# American Institute of Mathematical Sciences

November  2013, 7(4): 1215-1233. doi: 10.3934/ipi.2013.7.1215

## 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

Received  June 2012 Revised  January 2013 Published  November 2013

Rapid acquisition of magnetic resonance (MR) images via reconstruction from undersampled $k$-space data has the potential to greatly decrease MRI scan time on existing medical hardware. To this end, iterative image reconstruction based on the technique of compressed sensing has become the method choice for many researchers [1]. However, while conventional compressed sensing relies on random measurements from a discrete Fourier transform, actual MR scans often suffer from off-resonance effects and thus generate data by way of a non-Fourier operator [2]. Correcting for these effects requires that one employs more sophisticated image reconstruction methods and introduces computational bottlenecks that are not encountered in standard compressed sensing.
In this work, we demonstrate how one may accelerate the convergence of algorithms for solving the image reconstruction problem, $$\label{eq:caneq} (1) \underset{\rho}{argmin} J(\rho) subject to A \rho = s$$ by opting for a regularization of the form: $$\label{eq:hybrid} (2) J(\rho) = | \nabla \rho| + \nu | F \rho |$$ 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.
Citation: Ryan Compton, Stanley Osher, Louis-S. Bouchard. Hybrid regularization for MRI reconstruction with static field inhomogeneity correction. Inverse Problems & Imaging, 2013, 7 (4) : 1215-1233. doi: 10.3934/ipi.2013.7.1215
##### 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. 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. 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. [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. 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. 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. 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, 183 (2006), 228. 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). 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. 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. 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. 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, 200 (2009), 285. 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. 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. 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. 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. 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. 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. [21] Q. Yang, M. Smith and J. Wang, Magnetic susceptibility effects in high field MRI,, Biological Magnetic Resonance, 26 (2006), 249. 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). [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. 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. [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. 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. 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. 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. 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. [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. doi: 10.1109/TSP.2002.807005. [31] J. Fessler, Model-based image reconstruction for MRI,, IEEE Signal Processing Magazine, 27 (2010), 81. 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. 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. 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. 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. 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. doi: 10.1002/mrm.21703. [37] H. Schomberg, Off-resonance correction of MR images,, IEEE Transactions on Medical Imaging, 18 (1999), 481. 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. 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. 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. 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. doi: 10.1109/TSP.2005.853152. [42] L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast fourier transform,, SIAM Review, 46 (2004), 443. 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. 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. doi: 10.1002/mrm.1910370523. [45] D. Noll, Reconstruction Techniques for Magnetic Reasonance Imaging,, PhD thesis, (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. 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. 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. 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. doi: 10.1137/090771806. [50] E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin and M. Tygert, Randomized algorithms for the low-rank approximation of matrices,, Proceedings of the National Academy of Sciences of the United States of America, 104 (2007), 20167. doi: 10.1073/pnas.0709640104. [51] C. Papadimitriou, H. Tamaki and P. Raghavan, Latent semantic indexing: A probabilistic analysis,, Proceedings of the, (1998), 1. doi: 10.1145/275487.275505. [52] A. Çivril and M. Magdon-Ismail, On selecting a maximum volume sub-matrix of a matrix and related problems,, Theoretical Computer Science, 410 (2009), 4801. doi: 10.1016/j.tcs.2009.06.018. [53] G. Golub and C. V. Loan, Matrix Computations,, Johns Hopkins University Press, (1996). [54] M. Gu and S. C. Eisenstat, Efficient algorithm for computing a strong Rank-Revealing QR factorization,, SIAM Journal on Scientific Computing, 17 (1996), 848. doi: 10.1137/0917055. [55] E. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements,, Comm. Pure Appl. Math., 59 (2006), 1207. doi: 10.1002/cpa.20124. [56] E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,, IEEE Transactions on Information Theory, 52 (2006), 489. doi: 10.1109/TIT.2005.862083. [57] D. Liang, H. Wang and L. Ying, SENSE reconstruction with nonlocal TV regularization,, IEEE Engineering in Medicine and Biology Society, 2009 (2009), 1032. [58] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration,, Multiscale Modeling & Simulation, 4 (2005), 460. doi: 10.1137/040605412. [59] M. Frigo and S. G. Johnson, The design and implementation of FFTW3,, Proceedings of the IEEE, 93 (2005), 216. doi: 10.1109/JPROC.2004.840301. [60] A. K. Funai, J. A. Fessler, D. T. B. Yeo, V. T. Olafsson and D. C. Noll, Regularized field map estimation in MRI,, IEEE Transactions on Medical Imaging, 27 (2008), 1484. doi: 10.1109/TMI.2008.923956.

