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57
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 126 (22 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
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.
GROUP SPARSE OPTIMIZATION BY ALTERNATING DIRECTION METHOD
, 2011
"... Abstract. This paper proposes efficient algorithms for group sparse optimization with mixed ℓ2,1regularization, which arises from the reconstruction of group sparse signals in compressive sensing, and the group Lasso problem in statistics and machine learning. It is known that encoding the group in ..."
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Cited by 25 (3 self)
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Abstract. This paper proposes efficient algorithms for group sparse optimization with mixed ℓ2,1regularization, which arises from the reconstruction of group sparse signals in compressive sensing, and the group Lasso problem in statistics and machine learning. It is known that encoding the group information in addition to sparsity will lead to better signal recovery/feature selection. The ℓ2,1regularization promotes group sparsity, but the resulting problem, due to the mixednorm structure and possible grouping irregularity, is considered more difficult to solve than the conventional ℓ1regularized problem. Our approach is based on a variable splitting strategy and the classic alternating direction method (ADM). Two algorithms are presented, one derived from the primal and the other from the dual of the ℓ2,1regularized problem. The convergence of the proposed algorithms is guaranteed by the existing ADM theory. General group configurations such as overlapping groups and incomplete covers can be easily handled by our approach. Computational results show that on random problems the proposed ADM algorithms exhibit good efficiency, and strong stability and robustness.
Compressive MUSIC: a missing link between compressive sensing and array signal processing
, 2011
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Fast Algorithms for Image Reconstruction with Application to Partially Parallel MR Imaging
"... This paper presents two fast algorithms for total variationbased image reconstruction in partially parallel magnetic resonance imaging (PPI) where the inversion matrix is large and illconditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more eas ..."
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Cited by 6 (2 self)
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This paper presents two fast algorithms for total variationbased image reconstruction in partially parallel magnetic resonance imaging (PPI) where the inversion matrix is large and illconditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more easily solved subproblems. The first method reduces the image reconstruction problem to an unconstrained minimization problem, which is solved by an alternating proximal minimization algorithm. One phase of the algorithm solves a total variation (TV) denoising problem, and second phase solves an illconditioned linear system. Linear and sublinear convergence results are given, and an implementation based on a primaldual hybrid gradient (PDHG) scheme for the TV problem and a BarzilaiBorwein scheme for the linear inversion is proposed. The second algorithm exploits the special structure of the PPI reconstruction problem by decomposing it into one subproblem involving Fourier transforms and another subproblem that can be treated by the PDHG scheme. Numerical results and comparisons with recently developed methods indicate the efficiency of the proposed algorithms. Key words. Image reconstruction, Variable splitting, TV denoising, Nonlinear optimization 1
Regularizing GRAPPA using simultaneous sparsity to recover denoised images
 in Proc. SPIE Wavelets Sparsity XIV
"... To enable further acceleration of magnetic resonance (MR) imaging, compressed sensing (CS) is combined with GRAPPA, a parallel imaging method, to reconstruct images from highly undersampled data with significantly improved RMSE compared to reconstructions using GRAPPA alone. This novel combination o ..."
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Cited by 4 (4 self)
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To enable further acceleration of magnetic resonance (MR) imaging, compressed sensing (CS) is combined with GRAPPA, a parallel imaging method, to reconstruct images from highly undersampled data with significantly improved RMSE compared to reconstructions using GRAPPA alone. This novel combination of GRAPPA and CS regularizes the GRAPPA kernel computation step using a simultaneous sparsity penalty function of the coil images. This approach can be implemented by formulating the problem as the joint optimization of the least squares fit of the kernel to the ACS lines and the sparsity of the images generated using GRAPPA with the kernel.
Nuclear normregularized SENSE reconstruction
 Magnetic Reson Imaging
, 2012
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 4 (0 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Analytical form of SheppLogan phantom for parallel MRI
 in Proc. ISBI
"... ABSTRACT We present an analytical form of groundtruth kspace data for the 2D SheppLogan brain phantom in the presence of multiple and nonhomogeneous receiving coils. The analytical form allows us to conduct realistic simulations and validations of reconstruction algorithms for parallel MRI. Th ..."
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Cited by 2 (2 self)
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ABSTRACT We present an analytical form of groundtruth kspace data for the 2D SheppLogan brain phantom in the presence of multiple and nonhomogeneous receiving coils. The analytical form allows us to conduct realistic simulations and validations of reconstruction algorithms for parallel MRI. The key contribution of our work is to use a polynomial representation of the coil's sensitivity. We show that this method is particularly accurate and fast with respect to the conventional methods. The implementation is made available to the community.
Regularized MR coil sensitivity estimation using augmented Lagrangian methods
 in Proc. IEEE Intl. Symp. Biomed. Imag., 2012
"... ABSTRACT Several magnetic resonance (MR) parallel imaging techniques require explicit estimates of the receive coil sensitivity profiles. These estimates must be accurate over both the object and its surrounding regions to avoid generating artifacts in the reconstructed images. Statistical estimati ..."
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Cited by 2 (1 self)
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ABSTRACT Several magnetic resonance (MR) parallel imaging techniques require explicit estimates of the receive coil sensitivity profiles. These estimates must be accurate over both the object and its surrounding regions to avoid generating artifacts in the reconstructed images. Statistical estimation methods provide robust sensitivity estimates but can be computationally expensive. In this paper, we propose an augmented Lagrangian (AL) based method that estimates the coil sensitivity by minimizing a quadratic cost function. This method reformulates the finite differencing matrix to allow for exact alternating minimization steps. We also explore a variation of our algorithm that involves intermediate updating of the Lagrange multipliers. We demonstrate that our proposed algorithm converges in half the time of the traditional conjugate gradient method with a circulant preconditioner (PCG) on a real data set.