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Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization
Hybrid regularization for MRI reconstruction with static field inhomogeneity correction
1. | Department of Mathematics, University of California, Los Angeles, CA 90095, United States |
2. | Department of Chemistry and Biochemistry, University of California, Los Angeles, CA, 90095, United States |
In this work, we demonstrate how one may accelerate the convergence of algorithms for solving the image reconstruction problem, \begin{equation}\label{eq:caneq} (1)         \underset{\rho}{argmin} J(\rho) subject to A \rho = s \end{equation} by opting for a regularization of the form: \begin{equation}\label{eq:hybrid} (2)               J(\rho) = | \nabla \rho| + \nu | F \rho | \end{equation} when $F$ is a tight frame and $A$ is only approximately a Fourier transform. In our experiments, reconstructing field-corrected MR images with the hybrid regularization of 2 provides a speedup of roughly one order of magnitude when compared with an approach based solely on total-variation and may produce higher quality images than an approach based solely on tight frames.
References:
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M. Lustig, D. Donoho and J. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58 (2007), 1182-1195.
doi: 10.1002/mrm.21391. |
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J. Fessler, S. Lee, V. T. Olafsson, H. R. Shi and C. D. Noll, Toeplitz-based iterative image reconstruction for MRI with correction for magnetic field inhomogeneity, IEEE Transactions on Signal Processing, 53 (2005), 3393-3402.
doi: 10.1109/TSP.2005.853152. |
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P. J. Prado, Single sided imaging sensor, Magnetic Resonance Imaging, 21 (2003), 397-400. |
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J. Perlo, F. Casanova and B. Blümich, 3D imaging with a single-sided sensor: An open tomograph, Journal of Magnetic Resonance, 166 (2004), 228-235.
doi: 10.1016/j.jmr.2003.10.018. |
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J. Paulsen, J. Franck, V. Demas and L.-S. Bouchard, Least squares magnetic-field optimization for portable nuclear magnetic resonance magnet design, IEEE Transactions On, 44 (2008), 4582-4590.
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B. Blumich, P. Blumer, G. Eidman, A. Guthausen, R. Haken, U. Schmitz, K. Saito and G. Zimmer, The NMR-mouse: Construction, excitation, and applications, Magnetic Resonance Imaging, 16 (1998), 479-484.
doi: 10.1016/S0730-725X(98)00069-1. |
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A. E. Marble, I. V. Mastikhin, B. G. Colpitts and B. J. Balcom, A constant gradient unilateral magnet for near-surface MRI profiling, Journal of Magnetic Resonance (San Diego, Calif. : 1997), 183 (2006), 228-234.
doi: 10.1016/j.jmr.2006.08.013. |
[9] |
J. M. Franck, V. Demas, R. W. Martin, L.-S. Bouchard and A. Pines, Shimmed matching pulses: Simultaneous control of rf and static gradients for inhomogeneity correction, The Journal of Chemical Physics, 131 (2009), 234506.
doi: 10.1063/1.3243850. |
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D. Topgaard, R. W. Martin, D. Sakellariou, C. A. Meriles and A. Pines, "Shim pulses" for NMR spectroscopy and imaging, Proceedings of the National Academy of Sciences of the United States of America, 101 (2004), 17576-17581.
doi: 10.1073/pnas.0408296102. |
[11] |
C. A. Meriles, D. Sakellariou, H. Heise and A. Pines, Approach to High-Resolution ex Situ NMR spectroscopy, Science, 293 (2001), 82-85.
doi: 10.1126/science.1061498. |
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C. A. Meriles, D. Sakellariou, A. H. Trabesinger, V. Demas and A. Pines, Zero- to low-field MRI with averaging of concomitant gradient fields, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 1840-1842.
doi: 10.1073/pnas.0409115102. |
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N. Kelso, S.-K. Lee, L.-S. Bouchard, V. Demas, M. Mück, A. Pines and J. Clarke, Distortion-free magnetic resonance imaging in the zero-field limit, Journal of magnetic resonance (San Diego, Calif. : 1997), 200 (2009), 285-290.
