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22
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 83 (13 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 30 (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 2 (1 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.
SINGLE ECHO ACQUISTION OF MR IMAGES USING RF COIL ARRAYS
"... AbstractParallel imaging methods such as SMASH and SENSE reduce imaging time by using receiver coil sensitivity patterns to reduce requirements for gradient based image localization. This paper describes the first use of a coil array to completely eliminate gradient phase encoding. By using a 64 el ..."
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Cited by 1 (1 self)
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AbstractParallel imaging methods such as SMASH and SENSE reduce imaging time by using receiver coil sensitivity patterns to reduce requirements for gradient based image localization. This paper describes the first use of a coil array to completely eliminate gradient phase encoding. By using a 64 element planar array and a custom built 64 channel receiver system, 64x256 resolution images were constructed from a single line of kspace, demonstrating the ability to form complete images from a single echo acquisition. Together, the receiver and array are capable of generating complete images during successive echo acquisitions, potentially enabling extremely rapid frame rates.
UNSUPERVISED WAVELETBASED REGULARIZATION IN PARALLEL MRI RECONSTRUCTION
, 2009
"... Since the 1990s, parallel Magnetic Resonance Imaging (pMRI) has emerged as a powerful 3D imaging technique for reducing scanning time. To speed up acquisition, the acquired kspace is sampled R times under the Nyquist rate. Full Field of View images are then reconstructed from the aliased data acqu ..."
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Cited by 1 (0 self)
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Since the 1990s, parallel Magnetic Resonance Imaging (pMRI) has emerged as a powerful 3D imaging technique for reducing scanning time. To speed up acquisition, the acquired kspace is sampled R times under the Nyquist rate. Full Field of View images are then reconstructed from the aliased data acquired along complementary coils, by applying for instance the Sensitivity Encoding (SENSE) algorithm [1]. However, SENSEbased reconstructed images suffer from several artifacts because of noise and inaccurate coil sensitivity profiles. The inverse pMRI reconstruction problem being illconditioned, regularization tools allowed us to obtain a significant enhancement of the reconstructed image quality even at high reduction factors (e.g. R = 4) and low magnetic field (1.5 Tesla) [2,3]. In this talk, we summarize our recent advances for regularizing the pMRI reconstruction in the Wavelet Transform (WT) domain, which gives access to sparse image representations. Here, a special attention has to be paid about the regularization model since the data and the unknown image are complexvalued. Several penalty functions, which assume independence or not between real and imaginary parts of the wavelet coefficients, have been successively
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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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
A Maximum Likelihood Approach to Parallel Imaging With Coil Sensitivity Noise
 IEEE TRANSACTIONS ON MEDICAL IMAGING
"... Parallel imaging is a powerful technique to speed up Magnetic Resonance (MR) image acquisition via multiple coils. Both the received signal of each coil and its sensitivity map, which describes its spatial response, are needed during reconstruction. Widely used schemes such as SENSE assume that sens ..."
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Parallel imaging is a powerful technique to speed up Magnetic Resonance (MR) image acquisition via multiple coils. Both the received signal of each coil and its sensitivity map, which describes its spatial response, are needed during reconstruction. Widely used schemes such as SENSE assume that sensitivity maps of the coils are noiseless while the only errors are due to a noisy signal. In practice, however sensitivity maps are subject to a wide variety of errors. At first glance, sensitivity noise appears to result in an errorsinvariables problem of the kind that is typically solved using Total Least Squares (TLS). However, existing TLS algorithms are inappropriate for the specific type of block structure that arises in parallel imaging. In this paper we take a maximum likelihood approach to the problem of parallel imaging in the presence of independent Gaussian sensitivity noise. This results in a quasiquadratic objective function, which can be efficiently minimized. Experimental evidence suggests substantial gains over conventional SENSE, especially in nonideal imaging conditions like low SNR, high gfactors, large acceleration and misaligned sensitivity maps.
Image reconstruction from sensitivity encoded MRI data using extrapolated iterations of parallel projections onto convex sets
"... Parallel imaging techniques for MRI use differences in spatial sensitivity of multiple receiver coils to achieve additional encoding effect and significantly reduce data acquisition time. Recently, a projection onto convex sets (POCS) based method for reconstruction from sensitivityencoded data (PO ..."
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Parallel imaging techniques for MRI use differences in spatial sensitivity of multiple receiver coils to achieve additional encoding effect and significantly reduce data acquisition time. Recently, a projection onto convex sets (POCS) based method for reconstruction from sensitivityencoded data (POCSENSE) has been proposed. The main advantage of the POCSENCE in comparison with other iterative reconstruction techniques is that it offers a straightforward and computationally efficient way to incorporate nonlinear constraints into the reconstruction that can lead to improved image quality and/or reliable reconstruction for underdetermined problems. However, POCSENSE algorithm demonstrates slow convergence in cases of badly conditioned problems. In this work, we propose a novel method for image reconstruction from sensitivity encoded MRI data that overcomes the limitation of the original POCSENSE technique. In the proposed method, the convex combination of projections onto convex sets is used to obtain an updated estimate of the solution via relaxation. The new method converges very efficiently due to the use of an iterationdependent relaxation parameter that may extend far beyond the theoretical limits of POCS. The developed method was validated with phantom and volunteer MRI data and was demonstrated to have a much higher convergence rate than that of the original POCSENSE technique.
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"... Image reconstruction from sensitivity encoded MRI data using extrapolated iterations of parallel projections ..."
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Image reconstruction from sensitivity encoded MRI data using extrapolated iterations of parallel projections
NOISEADAPTIVE ANISOTROPIC DIFFUSION FILTERING OF MRI IMAGES RECONSTRUCTED BY SENSE (SENSITIVITY ENCODING) METHOD
"... SENSE (SENSitivity Encoding) imaging provides significant acquisition speedups in MRI. The main drawback of the method is that it generates images that have increased and spatially nonuniform noise levels and, hence, will often require retrospective filtering. In this paper, we show that standard an ..."
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SENSE (SENSitivity Encoding) imaging provides significant acquisition speedups in MRI. The main drawback of the method is that it generates images that have increased and spatially nonuniform noise levels and, hence, will often require retrospective filtering. In this paper, we show that standard anisotropic diffusion filtering, while being an effective technique for edgepreserving denoising of images with uniform noise levels, is often nonoptimal for SENSEreconstructed data. We have developed a modification of this filter for SENSE images using a robust statistical analysis of the anisotropic diffusion process. The new method utilizes the image noise matrix that is available from the SENSE reconstruction to automatically adjust filtering parameters with local noise levels. The effectiveness of the method and its advantage over standard anisotropic diffusion filtering for SENSE images were demonstrated with phantom and patient MRI data. 1.