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##### 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. 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. 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. [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. 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. 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. 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, 183 (2006), 228. 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). 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. 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. 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. 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, 200 (2009), 285. 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. 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. 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. 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. 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. 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. [21] Q. Yang, M. Smith and J. Wang, Magnetic susceptibility effects in high field MRI,, Biological Magnetic Resonance, 26 (2006), 249. 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). [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. 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. [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. 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. 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. 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. 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. [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. doi: 10.1109/TSP.2002.807005. [31] J. Fessler, Model-based image reconstruction for MRI,, IEEE Signal Processing Magazine, 27 (2010), 81. 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. 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. 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. 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. 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. doi: 10.1002/mrm.21703. [37] H. Schomberg, Off-resonance correction of MR images,, IEEE Transactions on Medical Imaging, 18 (1999), 481. 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. 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. 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. 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. doi: 10.1109/TSP.2005.853152. [42] L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast fourier transform,, SIAM Review, 46 (2004), 443. 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. 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. doi: 10.1002/mrm.1910370523. [45] D. Noll, Reconstruction Techniques for Magnetic Reasonance Imaging,, PhD thesis, (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. 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. 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. 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. doi: 10.1137/090771806. [50] E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin and M. Tygert, Randomized algorithms for the low-rank approximation of matrices,, Proceedings of the National Academy of Sciences of the United States of America, 104 (2007), 20167. doi: 10.1073/pnas.0709640104. [51] C. Papadimitriou, H. Tamaki and P. Raghavan, Latent semantic indexing: A probabilistic analysis,, Proceedings of the, (1998), 1. doi: 10.1145/275487.275505. [52] A. Çivril and M. Magdon-Ismail, On selecting a maximum volume sub-matrix of a matrix and related problems,, Theoretical Computer Science, 410 (2009), 4801. doi: 10.1016/j.tcs.2009.06.018. [53] G. Golub and C. V. Loan, Matrix Computations,, Johns Hopkins University Press, (1996). [54] M. Gu and S. C. Eisenstat, Efficient algorithm for computing a strong Rank-Revealing QR factorization,, SIAM Journal on Scientific Computing, 17 (1996), 848. doi: 10.1137/0917055. [55] E. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements,, Comm. Pure Appl. Math., 59 (2006), 1207. doi: 10.1002/cpa.20124. [56] E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,, IEEE Transactions on Information Theory, 52 (2006), 489. doi: 10.1109/TIT.2005.862083. [57] D. Liang, H. Wang and L. Ying, SENSE reconstruction with nonlocal TV regularization,, IEEE Engineering in Medicine and Biology Society, 2009 (2009), 1032. [58] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration,, Multiscale Modeling & Simulation, 4 (2005), 460. doi: 10.1137/040605412. [59] M. Frigo and S. G. Johnson, The design and implementation of FFTW3,, Proceedings of the IEEE, 93 (2005), 216. doi: 10.1109/JPROC.2004.840301. [60] A. K. Funai, J. A. Fessler, D. T. B. Yeo, V. T. Olafsson and D. C. Noll, Regularized field map estimation in MRI,, IEEE Transactions on Medical Imaging, 27 (2008), 1484. doi: 10.1109/TMI.2008.923956.
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