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J. Aelterman, H. Q. Luong, B. Goossens, A. Pižurica and W. Philips, Augmented Lagrangian based reconstruction of non-uniformly sub-Nyquist sampled MRI data, Signal Processing, 91 (2011), 2731-2742.
doi: 10.1016/j.sigpro.2011.04.033. |
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L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia and P. Ciuciu, A wavelet-based regularized reconstruction algorithm for SENSE parallel MRI with applications to neuroimaging, Medical Image Analysis, 15 (2011), 185-201. |
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Q. Yang, M. Smith and J. Wang, Magnetic susceptibility effects in high field MRI, Biological Magnetic Resonance, 26 (2006), 249-284.
doi: 10.1007/978-0-387-49648-1_9. |
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T.-k. Truong, D. W. Chakeres and P. Schmalbrock, Effects of B 0 and B 1 Inhomogeneity in Ultra-High Field MRI, Proc. Intl. Soc. Mag. Reson. Med., 11 (2004), 2170. |
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J. Reichenbach, R. Venkatesan, D. Yablonskiy, M. R. Thompson and E. M. Haacke, Theory and application of static field inhomogeneity effects in gradient- echo imaging, Journal of Magnetic Resonance Imaging, 7 (1997), 266-279.
doi: 10.1002/jmri.1880070203. |
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M. A. Moerland, R. Beersma, R. Bhagwandien, H. K. Wijrdeman and C. J. Bakker, Analysis and correction of geometric distortions in 1.5 T magnetic resonance images for use in radiotherapy treatment planning, Physics in Medicine and Biology, 40 (1995), 1651-1654. |
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A. Neufeld, Y. Assaf, M. Graif, T. Hendler and G. Navon, Susceptibility-matched envelope for the correction of EPI artifacts, Magnetic Resonance Imaging, 23 (2005), 947-951.
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[26] |
R. C. McKinstry and D. Y. Jarrett, Magnetic susceptibility artifacts on MRI: A hairy situation, American Journal of Roentgenology, 182 (2004), 532-535.
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doi: 10.2214/ajr.177.6.1771487. |
[28] |
F. Baselice, G. Ferraioli and A. Shabou, Field map reconstruction in magnetic resonance imaging using Bayesian estimation, Sensors, 10 (2010), 266-279.
doi: 10.3390/s100100266. |
[29] |
B. Kressler, T. Liu, P. Spincemaille, Q. Jiang and Y. Wang, Nonlinear regularization for per voxel estimation of magnetic susceptibility distributions from MRI field maps, IEEE Transactions on Medical Imaging, 29 (2010), 273-281. |
[30] |
J. A. Fessler, S. Member and B. P. Sutton, Nonuniform fast Fourier transforms using min-max interpolation, IEEE Trans. Signal Process, 51 (2003), 560-574.
doi: 10.1109/TSP.2002.807005. |
[31] |
J. Fessler, Model-based image reconstruction for MRI, IEEE Signal Processing Magazine, 27 (2010), 81-89.
doi: 10.1109/MSP.2010.936726. |
[32] |
B. P. Sutton, S. Member, D. C. Noll, J. A. Fessler and S. Member, Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities, IEEE Transactions on Medical Imaging, 22 (2003), 178-188.
doi: 10.1109/TMI.2002.808360. |
[33] |
K. T. Block, M. Uecker and J. Frahm, Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint, Magnetic Resonance in Medicine, 57 (2007), 1086-1098.
doi: 10.1002/mrm.21236. |
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S. Ramani and J. A. Fessler, An accelerated iterative reweighted least squares algorithm for compressed sensing MRI, IEEE ISBI, (2010), 257-260.
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B. J. Wilm, C. Barmet, M. Pavan and K. P. Pruessmann, Higher order reconstruction for MRI in the presence of spatiotemporal field perturbations, Magnetic Resonance in Medicine, 65 (2011), 1690-1701.
doi: 10.1002/mrm.22767. |
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W. Chen, C. T. Sica and C. H. Meyer, Fast conjugate phase image reconstruction based on a Chebyshev approximation to correct for B0 field inhomogeneity and concomitant gradients, Magnetic Resonance in Medicine, 60 (2008), 1104-1111.
doi: 10.1002/mrm.21703. |
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H. Schomberg, Off-resonance correction of MR images, IEEE Transactions on Medical Imaging, 18 (1999), 481-495.
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D. C. Noll, J. A. Fessler and B. P. Sutton, Conjugate phase MRI reconstruction with spatially variant sample density correction, IEEE Transactions on Medical Imaging, 24 (2005), 325-336.
doi: 10.1109/TMI.2004.842452. |
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T. Goldstein and S. Osher, The split bregman method for L1-Regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.
doi: 10.1137/080725891. |
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J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling & Simulation, 8 (2010), 337-369.
doi: 10.1137/090753504. |
[41] |
J. Fessler, S. Lee, V. Olafsson, H. Shi and D. Noll, Toeplitz-based iterative image reconstruction for MRI with correction for magnetic field inhomogeneity, IEEE Transactions on Signal Processing, 53 (2005), 3393-3402.
