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2015, 5(2): 127-134. doi: 10.3934/naco.2015.5.127

Rank-one and sparse matrix decomposition for dynamic MRI

1. 

Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, China, China

Received  November 2014 Revised  April 2015 Published  June 2015

We introduce a rank-one and sparse matrix decomposition model for dynamic magnetic resonance imaging (MRI). Since $l_p$-norm $(0 < p < 1)$ is generally nonconvex, nonsmooth, non-Lipschitz, we propose reweighted $l_1$-norm to surrogate $l_p$-norm. Based on this, we put forward a modified alternative direction method. Numerical experiments are also given to illustrate the efficiency of our algorithm.
Citation: Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127
References:
[1]

K. Amin, W. Xu, A. Avestimehr and B. Hassibi, Weighted $l_1$ minimization for sparse recovery with prior information,, IEEE International Symposium on Information Theory, 2 (2009), 483.   Google Scholar

[2]

O. Banerjee, L. Ghaouiand and A. D'Aspremont, Model selection through sparse maximum likelihood estimation for multivariate Gaussian or Binary data,, The Journal of Machine Learning Research, 9 (2008), 485.   Google Scholar

[3]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via alternating direction method of multipliers,, Foundations and Trends in Machine Learning, 3 (2011), 1.   Google Scholar

[4]

E. Candès, M. Wakin and S. Boyd, Enhancing sparsity by reweighted $l_1$ minimization,, Journal of Fourier Analysis and Applications, 14 (2008), 877.  doi: 10.1007/s00041-008-9045-x.  Google Scholar

[5]

V. Chandrasekaran, S. Sanghavi, P. Parrilo and A. Willsky, Rank-sparsity incoherence for matrix decomposition,, SIAM Journal on Optimization, 21 (2011), 572.  doi: 10.1137/090761793.  Google Scholar

[6]

X. Chen, D. Ge, Z. Wang and Y. Ye, Complexity of unconstrained $L_2-L_p$ minimization,, Mathematical Programming, 143 (2014), 371.  doi: 10.1007/s10107-012-0613-0.  Google Scholar

[7]

I. Daubechies, R. DeVore, M. Fornasier and C. Güntürk, Iteratively reweighted least squares minimization for sparse recovery,, Communications on Pure and Applied Mathematics, 63 (2010), 1.  doi: 10.1002/cpa.20303.  Google Scholar

[8]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_p$-minimization for 0 < q < 1,, Applied and Computational Harmonic Analysis, 26 (2009), 395.  doi: 10.1016/j.acha.2008.09.001.  Google Scholar

[9]

D. Ge, X. Jiang and Y. Ye, A note on the complexity of $l_p$ minimization,, Mathematical Programming, 129 (2011), 285.  doi: 10.1007/s10107-011-0470-2.  Google Scholar

[10]

X. Li, M. Ng and X. Yuan, Nuclear-norm-free variational models for background extraction from surveillance video,, submitted to IEEE Transactions on Image Processing, (2013).   Google Scholar

[11]

Z. Lin, M. Chen and Y. Ma, The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, Preprint, (2010).   Google Scholar

[12]

R. Otazo, E. Candès and D. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,, Magnetic Resonance in Medicine, 73 (2015), 1125.   Google Scholar

[13]

J. Wright, A. Ganesh, S. Rao, Y. Peng and Y. Ma, Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization,, Advances in Neural Information Processing Systems, (2009), 2080.   Google Scholar

[14]

S. Wright, R. Nowak and M. Figueiredo, Sparse reconstruction by separable approximation,, IEEE Transactions on Signal Processing, 57 (2009), 2479.  doi: 10.1109/TSP.2009.2016892.  Google Scholar

[15]

X. Xiu, L. Kong and S. Zhou, Modified iterative reweighted $l_1$ algorithm for surveillance video,, Preprint, (2014).   Google Scholar

[16]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pacific Journal of Optimization, 9 (2013), 167.   Google Scholar

[17]

