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Split Bregman algorithms for multiple measurement vector problem. (English) Zbl 1349.94098

Summary: The standard sparse representation aims to reconstruct sparse signal from single measurement vector which is known as SMV model. In some applications, the SMV model extend to the multiple measurement vector (MMV) model, in which the signal consists of a set of jointly sparse vectors. In this paper, efficient algorithms based on split Bregman iteration are proposed to solve the MMV problems with both constrained form and unconstrained form. The convergence of the proposed algorithms is also discussed. Moreover, the proposed algorithms are used in magnetic resonance imaging reconstruction. Numerical results show the effectiveness of the proposed algorithms.

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Software:

SLEP; Yall1
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Full Text: DOI

References:

[1] Bazerque, J. A., & Giannakis, G. B. (2010). Distributed spectrum sensing for cognitive radio networks by exploiting sparsity. IEEE Transactions on Signal Processing, 58(3), 1847-1862. · Zbl 1392.94010 · doi:10.1109/TSP.2009.2038417
[2] Berg, E., & Friedlander, M. (2010). Theoretical and empirical results for recovery from multiple measurements. IEEE Transactions on Information Theory, 56(5), 2516-2527. · Zbl 1366.94138 · doi:10.1109/TIT.2010.2043876
[3] Berg, E., Schmidt, M., Friedlander ,M. P., & Murphy, K. (2008). Group sparsity via linear-time projection. Technical Report TR-2008-09, Department of Computer Science, University of British Columbia. · Zbl 1163.94395
[4] Bilen, C., Wang, Y., & Selesnick, I. W. (2012). High-speed compressed sensing reconstruction in dynamic parallel MRI using augmented lagrangian and parallel processing. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2(3), 370-379. · doi:10.1109/JETCAS.2012.2217032
[5] Cai, J. F., Osher, S., & Shen, Z. (2009). Split bregman methods and frame based image restoration. Multiscale Modeling & Simulation, 8(2), 337-369. · Zbl 1189.94014 · doi:10.1137/090753504
[6] Chen, J., & Huo, X. (2006). Theoretical results on sparse representations of multiple-measurement vectors. IEEE Transactions on Signal Processing, 54(12), 4634-4643. · Zbl 1375.94051 · doi:10.1109/TSP.2006.881263
[7] Cotter, S. F., Rao, B. D., Engan, K., & Kreutz-Delgado, K. (2005). Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Transactions on Signal Processing, 53(7), 2477-2488. · Zbl 1372.65123 · doi:10.1109/TSP.2005.849172
[8] Davies, M. E., & Eldar, Y. C. (2012). Rank awareness in joint sparse recovery. IEEE Transactions on Information Theory, 58(2), 1135-1146. · Zbl 1365.94175 · doi:10.1109/TIT.2011.2173722
[9] Deng, W., Yin, W., & Zhang, Y. (2011). Group sparse optimization by alternating direction method. TR11-06, Department of Computational and Applied Mathematics, Rice University.
[10] Duarte, M. F., & Eldar, Y. C. (2011). Structured compressed sensing: From theory to applications. IEEE Transactions on Signal Processing, 59(9), 4053-4085. · Zbl 1392.94188 · doi:10.1109/TSP.2011.2161982
[11] Eldar, Y. C., Kuppinger, P., & Bolcskei, H. (2010). Block-sparse signals: Uncertainty relations and efficient recovery. IEEE Transactions on Signal Processing, 58(6), 3042-3054. · Zbl 1392.94195 · doi:10.1109/TSP.2010.2044837
[12] Goldstein, T., & Osher, S. (2009). The split bregman method for l1-regularized problems. SIAM Journal on Imaging Sciences, 2(2), 323-343. · Zbl 1177.65088 · doi:10.1137/080725891
[13] He, Z., Cichocki, A., Cichocki, R., & Cao, J. (2008). CG-M-FOCUSS and its application to distributed compressed sensing. In: Advances in Neural Networks-ISNN 2008, pp. 237-245. · Zbl 1202.62052
[14] Huang, J., & Zhang, T. (2010). The benefit of group sparsity. The Annals of Statistics, 38(4), 1978-2004. · Zbl 1202.62052 · doi:10.1214/09-AOS778
[15] Jiang, L., & Yin, H. (2012). Bregman iteration algorithm for sparse nonnegative matrix factorizations via alternating l1-norm minimization. Multidimensional Systems and Signal Processing, 23(3), 315-328. · Zbl 1337.65038 · doi:10.1007/s11045-011-0147-2