doi: 10.1109/TSP.2005.853152. |
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L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast fourier transform, SIAM Review, 46 (2004), 443-454.
doi: 10.1137/S003614450343200X. |
[43] |
P. Irarrazabal, C. H. Meyer, D. G. Nishimura and A. Macovski, Inhomogeneity correction using an estimated linear field map, Magnetic Resonance in Medicine, 35 (1996), 278-282.
doi: 10.1002/mrm.1910350221. |
[44] |
L.-C. Man, J. M. Pauly and A. Macovski, Multifrequency interpolation for fast off-resonance correction, Magnetic Resonance in Medicine, 37 (1997), 785-792.
doi: 10.1002/mrm.1910370523. |
[45] |
D. Noll, Reconstruction Techniques for Magnetic Reasonance Imaging, PhD thesis, Stanford University, 1991. |
[46] |
H. Moriguchi, B. M. Dale, J. S. Lewin and J. L. Duerk, Block regional off-resonance correction (BRORC): A fast and effective deblurring method for spiral imaging, Magnetic Resonance in Medicine, 50 (2003), 643-648.
doi: 10.1002/mrm.10570. |
[47] |
V. Rokhlin, A. Szlam and M. Tygert, A randomized algorithm for PCA, SIAM J. Matrix anal. appl., 31 (2009), 1100-1124.
doi: 10.1137/080736417. |
[48] |
P.-G. Martinsson, V. Rokhlin and M. Tygert, A randomized algorithm for the decomposition of matrices, Applied and Computational Harmonic Analysis, 30 (2011), 47-68.
doi: 10.1016/j.acha.2010.02.003. |
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N. Halko, P. Martinsson and J. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Review, 53 (2011), 217-288.
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show all references
References:
[1] |
M. Lustig, D. Donoho and J. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58 (2007), 1182-1195.
doi: 10.1002/mrm.21391. |
[2] |
J. Fessler, S. Lee, V. T. Olafsson, H. R. Shi and C. D. Noll, Toeplitz-based iterative image reconstruction for MRI with correction for magnetic field inhomogeneity, IEEE Transactions on Signal Processing, 53 (2005), 3393-3402.
doi: 10.1109/TSP.2005.853152. |
[3] |
P. Mansfield, NMR Imaging in Biomedicine (Advances in Magnetic Resonance), Academic Press, 1982. |
[4] |
P. J. Prado, Single sided imaging sensor, Magnetic Resonance Imaging, 21 (2003), 397-400. |
[5] |
J. Perlo, F. Casanova and B. Blümich, 3D imaging with a single-sided sensor: An open tomograph, Journal of Magnetic Resonance, 166 (2004), 228-235.
doi: 10.1016/j.jmr.2003.10.018. |
[6] |
J. Paulsen, J. Franck, V. Demas and L.-S. Bouchard, Least squares magnetic-field optimization for portable nuclear magnetic resonance magnet design, IEEE Transactions On, 44 (2008), 4582-4590.
doi: 10.1109/TMAG.2008.2001697. |
[7] |
B. Blumich, P. Blumer, G. Eidman, A. Guthausen, R. Haken, U. Schmitz, K. Saito and G. Zimmer, The NMR-mouse: Construction, excitation, and applications, Magnetic Resonance Imaging, 16 (1998), 479-484.
doi: 10.1016/S0730-725X(98)00069-1. |
[8] |
A. E. Marble, I. V. Mastikhin, B. G. Colpitts and B. J. Balcom, A constant gradient unilateral magnet for near-surface MRI profiling, Journal of Magnetic Resonance (San Diego, Calif. : 1997), 183 (2006), 228-234.
doi: 10.1016/j.jmr.2006.08.013. |
[9] |
J. M. Franck, V. Demas, R. W. Martin, L.-S. Bouchard and A. Pines, Shimmed matching pulses: Simultaneous control of rf and static gradients for inhomogeneity correction, The Journal of Chemical Physics, 131 (2009), 234506.
doi: 10.1063/1.3243850. |
[10] |
D. Topgaard, R. W. Martin, D. Sakellariou, C. A. Meriles and A. Pines, "Shim pulses" for NMR spectroscopy and imaging, Proceedings of the National Academy of Sciences of the United States of America, 101 (2004), 17576-17581.
doi: 10.1073/pnas.0408296102. |
[11] |
C. A. Meriles, D. Sakellariou, H. Heise and A. Pines, Approach to High-Resolution ex Situ NMR spectroscopy, Science, 293 (2001), 82-85.
