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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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [11] Z. Lin, M. Chen and Y. Ma, The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, Preprint, (2010). [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. [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. [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. [15] X. Xiu, L. Kong and S. Zhou, Modified iterative reweighted $l_1$ algorithm for surveillance video,, Preprint, (2014). [16] X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pacific Journal of Optimization, 9 (2013), 167. [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. [18] S. Zhou, N. Xiu, Y. Wang and L. Kong, Exact recovery for sparse signal via weighted $l_1$ minimization,, Preprint, (2014).

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. [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. [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. [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. [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. [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. [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. [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. [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. [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). [11] Z. Lin, M. Chen and Y. Ma, The augmented Lagrange multiplier method for exact recovery of a corrupted low-rank matrices,, Preprint, (2010). [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. [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. [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. [15] X. Xiu, L. Kong and S. Zhou, Modified iterative reweighted $l_1$ algorithm for surveillance video,, Preprint, (2014). [16] X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction methods,, Pacific Journal of Optimization, 9 (2013), 167. [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. [18] S. Zhou, N. Xiu, Y. Wang and L. Kong, Exact recovery for sparse signal via weighted $l_1$ minimization,, Preprint, (2014).
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