Abstract
In this paper, we propose a new model for MR image reconstruction based on second order total variation (\(\text {TV}^{2}\)) regularization and wavelet, which can be considered as requiring the image to be sparse in both the spatial finite differences and wavelet transforms. Furthermore, by applying the variable splitting technique twice, augmented Lagrangian method and the Barzilai-Borwein step size selection scheme, an ADMM algorithm is designed to solve the proposed model. It reduces the reconstruction problem to several unconstrained minimization subproblems, which can be solved by shrinking operators and alternating minimization algorithms. The proposed algorithm needs not to solve a fourth-order PDE but to solve several second-order PDEs so as to improve calculation efficiency. Numerical results demonstrate the effectiveness of the presented algorithm and illustrate that the proposed model outperforms some reconstruction models in the quality of reconstructed images.
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References
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)
Bredies, K., Lorenz, D.: Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14(5-6), 813–837 (2008)
Cai, J., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)
Candes, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)
Chamblle, A.: An algorithm for total variation minimization and application. J. Math. Imaging Vis. 20(1-2), 89–97 (2004)
Chan, R., Yang, J., Yuan, X.: Alternating direction method for image inpainting in wavelet domanis. SIAM J. Imaging Sci. 4(3), 807–826 (2011)
Chen, H., Song, J., Tai, X.: A dual algorithm for minimization of the LLT model. Adv. Comput. Math. 31(1-3), 115–130 (2009)
Chen, Y., Hager, W., Huang, F., Phan, D., Ye, X.: A fast algorithm for image reconstruction with application to partially parallel MR imaging. SIAM J. Imaging Sci. 5(1), 90–118 (2012)
Deng, W., Yin, W.: On the global and linear convergence of the generalized alternating direction method of multipliers. Submitted to SIAM Journal on Optimization. Rice univ. Houston Tx. dept of computational and applied mathematics (2012)
Goldstein, T., Osher, S.: The split Bregman method for l\(_{1}\) regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)
Guo, W., Qin, J., Yin, W.: A new detail-preserving regularity scheme. Rice CAAM technical report. 13-01, 2013
Guo, W., Yin, W.: Edge guided reconstruction for compressive imaging. SIAM J. Imaging Sci. 5(3), 809–834 (2012)
He, B., Liao, L., Han, D., Yang, H.: A new inexact alternating directions method for monotone variational inequalities.Mat. Program. 92(1), 103–118 (2002)
Hong, M., Luo, Z.: On the linear convergence of the alternating direction method of multipliers. arXiv preprint arXiv:1208.3922 (2012)
Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation for MRI. Magn. Reson. Med. 65(2), 480–491 (2011)
Lustig, M., Donoho, D., Pauly, J.: Sparse MRI: the application of comressed sensing for rapid MR imaging. Magn. Reson. Med. 58(6), 1182–1195 (2007)
Lysaker, M., Lundervold, A., Tai, X.: Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)
Rudin, L., Osher, S., Fatemi, E.: Non-linear total variation noise removal algorithm. Phys. D 60(1), 259–268 (1992)
Wang, Y., Yin,W., Zhang, Y.: A fast algorithm for image debluring with total variation regularization. CAAM technical reports (2007)
Wright, S., Nowak, R., Figueiredo, M.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57(7), 2479–2493 (2009)
Wu, C., Tai, X.: Augmented Lagrangian method, dual Methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)
Yang, J., Zhang, Y., Yin, W.: A fast TVl1-l2 minimization algorithm for signal reconstruction from partial Fourier data. CAAM, Rice Univ., Tech. Rep. TR08-29 (2008)
Ye, X., Chen, Y., Huang, F.: Computational acceleration for MR image reconstruction in partially parallel imaging. IEEE Trans. Med. Imaging 30(5), 1055–1063 (2011)
Zhang, J., Yang, Y.: Nonlinear multigrid method for solving the anisotropic image denoising models. Numer. Algorith. 63, 291–315 (2013)
Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. CAM UCLA, Tech. Rep. 09-03 (2009)
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The research has been supported by the NNSF of China (No.60872129) and the Science and Technology Project of Changsha City of China (No.K1207023-31).
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Xie, WS., Yang, YF. & Zhou, B. An ADMM algorithm for second-order TV-based MR image reconstruction. Numer Algor 67, 827–843 (2014). https://doi.org/10.1007/s11075-014-9826-z
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DOI: https://doi.org/10.1007/s11075-014-9826-z