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31
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 88 (15 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 34 (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 4 (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.
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 1 (0 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
3D waveletbased regularization for parallel MRI reconstruction: impact on subject and grouplevel statistical sensitivity in fMRI
 In Proc. 8th IEEE Int. Symp. Biomed. Imaging (ISBI’11
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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 1 (1 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.
Single Echo Acquistion of MR Images Using RF Coil Arrays
 in Proceedings of the Second Joint EMBS/BMES Conference
, 2002
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An Iterative Method for Parallel MRI SENSEbased Reconstruction in the Wavelet Domain
, 909
"... To reduce scanning time and/or improve spatial/temporal resolution in some MRI applications, parallel MRI (pMRI) acquisition techniques with multiple coils acquisition have emerged since the early 1990’s as powerful 3D imaging methods that allow faster acquisition of reduced Field of View (FOV) imag ..."
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To reduce scanning time and/or improve spatial/temporal resolution in some MRI applications, parallel MRI (pMRI) acquisition techniques with multiple coils acquisition have emerged since the early 1990’s as powerful 3D imaging methods that allow faster acquisition of reduced Field of View (FOV) images. In these techniques, the full FOV image has to be reconstructed from the resulting acquired undersampled kspace data. To this end, several reconstruction techniques have been proposed such as the widelyused SENSE method. However, the reconstructed image generally presents artifacts when perturbations occur in both the measured data and the estimated coil sensitivity maps. In this paper, we aim at achieving good reconstructed image quality when using low magnetic field and high reduction factor. Under these severe experimental conditions, neither the SENSE method nor the Tikhonov regularization in the image domain give convincing results. To this aim, we present a novel method for SENSEbased reconstruction which proceeds with regularization in the complex wavelet domain. To further en
MULTICHANNEL METHODS FOR RESTORATION IN COMPUTED IMAGING
, 2007
"... This dissertation addresses datadriven image restoration for computed imaging systems. The work is focused on problems in two imaging modalities: the autofocus problem in synthetic aperture radar (SAR), and the problem of estimating coil sensitivities in parallel magnetic resonance imaging (PMRI). ..."
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This dissertation addresses datadriven image restoration for computed imaging systems. The work is focused on problems in two imaging modalities: the autofocus problem in synthetic aperture radar (SAR), and the problem of estimating coil sensitivities in parallel magnetic resonance imaging (PMRI). A common thread in both problems is their inherent multichannel nature, i.e., both exhibit special structure due to the redundancy provided by multiple signal measurements. By explicitly exploiting the multichannel structure, novel algorithms are developed offering improved restoration performance. We first present a theoretical study providing more insight into metricbased SAR autofocus techniques. Our analytical results show how metricbased methods implicitly rely on the multichannel defocusing model of SAR autofocus to form wellfocused restorations. Utilizing the multichannel structure of the SAR autofocus problem explicitly, we develop a new noniterative restoration approach termed the MuliChannel Autofocus (MCA) algorithm. In this approach, the focused image is directly recovered using a linear algebraic formulation. Experimental results using actual and simulated SAR data demonstrate that MCA provides superior performance in comparison with existing autofocus methods. Lastly, we develop a new subspacebased approach for estimating receiver coil sensitivity functions used in PMRI reconstruction. Our approach does not rely on sumofsquares assumptions used in previous PMRI techniques, thus avoiding potential problems such as poor image contrast and aliasing artifacts.
Nonlocal transformdomain denoising of volumetric data with groupwise adaptive variance estimation
"... We propose an extension of the BM4D volumetric filter to the denoising of data corrupted by spatially nonuniform noise. BM4D implements the grouping and collaborative filtering paradigm, where similar cubes of voxels are stacked into a fourdimensional “group”. Each group undergoes a sparsifying fou ..."
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We propose an extension of the BM4D volumetric filter to the denoising of data corrupted by spatially nonuniform noise. BM4D implements the grouping and collaborative filtering paradigm, where similar cubes of voxels are stacked into a fourdimensional “group”. Each group undergoes a sparsifying fourdimensional transform, that exploits the local correlation among voxels in each cube and the nonlocal correlation between corresponding voxels of different cubes. Thus, signal and noise are effectively separated in transform domain. In this work we take advantage of the sparsity induced by the fourdimensional transform to provide a spatially adaptive estimation of the local noise variance by applying a robust median estimator of the absolute deviation to the spectrum of each filtered group. The adaptive variance estimates are then used during coefficients shrinkage. Finally, the inverse fourdimensional transform is applied to the filtered group, and each individual cube estimate is adaptively aggregated at its original location. Experiments on medical data corrupted by spatially varying Gaussian and Rician noise demonstrate the efficacy of the proposed approach in volumetric data denoising. In case of magnetic resonance signals, the adaptive variance estimate can be also used to compensate the estimation bias due to the nonzeromean errors of the Riciandistributed data.