# American Institute of Mathematical Sciences

2014, 8(3): 925-937. doi: 10.3934/ipi.2014.8.925

## Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing

 1 School of Science, Communication University of China, Beijing, 100024, China, China, China 2 Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

Received  August 2012 Revised  August 2013 Published  August 2014

The problem of compressive-sensing (CS) L2-L1-TV reconstruction of magnetic resonance (MR) scans from undersampled $k$-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regularization parameters are given, the solution for an MR reconstruction model is fixed and is less effective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the proposed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via flexible parameter selection.
Citation: Yonggui Zhu, Yuying Shi, Bin Zhang, Xinyan Yu. Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing. Inverse Problems & Imaging, 2014, 8 (3) : 925-937. doi: 10.3934/ipi.2014.8.925
##### References:
 [1] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,, IEEE Transaction on Image Processing, 5 (2009), 2419. doi: 10.1109/TIP.2009.2028250. [2] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Sciences, 2 (2009), 183. doi: 10.1137/080716542. [3] E. J. Candès, 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. [4] E. J. Candès and J. Romberg, Sparsity and incoherence in compressive sampling,, Inverse Problems, 23 (2007), 969. doi: 10.1088/0266-5611/23/3/008. [5] Y. Chen, X. Ye and F. Huang, A novel method and fast algorithm for MR image reconstruction with significantly under-sampled data,, Inverse Problems and Imaging, 4 (2010), 223. doi: 10.3934/ipi.2010.4.223. [6] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Modeling and Simulation, 4 (2005), 1168. doi: 10.1137/050626090. [7] P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators,, Journal of Convex Analysis, 16 (2009), 727. [8] P. L. Combettes and J.-C. Pesquet, A proximal decomposition method for solving convex variational in verse problems,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/6/065014. [9] D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. [10] D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decompositions,, IEEE Transactions on Information Theory, 47 (2001), 2845. doi: 10.1109/18.959265. [11] J. Eckstein and B. F. Svaiter, General projective splitting methods for sums of maximal monotone operators,, SIAM Journal on Control Optimization, 48 (2009), 787. doi: 10.1137/070698816. [12] M. Figueiredo, R. Nowak and S. Wright, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 586. doi: 10.1109/JSTSP.2007.910281. [13] J.-J. Fuchs, On sparse representations in arbitrary redundant bases,, IEEE Transactions on Information Theory, 50 (2004), 1341. doi: 10.1109/TIT.2004.828141. [14] D. Gabay, Chapter IX applications of the method of multipliers to variational inequalities,, Studies in Mathematics and its Applications, 15 (1983), 299. doi: 10.1016/S0168-2024(08)70034-1. [15] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations,, Computers and Mathematics with Applications, 2 (1976), 17. doi: 10.1016/0898-1221(76)90003-1. [16] R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM Studies in Applied Mathematics, (1989). doi: 10.1137/1.9781611970838. [17] D. Goldfarb and S. Ma, Fast multiple-splitting algorithms for convex optimization,, SIAM Journal on Optimization, 22 (2012), 533. doi: 10.1137/090780705. [18] T. Goldstein and O. Osher, The split Bregman method for L1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. [19] J. Huang, S. Zhang and D. Metaxas, Efficient MR image reconstruction for compressed MR imaging,, in Medical Image Computing and Computer-Assisted Intervention-MICCAI 2010, (2010), 135. doi: 10.1007/978-3-642-15705-9_17. [20] M. Lustig, D. L. 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. [21] J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. [22] B. K. Natarajan, Sparse approximate solutions to linear systems,, SIAM Journal on Computing, 24 (1995), 227. doi: 10.1137/S0097539792240406. [23] Z. Opal, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 591. doi: 10.1090/S0002-9904-1967-11761-0. [24] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [25] J. E. Spingarn, Partial inverse of monotone operator,, Applied Mathematics and Optimization, (1983), 247. doi: 10.1007/BF01448388. [26] X. C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,, in Scale Space and Variational Methods in Computer Vision, (5567), 502. [27] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings,, SIAM Journal on Control and Optimization, 38 (2000), 431. doi: 10.1137/S0363012998338806. [28] P. Tseng and S. Yun, A coordinate gradient descent method for nonsmooth separable minimization,, Math. Prog. B., 117 (2009), 387. doi: 10.1007/s10107-007-0170-0. [29] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM Journal on Imaging Sciences, 1 (2008), 248. doi: 10.1137/080724265. [30] J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TV-L1-L2 Signal reconstruction from partial Fourier data,, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288. [31] W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman Iterative algorithms for $l_1$-minimization with applications to compressed sensing,, SIAM Journal on Imaging Sciences, 1 (2008), 143. doi: 10.1137/070703983. [32] Y. Zhu and I. Chern, Fast alternating minimization method for compressive sensing MRI under wavelet sparsity and TV sparsity,, in 2011 Sixth International Conference on Image and Graphics, (2011), 356. doi: 10.1109/ICIG.2011.23. [33] Y. Zhu and I. Chern, Convergence of the alternating minimization method for sparse MR image reconstruction,, Journal of Information and Computational Science, 8 (2011), 2067. [34] Y. Zhu and Y. Shi, A fast method for reconstruction of total-variation MR images with a periodic boundary condition,, IEEE Signal Processing Letters, 20 (2013), 291. doi: 10.1109/LSP.2013.2245502.