Y. Zhao and D. Li, Reweighted $l_1$-minimization for sparse solutions to underdetermined linear systems,, SIAM Journal on Optimization, 22 (2012), 1065.  doi: 10.1137/110847445.  Google Scholar

[18]

S. Zhou, N. Xiu, Y. Wang and L. Kong, Exact recovery for sparse signal via weighted $l_1$ minimization,, Preprint, (2014).   Google Scholar

show all references

References:
[1]

K. Amin, W. Xu, A. Avestimehr and B. Hassibi, Weighted $l_1$ minimization for sparse recovery with prior information,, IEEE International Symposium on Information Theory, 2 (2009), 483.   Google Scholar

[2]

O. Banerjee, L. Ghaouiand and A. D'Aspremont, Model selection through sparse maximum likelihood estimation for multivariate Gaussian or Binary data,, The Journal of Machine Learning Research, 9 (2008), 485.   Google Scholar

[3]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via alternating direction method of multipliers,, Foundations and Trends in Machine Learning, 3 (2011), 1.   Google Scholar

[4]

E. Candès, M. Wakin and S. Boyd, Enhancing sparsity by reweighted $l_1$ minimization,, Journal of Fourier Analysis and Applications, 14 (2008), 877.  doi: 10.1007/s00041-008-9045-x.  Google Scholar

[5]

V. Chandrasekaran, S. Sanghavi, P. Parrilo and A. Willsky, Rank-sparsity incoherence for matrix decomposition,, SIAM Journal on Optimization, 21 (2011), 572.  doi: 10.1137/090761793.  Google Scholar

[6]

X. Chen, D. Ge, Z. Wang and Y. Ye, Complexity of unconstrained $L_2-L_p$ minimization,, Mathematical Programming, 143 (2014), 371.  doi: 10.1007/s10107-012-0613-0.  Google Scholar

[7]

I. Daubechies, R. DeVore, M. Fornasier and C. Güntürk, Iteratively reweighted least squares minimization for sparse recovery,, Communications on Pure and Applied Mathematics, 63 (2010), 1.  doi: 10.1002/cpa.20303.  Google Scholar

[8]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_p$-minimization for 0 < q < 1,, Applied and Computational Harmonic Analysis, 26 (2009), 395.  doi: 10.1016/j.acha.2008.09.001.  Google Scholar

[9]

D. Ge, X. Jiang and Y. Ye, A note on the complexity of $l_p$ minimization,, Mathematical Programming, 129 (2011), 285.  doi: 10.1007/s10107-011-0470-2.  Google Scholar

[10]

X. Li, M. Ng and X. Yuan, Nuclear-norm-free variational models for background extraction from surveillance video,, submitted to IEEE Transactions on Image Processing, (2013).   Google Scholar

[11]

Z. Lin, M. Chen and Y. Ma, The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, Preprint, (2010).   Google Scholar

[12]

R. Otazo, E. Candès and D. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,, Magnetic Resonance in Medicine, 73 (2015), 1125.   Google Scholar

[13]

J. Wright, A. Ganesh, S. Rao, Y. Peng and Y. Ma, Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization,, Advances in Neural Information Processing Systems, (2009), 2080.   Google Scholar

[14]

S. Wright, R. Nowak and M. Figueiredo, Sparse reconstruction by separable approximation,, IEEE Transactions on Signal Processing, 57 (2009), 2479.  doi: 10.1109/TSP.2009.2016892.  Google Scholar

[15]

X. Xiu, L. Kong and S. Zhou, Modified iterative reweighted $l_1$ algorithm for surveillance video,, Preprint, (2014).   Google Scholar

[16]

X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pacific Journal of Optimization, 9 (2013), 167.   Google Scholar

[17]

Y. Zhao and D. Li, Reweighted $l_1$-minimization for sparse solutions to underdetermined linear systems,, SIAM Journal on Optimization, 22 (2012), 1065.  doi: 10.1137/110847445.  Google Scholar

[18]

S. Zhou, N. Xiu, Y. Wang and L. Kong, Exact recovery for sparse signal via weighted $l_1$ minimization,, Preprint, (2014).   Google Scholar

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