American Institute of Mathematical Sciences

Advanced
August  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

Yonggui Zhu 1, , Yuying Shi 2, , Bin Zhang 1, and  Xinyan Yu 1,

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.
Keywords: magnetic resonance image reconstruction., weighted average, alternating minimization method, Compressive sensing.
Mathematics Subject Classification: Primary: 65N20, 97M10, 97M60; Secondary: 65Q3.
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators,, Journal of Convex Analysis, 16 (2009), 727. Google Scholar

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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. Google Scholar

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D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[17]

D. Goldfarb and S. Ma, Fast multiple-splitting algorithms for convex optimization,, SIAM Journal on Optimization, 22 (2012), 533. doi: 10.1137/090780705. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. Google Scholar

[22]

B. K. Natarajan, Sparse approximate solutions to linear systems,, SIAM Journal on Computing, 24 (1995), 227. doi: 10.1137/S0097539792240406. Google Scholar

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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. Google Scholar

[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. Google Scholar

[25]

J. E. Spingarn, Partial inverse of monotone operator,, Applied Mathematics and Optimization, (1983), 247. doi: 10.1007/BF01448388. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators,, Journal of Convex Analysis, 16 (2009), 727. Google Scholar

[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. Google Scholar

[9]

D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

D. Goldfarb and S. Ma, Fast multiple-splitting algorithms for convex optimization,, SIAM Journal on Optimization, 22 (2012), 533. doi: 10.1137/090780705. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. Google Scholar

[22]

B. K. Natarajan, Sparse approximate solutions to linear systems,, SIAM Journal on Computing, 24 (1995), 227. doi: 10.1137/S0097539792240406. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

J. E. Spingarn, Partial inverse of monotone operator,, Applied Mathematics and Optimization, (1983), 247. doi: 10.1007/BF01448388. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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