limited by the large number of samples needed along the phase encoding direction. In conventional MRI using Fourier encoding, the required number of samples is determined by
A fast waveletbased reconstruction method for magnetic resonance imaging
 IEEE Trans. Med. Imag
, 2011
"... Abstract—In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary kspace trajectories. Reconstruction is pose ..."
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Cited by 19 (3 self)
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Abstract—In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary kspace trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific kspace trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality. Index Terms—Compressed sensing, fast iterative shrinkage/ thresholding algorithm (FISTA), fast weighted iterative shrinkage/ thresholding algorithm (FWISTA), iterative shrinkage/thresholding algorithm (ISTA), magnetic resonance imaging (MRI), nonCartesian, nonlinear reconstruction, sparsity, thresholded Landweber, total variation, undersampled spiral, wavelets. I.
Optimization of kspace trajectories for compressed sensing by Bayesian experimental design
 MAGNETIC RESONANCE IN MEDICINE
, 2009
"... The optimization of kspace sampling for nonlinear sparse MRI reconstruction is phrased as Bayesian experimental design problem. Bayesian inference is approximated by a novel relaxation to standard signal processing primitives, resulting in an efficient optimization algorithm for Cartesian and spi ..."
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Cited by 16 (3 self)
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The optimization of kspace sampling for nonlinear sparse MRI reconstruction is phrased as Bayesian experimental design problem. Bayesian inference is approximated by a novel relaxation to standard signal processing primitives, resulting in an efficient optimization algorithm for Cartesian and spiral trajectories. On clinical resolution brain image data from a Siemens 3T scanner, automatically optimized trajectories lead to significantly improved images, compared to standard lowpass, equispaced or variable density randomized designs. Insights into the nonlinear design optimization problem for MR imaging are given.
Nayak, “Accelerated threedimensional upper airway MRI using compressed sensing
 Magn. Reson. Med
, 2009
"... upper airway has provided insights into vocal tract shaping and data for its modeling. Small movements of articulators can lead to large changes in the produced sound, therefore improving the resolution of these data sets, within the constraints of a sustained speech sound (6–12 s), is an important ..."
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Cited by 13 (5 self)
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upper airway has provided insights into vocal tract shaping and data for its modeling. Small movements of articulators can lead to large changes in the produced sound, therefore improving the resolution of these data sets, within the constraints of a sustained speech sound (6–12 s), is an important area for investigation. The purpose of the study is to provide a first application of compressed sensing (CS) to highresolution 3D upper airway MRI using spatial finite difference as the sparsifying transform, and to experimentally determine the benefit of applying constraints on image phase. Estimates of image phase are incorporated into the CS reconstruction to improve the sparsity of the finite difference of the solution. In a retrospective subsampling experiment with no sound production, 5 � and 4 � were the highest acceleration factors that produced acceptable image quality when using a phase constraint and when not using a phase constraint, respectively.
Sparse stochastic processes and discretization of linear inverse problems
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2013
"... We present a novel statisticallybased discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving illconditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuousdomain signals as solutions of line ..."
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Cited by 10 (6 self)
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We present a novel statisticallybased discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving illconditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuousdomain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the wellstudied methods of Tikhonov and 1type regularizations as particular cases, but also open the door to a broader class of sparsitypromoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, magnetic resonance imaging, and Xray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.
Iterative thresholding compressed sensing MRI based on contourlet transform. Inverse Probl
 Sci. En
"... Abstract: Reducing the acquisition time is important for clinical magnetic resonance imaging (MRI). Compressed sensing has recently emerged as a theoretical foundation for the reconstruction of magnetic resonance (MR) images from undersampled kspace measurements, assuming those images are sparse in ..."
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Cited by 8 (3 self)
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Abstract: Reducing the acquisition time is important for clinical magnetic resonance imaging (MRI). Compressed sensing has recently emerged as a theoretical foundation for the reconstruction of magnetic resonance (MR) images from undersampled kspace measurements, assuming those images are sparse in a certain transform domain. However, most realworld signals are compressible rather than exactly sparse. For example, the commonly used 2D wavelet for compressed sensing MRI (CSMRI) does not sparsely represent curves and edges. In this paper, we introduce a geometric image transform, the contourlet, to overcome this shortage. In addition, the improved redundancy provided by the contourlet can successfully suppress the pseudoGibbs phenomenon, a tiresome artifact produced by undersampling of kspace, around the singularities of images. For numerical calculation, a simple but effective iterative thresholding algorithm is employed to solve 1l norm optimization for CSMRI. Considering the recovered information and image features, we introduce three objective criteria, which are the peak signaltonoise ratio (PSNR), mutual information (MI) and transferred edge information (TEI), to evaluate the performance of different image transforms. Simulation results demonstrate that contourletbased CSMRI can better reconstruct the curves and edges than traditional waveletbased methods, especially at low kspace sampling rate.