doi: 10.1126/science.1061498. |
[12] |
C. A. Meriles, D. Sakellariou, A. H. Trabesinger, V. Demas and A. Pines, Zero- to low-field MRI with averaging of concomitant gradient fields, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 1840-1842.
doi: 10.1073/pnas.0409115102. |
[13] |
N. Kelso, S.-K. Lee, L.-S. Bouchard, V. Demas, M. Mück, A. Pines and J. Clarke, Distortion-free magnetic resonance imaging in the zero-field limit, Journal of magnetic resonance (San Diego, Calif. : 1997), 200 (2009), 285-290.
doi: 10.1016/j.jmr.2009.07.016. |
[14] |
L.-S. Bouchard, Unidirectional magnetic-field gradients and geometric-phase errors during Fourier encoding using orthogonal ac fields, Physical Review B, 74 (2006), 1-11.
doi: 10.1103/PhysRevB.74.054103. |
[15] |
L.-S. Bouchard and M. Anwar, Synthesis of matched magnetic fields for controlled spin precession, Physical Review B, 76 (2007), 1-10.
doi: 10.1103/PhysRevB.76.014430. |
[16] |
E. M. Haacke and R. Brown, Magnetic Resonance Imaging Physical Principles and Sequence Design, 1999. |
[17] |
J. Romberg, Imaging via compressive sampling, IEEE Signal Processing Magazine, 25 (2008), 14-20.
doi: 10.1109/MSP.2007.914729. |
[18] |
M. Guerquin-Kern, M. Häberlin, K. P. Pruessmann and M. Unser, A fast wavelet-based reconstruction method for magnetic resonance imaging, IEEE transactions on medical imaging, 30 (2011), 1649-1460.
doi: 10.1109/TMI.2011.2140121. |
[19] |
J. Aelterman, H. Q. Luong, B. Goossens, A. Pižurica and W. Philips, Augmented Lagrangian based reconstruction of non-uniformly sub-Nyquist sampled MRI data, Signal Processing, 91 (2011), 2731-2742.
doi: 10.1016/j.sigpro.2011.04.033. |
[20] |
L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia and P. Ciuciu, A wavelet-based regularized reconstruction algorithm for SENSE parallel MRI with applications to neuroimaging, Medical Image Analysis, 15 (2011), 185-201. |
[21] |
Q. Yang, M. Smith and J. Wang, Magnetic susceptibility effects in high field MRI, Biological Magnetic Resonance, 26 (2006), 249-284.
doi: 10.1007/978-0-387-49648-1_9. |
[22] |
T.-k. Truong, D. W. Chakeres and P. Schmalbrock, Effects of B 0 and B 1 Inhomogeneity in Ultra-High Field MRI, Proc. Intl. Soc. Mag. Reson. Med., 11 (2004), 2170. |
[23] |
J. Reichenbach, R. Venkatesan, D. Yablonskiy, M. R. Thompson and E. M. Haacke, Theory and application of static field inhomogeneity effects in gradient- echo imaging, Journal of Magnetic Resonance Imaging, 7 (1997), 266-279.
doi: 10.1002/jmri.1880070203. |
[24] |
M. A. Moerland, R. Beersma, R. Bhagwandien, H. K. Wijrdeman and C. J. Bakker, Analysis and correction of geometric distortions in 1.5 T magnetic resonance images for use in radiotherapy treatment planning, Physics in Medicine and Biology, 40 (1995), 1651-1654. |
[25] |
A. Neufeld, Y. Assaf, M. Graif, T. Hendler and G. Navon, Susceptibility-matched envelope for the correction of EPI artifacts, Magnetic Resonance Imaging, 23 (2005), 947-951.
doi: 10.1016/j.mri.2005.07.011. |
[26] |
R. C. McKinstry and D. Y. Jarrett, Magnetic susceptibility artifacts on MRI: A hairy situation, American Journal of Roentgenology, 182 (2004), 532-535.
doi: 10.2214/ajr.182.2.1820532. |
[27] |
I. C. Duncan, The "Aura'' sign: An unusual cultural variant affecting MR imaging, American Journal of Roentgenology, 177 (2001), 1485-1489.
doi: 10.2214/ajr.177.6.1771487. |
[28] |
F. Baselice, G. Ferraioli and A. Shabou, Field map reconstruction in magnetic resonance imaging using Bayesian estimation, Sensors, 10 (2010), 266-279.