[16] Lee, D. H., Hong, C. P., & Lee, M. W. (2013). Sparse magnetic resonance imaging reconstruction using the bregman iteration. Journal of the Korean Physical Society, 62(2), 328-332. · doi:10.3938/jkps.62.328
[17] Lee, K., Bresler, Y., & Junge, M. (2012). Subspace methods for joint sparse recovery. IEEE Transactions on Information Theory, 58(6), 3613-3641. · Zbl 1365.94179 · doi:10.1109/TIT.2012.2189196
[18] Liu, B., King, K., Steckner, M., Xie, J., Sheng, J., & Ying, L. (2009a). Regularized sensitivity encoding (sense) reconstruction using bregman iterations. Magnetic Resonance in Medicine, 61(1), 145-152. · doi:10.1002/mrm.21799
[19] Liu, J., Ji, S., & Ye, J. (2009b). SLEP: Sparse Learning with Efficient Projections. Arizona State University. · Zbl 1203.90153
[20] Ma, S., Goldfarb, D., & Chen, L. (2011). Fixed point and bregman iterative methods for matrix rank minimization. Mathematical Programming, 128(1-2), 321-353. · Zbl 1221.65146 · doi:10.1007/s10107-009-0306-5
[21] Majumdar, A., & Ward, R. (2012) Face recognition from video: An MMV recovery approach. In: 2012 IEEE international conference on acoustics, speech and signal processing (ICASSP), IEEE, pp. 2221-2224. · Zbl 1392.94188
[22] Majumdar, A., & Ward, R. (2013). Rank awareness in group-sparse recovery of multi-echo mr images. Sensors, 13(3), 3902-3921. · doi:10.3390/s130303902
[23] Majumdar, A., & Ward, R. K. (2011). Joint reconstruction of multiecho mr images using correlated sparsity. Magnetic Resonance Imaging, 29(7), 899-906. · doi:10.1016/j.mri.2011.03.008
[24] Mishali, M., & Eldar, Y. C. (2008). Reduce and boost: Recovering arbitrary sets of jointly sparse vectors. IEEE Transactions on Signal Processing, 56(10), 4692-4702. · Zbl 1390.94306 · doi:10.1109/TSP.2008.927802
[25] Qin, Z., Scheinberg, K., & Goldfarb, D. (2010). Efficient block-coordinate descent algorithms for the group lasso. Technical Report 2806, Optimization Online. · Zbl 1275.90059
[26] Smith, D. S., Gore, J. C., Yankeelov, T. E., & Welch, E. B. (2012). Real-time compressive sensing MRI reconstruction using GPU computing and split bregman methods. International Journal of Biomedical Imaging, 2012, 1-6.
[27] Stone, S. S., Haldar, J. P., Tsao, S. C., Sutton, B. P., Liang, Z. P., et al. (2008). Accelerating advanced MRI reconstructions on GPUs. Journal of Parallel and Distributed Computing, 68(10), 1307-1318. · doi:10.1016/j.jpdc.2008.05.013
[28] Tropp, J. A. (2006). Algorithms for simultaneous sparse approximation. Part ii: Convex relaxation. Signal Processing, 86(3), 589-602. · Zbl 1163.94395 · doi:10.1016/j.sigpro.2005.05.031
[29] Tropp, J. A., Gilbert, A. C., & Strauss, M. J. (2006). Algorithms for simultaneous sparse approximation. Part i: Greedy pursuit. Signal Processing, 86(3), 572-588. · Zbl 1163.94396 · doi:10.1016/j.sigpro.2005.05.030
[30] Tseng, P., & Yun, S. (2009). A coordinate gradient descent method for nonsmooth separable minimization. Mathematical Programming, 117(1-2), 387-423. · Zbl 1166.90016 · doi:10.1007/s10107-007-0170-0
[31] Xu, J., Feng, X., & Hao, Y. (2012). A coupled variational model for image denoising using a duality strategy and split bregman. Multidimensional Systems and Signal Processing, pp. 1-12. doi:10.1007/s11045-012-0190-7. · Zbl 1202.62052
[32] Yin, W., Osher, S., Goldfarb, D., & Darbon, J. (2008). Bregman iterative algorithms for \[\ell 1\] ℓ1-minimization with applications to compressed sensing. SIAM Journal on Imaging Sciences, 1(1), 143-168. · Zbl 1203.90153 · doi:10.1137/070703983
[33] Yuan, M., & Lin, Y. (2006). Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(1), 49-67. · Zbl 1141.62030 · doi:10.1111/j.1467-9868.2005.00532.x
[34] Zou, J., Fu, Y., & Xie, S. (2012). A block fixed point continuation algorithm for block-sparse reconstruction. IEEE Signal Processing Letters, 19(6), 364-367. · doi:10.1109/LSP.2012.2195488
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