Abstract
Motivated by the ideas from two step model and its deformation, we propose a coupling model for MR image reconstruction, based on the advantages of TGV and shearlet regularization terms. By using variable splitting technique, splitting Bregman iteration scheme and alternating minimization method, the proposed model can be decomposed into several subproblems to avoid solving high-order PDEs. The u subproblem can be solved by Cramer’s rule and the diagonalization technique of the Fourier transform. The other subproblems can be solved simply by some shrinkage formulas. We also use the Barzilai-Borwein step selection scheme to accelerate these subproblem’s solutions. Finally, an ADMM algorithm is proposed to solve the coupling model. The numerical results show that the proposed coupling model and algorithm are feasible and effective.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bredies K, Kunisch K, Pock TT (2010) Total Generalized Variation. Siam J Imaging Sci 3(3):492–526
Chan RH, Yang J, Yuan X (2011) Alternating direction method for image inpainting in wavelet domains. SIAM J Imaging Sci 4(4):807–826
Dong F, Liu Z, Kong D, Liu K (2009) An improved LOT Model for image restoration. J Math Imaging Vis 34(1):89–97
Fei YY, Pang ZF, Shi BL, Wang ZG (2011) Split Bregman method for the modified LOT model in image denoising. Appl Math Comput 217(12):5392–5403
Guo W, Qin J, Yin W (2014) A new detail-preserving regularization scheme. Siam J Imaging Sci 7(2):1309–1334
Hao Y, Feng XC, Xu JL (2012) Split Bregman algorithm for a novel denoising model. J Electron Inf Technol 34(3):557–563
Hong LI, Wang JY, Hou-Biao LI (2017) Research on Poisson noise image restoration problems based on shearlet transform. J Univ Electron Sci Technol China 46(3):511–515
Lei H, Feng W (2014) Single-image super-resolution with total generalised variation and shearlet regularisations. IET Image Proc 8(12):833–845
Liu X (2018) A new TGV-Gabor model for cartoon-texture image decomposition. IEEE Signal Process Lett 25(8):1221–1225
Lustig M, Donoho D, Pauly JM (2007) Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn Reson Med 58(6):1182–1195
Lysaker OM, Lundervold A, Tai X-C (2003) Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans Image Process 12(12):1579–1590
Lysaker M, Osher S, Tai XC (2004) Noise removal using smoothed normals and surface fitting. IEEE Trans Image Process 13(10):1345–1357
Steidl G (2006) A note on the dual treatment of higher order regularization functionals. Computing 76(1–2):135–148
Ville DVD, Blu T, Unser M (2004) Integrated wavelet processing and spatial statistical testing of fMRI data. Neuroimage 23(4):1472–1485
Wang M, Lu CW (2016) Alternating direction method for TGV-TGV* based cartoon-texture image decomposition. IET Image Proc 10(6):495–504
Wang Y, Yin W, Zhang Y (2007) A fast algorithm for image deblurring with total variation regularization. In: CAAM Technical Report, pp TR07–10
Wu T, Zhang W, Wang DZW, Sun Y (2018) An efficient Peaceman-Rachford splitting method for constrained TGV-shearlet-based MRI reconstruction. In: Inverse problems in science and engineering, pp 1–19
Xie WS, Yu Fei Y, Zhou B (2014) An ADMM algorithm for second-order TV-based MR image reconstruction. Numer Algor 67(4):827–843
You YL, Kaveh M (2000) Fourth-order partial differential equations for noise removal. IEEE Trans Image Process 9(10):1723–1730
Zhang CC, Wang Yu, Bing XH (2016) Fluorescence microscopic image three dimensional reconstruction based on hybrid regularizer. Comput Eng Appl 52(24):205–209
Acknowledgements
The research has been supported by the Natural Science Foundation of the Hunan Province of China (no. 2019JJ40323).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Antonio C.G. Leitao.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhou, B., Yang, YF. A coupling model and ADMM algorithm based on TGV and shearlet regularization term for MRI reconstruction. Comp. Appl. Math. 40, 75 (2021). https://doi.org/10.1007/s40314-021-01466-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-021-01466-x