Skip to main content

Solving Constrained TV2L1-L2 MRI Signal Reconstruction via an Efficient Alternating Direction Method of Multipliers

  • Tingting Wu (a1) (a2), David Z. W. Wang (a3), Zhengmeng Jin (a2) and Jun Zhang (a4)

High order total variation (TV2) and 1 based (TV2L1) model has its advantage over the TVL1 for its ability in avoiding the staircase; and a constrained model has the advantage over its unconstrained counterpart for simplicity in estimating the parameters. In this paper, we consider solving the TV2L1 based magnetic resonance imaging (MRI) signal reconstruction problem by an efficient alternating direction method of multipliers. By sufficiently utilizing the problem's special structure, we manage to make all subproblems either possess closed-form solutions or can be solved via Fast Fourier Transforms, which makes the cost per iteration very low. Experimental results for MRI reconstruction are presented to illustrate the effectiveness of the new model and algorithm. Comparisons with its recent unconstrained counterpart are also reported.

Corresponding author
*Corresponding author. Email address: (T. T. Wu)
Hide All
[1] Afonso M. V., Bioucasdias J. M. and Figueiredo M. A., Fast image recovery using variable splitting and constrained optimization, IEEE T. Image Process. A, 19(9) (2010), pp. 23452356.
[2] Afonso M. V., Bioucasdias J. M. and Figueiredo M. A., An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems, IEEE T. Image Process., 20(3) (2011), pp. 681695.
[3] Cai X., Han D. and Yuan X., On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function, Comput. Optim. Appl., 66(1) (2017), pp. 3973.
[4] Candès E. J., Romberg J. and Tao T., Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE T. Inform. Theory, 52(2) (2006), pp. 489509.
[5] Chen C., He B., Ye Y. and Yuan X., The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program., 155(1-2) (2016), pp. 5779.
[6] Chen C., Li M., Liu X. and Ye Y., On the Convergence of Multi-Block Alternating Direction Method of Multipliers and Block Coordinate Descent Method, Mathemtics, 2016.
[7] Chen C., Ng M. K. and Zhao X., Alternating direction method of multipliers for nonlinear image restoration problems, IEEE T. Image Process., 24(1) (2015), pp. 3343.
[8] Chen H., Song J. and Tai X., A dual algorithm for minimization of the LLT model, Adv. Comput. Math., 31(1-3) (2009), pp. 115130.
[9] Chen Y., Hager W., Huang F., Phan D., Ye X. and Yin W., Fast algorithms for image reconstruction with application to partially parallel MR imaging, SIAM Journal on Imaging Sciences, 5(1) (2012), pp. 90118.
[10] Chen Y., Ye X. and Huang F., A novel method and fast algorithm for MR image reconstruction with significantly under-sampled data, Inverse Probl. Imag., 4(2) (2010), pp. 223240.
[11] Compton R., Osher S. and Bouchard L., Hybrid regularization for MRI reconstruction with static field inhomogeneity correction, IEEE International Symposium on Biomedical Imaging (ISBI), (2012), pp. 650655.
[12] Dai Y., Han D., Yuan X. and Zhang W., A sequential updating scheme of the Lagrange multiplier for separable convex programming, Math. Comput., 86(303) (2017), pp. 315343.
[13] Esser E., Zhang X. and Chan T. F., A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM Journal on Imaging Sciences, 3(4) (2010), pp. 10151046.
[14] Gabay D. and Mercier B., A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2(1) (1976), pp. 1740.
[15] Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer, 1984.
[16] Glowinski R. and Marroco A., Sur l'approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires, Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique, 9(2) (1975), pp. 4176.
[17] Han D., He H., Yang H. and Yuan X., A customized Douglas–Rachford splitting algorithm for separable convex minimization with linear constraints, Numer Math, 127(1) (2014), pp. 167200.
[18] Han D. and Yuan X., Local linear convergence of the alternating direction method of multipliers for quadratic programs, SIAM J. Numer. Anal, 51(6) (2013), pp. 34463457.
[19] Han D., Yuan X. and Zhang W., An augmented Lagrangian based parallel splitting method for separable convex minimization with applications to image processing, Math. Comput., 83(289) (2014), pp. 22632291.
[20] He B., Liao L., Han D. and Yang H., A new inexact alternating directions method for monotone variational inequalities, Math. Program., 92(1) (2002), pp. 103118.
[21] He B., Tao M. and Yuan X., Alternating direction method with Gaussian back substitution for separable convex programming, SIAM J. Optimiz, 22(2) (2012), pp. 313340.
[22] He C., Hu C., Li X., Yang X. and Zhang W., A parallel alternating direction method with application to compound l1-regularized imaging inverse problems, Inform. Sciences, 348 (2016), pp. 179197.
[23] Lustig M., Donoho D. and Pauly J. M., Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Reson. Med., 58(6) (2007), pp. 11821195.
[24] Lysaker M., Lundervold A. and Tai X., Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE T. Image Process., 12(12) (2003), pp. 15791590.
[25] Ng M. K., Weiss P. and Yuan X., Solving constrained total-variation image restoration and reconstruction problems via alternating directionmethods, SIAM J. Sci. Comput., 32(5) (2010), pp. 27102736.
[26] Rudin L. I., Osher S. and Fatemi E., Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60(1) (1992), pp. 259268.
[27] Wu C. and Tai X., Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3(3) (2010), pp. 300339.
[28] Wu T., Variable splitting based method for image restoration with impulse plus Gaussian noise, Math. Probl. Eng., (2016), 3151303.
[29] Xie W., Yang Y. and Zhou B., An ADMM algorithm for second-order TV-based MR image reconstruction, Numer. Algorithms, 67(4) (2014), pp. 827843.
[30] Yang J., Zhang Y. and Yin W., A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE J. Sel. Top. Signa., 4(2) (2010), pp. 288297.
[31] Yang W. and Han D., Linear convergence of the alternating direction method of multipliers for a class of convex optimization problems, SIAM J. Numer. Anal., 54(2) (2016), pp. 625640.
[32] Ye X., Chen Y. and Huang F., Computational acceleration for MR image reconstruction in partially parallel imaging, IEEE T. Med. Imaging, 30(5) (2011), pp. 10551063.
[33] Zhang J., Chen R., Deng C. and Wang S., Fast Linearized Augmented Lagrangian Method for Euler's Elastica Model, Numer. Math. Theor. Meth. Appl., 10(1) (2017), pp. 98115.
[34] Zhang J., Wei Z. and Xiao L., Bi-component decomposition based hybrid regularization method for partly-textured CS-MR image reconstruction, Signal Process., 128 (2016), pp. 274290.
[35] Zhi Z., Sun Y. and Pang Z., Two-Stage Image Segmentation Scheme Based on Inexact Alternating Direction Method, Numer. Math. Theor. Meth. Appl., 9(3) (2016), pp. 451469.
[36] Zhu Z., Cai G. and Wen Y., Adaptive Box-Constrained Total Variation Image Restoration Using Iterative Regularization Parameter Adjustment Method, Int. J. Pattern Recogn., 29(7) (2015), 1554003.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 1
Total number of PDF views: 14 *
Loading metrics...

Abstract views

Total abstract views: 98 *
Loading metrics...

* Views captured on Cambridge Core between 12th September 2017 - 21st November 2017. This data will be updated every 24 hours.