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##### References:
 [1] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,, IEEE Transaction on Image Processing, 5 (2009), 2419. doi: 10.1109/TIP.2009.2028250. [2] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems,, SIAM Journal on Imaging Sciences, 2 (2009), 183. doi: 10.1137/080716542. [3] E. J. Candès, 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. [4] E. J. Candès and J. Romberg, Sparsity and incoherence in compressive sampling,, Inverse Problems, 23 (2007), 969. doi: 10.1088/0266-5611/23/3/008. [5] Y. Chen, X. Ye and F. Huang, A novel method and fast algorithm for MR image reconstruction with significantly under-sampled data,, Inverse Problems and Imaging, 4 (2010), 223. doi: 10.3934/ipi.2010.4.223. [6] P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting,, Multiscale Modeling and Simulation, 4 (2005), 1168. doi: 10.1137/050626090. [7] P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators,, Journal of Convex Analysis, 16 (2009), 727. [8] P. L. Combettes and J.-C. Pesquet, A proximal decomposition method for solving convex variational in verse problems,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/6/065014. [9] D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. [10] D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decompositions,, IEEE Transactions on Information Theory, 47 (2001), 2845. doi: 10.1109/18.959265. [11] J. Eckstein and B. F. Svaiter, General projective splitting methods for sums of maximal monotone operators,, SIAM Journal on Control Optimization, 48 (2009), 787. doi: 10.1137/070698816. [12] M. Figueiredo, R. Nowak and S. Wright, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 586. doi: 10.1109/JSTSP.2007.910281. [13] J.-J. Fuchs, On sparse representations in arbitrary redundant bases,, IEEE Transactions on Information Theory, 50 (2004), 1341. doi: 10.1109/TIT.2004.828141. [14] D. Gabay, Chapter IX applications of the method of multipliers to variational inequalities,, Studies in Mathematics and its Applications, 15 (1983), 299. doi: 10.1016/S0168-2024(08)70034-1. [15] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations,, Computers and Mathematics with Applications, 2 (1976), 17. doi: 10.1016/0898-1221(76)90003-1. [16] R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM Studies in Applied Mathematics, (1989). doi: 10.1137/1.9781611970838. [17] D. Goldfarb and S. Ma, Fast multiple-splitting algorithms for convex optimization,, SIAM Journal on Optimization, 22 (2012), 533. doi: 10.1137/090780705. [18] T. Goldstein and O. Osher, The split Bregman method for L1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. [19] J. Huang, S. Zhang and D. Metaxas, Efficient MR image reconstruction for compressed MR imaging,, in Medical Image Computing and Computer-Assisted Intervention-MICCAI 2010, (2010), 135. doi: 10.1007/978-3-642-15705-9_17. [20] M. Lustig, D. L. 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. [21] J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. [22] B. K. Natarajan, Sparse approximate solutions to linear systems,, SIAM Journal on Computing, 24 (1995), 227. doi: 10.1137/S0097539792240406. [23] Z. Opal, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 591. doi: 10.1090/S0002-9904-1967-11761-0. [24] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [25] J. E. Spingarn, Partial inverse of monotone operator,, Applied Mathematics and Optimization, (1983), 247. doi: 10.1007/BF01448388. [26] X. C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,, in Scale Space and Variational Methods in Computer Vision, (5567), 502. [27] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings,, SIAM Journal on Control and Optimization, 38 (2000), 431. doi: 10.1137/S0363012998338806. [28] P. Tseng and S. Yun, A coordinate gradient descent method for nonsmooth separable minimization,, Math. Prog. B., 117 (2009), 387. doi: 10.1007/s10107-007-0170-0. [29] Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM Journal on Imaging Sciences, 1 (2008), 248. doi: 10.1137/080724265. [30] J. Yang, Y. Zhang and W. Yin, A fast alternating direction method for TV-L1-L2 Signal reconstruction from partial Fourier data,, IEEE Journal of Selected Topics in Signal Processing, 4 (2010), 288. [31] W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman Iterative algorithms for $l_1$-minimization with applications to compressed sensing,, SIAM Journal on Imaging Sciences, 1 (2008), 143. doi: 10.1137/070703983. [32] Y. Zhu and I. Chern, Fast alternating minimization method for compressive sensing MRI under wavelet sparsity and TV sparsity,, in 2011 Sixth International Conference on Image and Graphics, (2011), 356. doi: 10.1109/ICIG.2011.23. [33] Y. Zhu and I. Chern, Convergence of the alternating minimization method for sparse MR image reconstruction,, Journal of Information and Computational Science, 8 (2011), 2067. [34] Y. Zhu and Y. Shi, A fast method for reconstruction of total-variation MR images with a periodic boundary condition,, IEEE Signal Processing Letters, 20 (2013), 291. doi: 10.1109/LSP.2013.2245502.
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