doi: 10.3390/s100100266. |
[29] |
B. Kressler, T. Liu, P. Spincemaille, Q. Jiang and Y. Wang, Nonlinear regularization for per voxel estimation of magnetic susceptibility distributions from MRI field maps, IEEE Transactions on Medical Imaging, 29 (2010), 273-281. |
[30] |
J. A. Fessler, S. Member and B. P. Sutton, Nonuniform fast Fourier transforms using min-max interpolation, IEEE Trans. Signal Process, 51 (2003), 560-574.
doi: 10.1109/TSP.2002.807005. |
[31] |
J. Fessler, Model-based image reconstruction for MRI, IEEE Signal Processing Magazine, 27 (2010), 81-89.
doi: 10.1109/MSP.2010.936726. |
[32] |
B. P. Sutton, S. Member, D. C. Noll, J. A. Fessler and S. Member, Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities, IEEE Transactions on Medical Imaging, 22 (2003), 178-188.
doi: 10.1109/TMI.2002.808360. |
[33] |
K. T. Block, M. Uecker and J. Frahm, Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint, Magnetic Resonance in Medicine, 57 (2007), 1086-1098.
doi: 10.1002/mrm.21236. |
[34] |
S. Ramani and J. A. Fessler, An accelerated iterative reweighted least squares algorithm for compressed sensing MRI, IEEE ISBI, (2010), 257-260.
doi: 10.1109/ISBI.2010.5490364. |
[35] |
B. J. Wilm, C. Barmet, M. Pavan and K. P. Pruessmann, Higher order reconstruction for MRI in the presence of spatiotemporal field perturbations, Magnetic Resonance in Medicine, 65 (2011), 1690-1701.
doi: 10.1002/mrm.22767. |
[36] |
W. Chen, C. T. Sica and C. H. Meyer, Fast conjugate phase image reconstruction based on a Chebyshev approximation to correct for B0 field inhomogeneity and concomitant gradients, Magnetic Resonance in Medicine, 60 (2008), 1104-1111.
doi: 10.1002/mrm.21703. |
[37] |
H. Schomberg, Off-resonance correction of MR images, IEEE Transactions on Medical Imaging, 18 (1999), 481-495.
doi: 10.1109/42.781014. |
[38] |
D. C. Noll, J. A. Fessler and B. P. Sutton, Conjugate phase MRI reconstruction with spatially variant sample density correction, IEEE Transactions on Medical Imaging, 24 (2005), 325-336.
doi: 10.1109/TMI.2004.842452. |
[39] |
T. Goldstein and S. Osher, The split bregman method for L1-Regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.
doi: 10.1137/080725891. |
[40] |
J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling & Simulation, 8 (2010), 337-369.
doi: 10.1137/090753504. |
[41] |
J. Fessler, S. Lee, V. Olafsson, H. Shi and D. Noll, Toeplitz-based iterative image reconstruction for MRI with correction for magnetic field inhomogeneity, IEEE Transactions on Signal Processing, 53 (2005), 3393-3402.
doi: 10.1109/TSP.2005.853152. |
[42] |
L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast fourier transform, SIAM Review, 46 (2004), 443-454.
doi: 10.1137/S003614450343200X. |
[43] |
P. Irarrazabal, C. H. Meyer, D. G. Nishimura and A. Macovski, Inhomogeneity correction using an estimated linear field map, Magnetic Resonance in Medicine, 35 (1996), 278-282.
doi: 10.1002/mrm.1910350221. |
[44] |
L.-C. Man, J. M. Pauly and A. Macovski, Multifrequency interpolation for fast off-resonance correction, Magnetic Resonance in Medicine, 37 (1997), 785-792.
doi: 10.1002/mrm.1910370523. |
[45] |
D. Noll, Reconstruction Techniques for Magnetic Reasonance Imaging, PhD thesis, Stanford University, 1991. |
[46] |
H. Moriguchi, B. M. Dale, J. S. Lewin and J. L. Duerk, Block regional off-resonance correction (BRORC): A fast and effective deblurring method for spiral imaging, Magnetic Resonance in Medicine, 50 (2003), 643-648.
doi: 10.1002/mrm.10570. |
[47] |
V. Rokhlin, A. Szlam and M. Tygert, A randomized algorithm for PCA, SIAM J. Matrix anal. appl., 31 (2009), 1100-1124.
doi: 10.1137/080736417. |
[48] |
P.-G. Martinsson, V. Rokhlin and M. Tygert, A randomized algorithm for the decomposition of matrices, Applied and Computational Harmonic Analysis, 30 (2011), 47-68.
doi: 10.1016/j.acha.2010.02.003. |
[49] |
N. Halko, P. Martinsson and J. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Review, 53 (2011), 217-288.
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