Magnetic resonance image reconstruction algorithm via non-convex total variation regularization

doi:10.11772/j.issn.1001-9081.2019122187

Abstract

Abstract: To solve the problems of incomplete reconstruction, blurred boundary and residual noise in Magnetic Resonance (MR) image reconstruction, a non-convex total variation regularization reconstruction model based on L₂ regularization was proposed. First, Moreau envelope and minmax-concave penalty function were used to construct the non-convex regularization of L₂ norm, then it was applied into the total variation regularization to construct the sparse reconstruction model based on the isotropic non-convex total variation regularization. The proposed non-convex regularization was able to effectively avoid the underestimation of larger non-zero elements in convex regularization, so as to reconstruct the edge contour of the target more effectively. At the same time, it was able to guarantee the global convexity of objective function under certain conditions. Therefore, Alternating Direction Method of Multipliers (ADMM) was able to be used to solve the model. Simulation experiments were carried out to reconstruct several MR images under different sampling templates and sampling rates. Experimental results show that compared with several typical image reconstruction methods, the proposed model has better performance and lower relative error, its Peak Signal-to-Noise Ratio (PSNR) is significantly improved, which is 4 dB higher than that of traditional reconstruction method based on the non-convex regularization of L₁ norm; in addition, the visual effects of the reconstructed images are promoted significantly, effectively maintaining the edge details of the original images.

Key words: Magnetic Resonance Imaging (MRI), total variation regularization, Moreau envelope, minmax-concave penalty function, Alternating Direction Method of Multipliers (ADMM)

摘要： 针对于核磁共振（MR）图像重构中由于欠采样导致的重构图像不够完整、边缘模糊以及噪声残留等问题，提出了一种基于L₂正则的非凸全变差正则重构模型。首先，以Moreau包络和最小最大凹罚函数为工具构造L₂范数的非凸正则；然后，将其应用于全变差正则上来构造各向同性的非凸全变差正则稀疏重构模型。所提的非凸正则可以有效地避免凸正则中对较大非零元欠估计现象，能够更有效地重构目标的边缘轮廓；同时，在一定条件下可以保证目标函数的整体凸性，从而最后可以利用交替方向乘子法（ADMM）对模型进行求解。仿真实验对若干MR图像在不同的采样模板和采样率下进行了重构。实验结果均表明，与几种典型的图像重构方法相比，所提模型性能更优，相对误差明显降低，峰值信噪比（PSNR）有明显改善，较经典的L₁非凸正则重构模型提升了大约4 dB，并且重构后的图像视觉效果显著提升，有效地保留了原始图像的边缘细节。

关键词: 核磁共振成像, 全变差正则, Moreau包络, 最小最大凹罚函数, 交替方向乘子法

CLC Number:

TP301.6

SHEN Marui, LI Jincheng, ZHANG Ya, ZOU Jian. Magnetic resonance image reconstruction algorithm via non-convex total variation regularization[J]. Journal of Computer Applications, 2020, 40(8): 2358-2364.

沈马锐, 李金城, 张亚, 邹健. 基于非凸全变差正则的核磁共振图像重构算法[J]. 计算机应用, 2020, 40(8): 2358-2364.

References

[1] 贺超. 核磁共振成像系统原理及MR图像研究[J]. 云南大学学报(自然科学版), 2010, 32(S1):245-248. (HE C. The study of principle of nuclear magnetic resonance imaging and MR image[J]. Journal of Yunnan University (Natural Sciences Edition), 2010, 32(S1):245-248.)
[2] 樊晓宇,练秋生. 基于双稀疏模型的压缩感知核磁共振图像重构[J]. 生物医学工程学杂志, 2018, 35(5):688-696. (FAN X Y, LIAN Q S. Compressed sensing magnetic resonance image reconstruction based on double sparse model[J]. Journal of Biomedical Engineering, 2018, 35(5):688-696.)
[3] 程晓良,郑璇,韩渭敏. 求解欠定线性方程组稀疏解的算法[J]. 高校应用数学学报, 2013, 28(2):235-248. (CHENG X L, ZHENG X, HAN W M. Algorithms on sparse solution of under-determined linear systems[J]. Applied Mathematics A Journal of Chinese Universities, 2013, 28(2):235-248.)
[4] DONEVA M. Mathematical models for magnetic resonance imaging reconstruction:an overview of the approaches, problems, and future research areas[J]. IEEE Signal Processing Magazine, 2020, 37(1):24-32.
[5] 占美全,邓志良. 基于L1范数的总变分正则化超分辨率图像重建[J]. 科学技术与工程, 2010, 10(28):6903-6906. (ZHAN M Q, DENG Z L. L1 norm of total variation regularization based super resolution reconstruction for images[J]. Science Technology and Engineering, 2010, 10(28):6903-6906.)
[6] SELESNICK I W, PAREKH A, BAYRAM I. Convex 1-D total variation denoising with non-convex regularization[J]. IEEE Signal Processing Letters, 2015, 22(2):141-144.
[7] SELESNICK I W, FARSHCHIAN M. Sparse signal approximation via nonseparable regularization[J]. IEEE Transactions on Signal Processing, 2017, 65(10):2561-2575.
[8] SELESNICK I W. Total variation denoising via the Moreau envelope[J]. IEEE Signal Processing Letters, 2017, 24(2):216-220.
[9] SELESNICK I W. Sparse regularization via convex analysis[J]. IEEE Transactions on Signal Processing, 2017, 65(17):4481-4494.
[10] 刘晓光,高兴宝. 一种具有非凸非光滑组合正则的图像恢复方法[J]. 科学技术与工程, 2018, 18(7):197-202. (LIU X G, GAO X B. One image restoration method with the combined non-convex non-smooth regularization[J]. Science Technology and Engineering, 2018, 18(7):197-202.)
[11] ZHANG X J, XU C, LI M, et al. Sparse and low-rank coupling image segmentation model via nonconvex regularization[J]. International Journal of Pattern Recognition and Artificial Intelligence, 2015, 29(2):No.1555004.
[12] RUDIN L I, OSHER S, FATEMI E. Nonlinear total variation based noise removal algorithms[J]. Physica D:Nonlinear Phenomena, 1992, 60(1/2/3/4):259-268.
[13] ZHANG J, CHEN K, YU B. An iterative Lagrange multiplier method for constrained total-variation-based image denoising[J]. SIAM Journal on Numerical Analysis, 2012, 50(3):983-1003.
[14] XU Y P. A new algorithm for total variation based image denoising[J]. Acta Mathematicae Applicatae Sinica, English Series, 2012, 28(4):721-730.
[15] WANG Y, YANG J, YIN W, et al. A new alternating minimization algorithm for total variation image reconstruction[J]. SIAM Journal on Imaging Sciences, 2008, 1(4):248-272.
[16] 杨俊锋. 图像处理中全变差正则化数据拟合问题算法回顾[J]. 运筹学学报, 2017, 21(4):69-83. (YANG J F. An algorithmic review for total variation regularized data fitting problems in image processing[J]. Operations Research Transactions, 2017, 21(4):69-83.)
[17] LIU Y L, DU H, WANG Z, et al. Convex MR brain image reconstruction via non-convex total variation minimization[J]. International Journal of Imaging Systems and Technology, 2018, 28(4):246-253.
[18] BARBERO Á, SRA S. Modular proximal optimization for multidimensional total-variation regularization[J]. The Journal of Machine Learning Research, 2018, 19(1):2232-2313.
[19] FESSLER J A. Optimization methods for magnetic resonance image reconstruction:key models and optimization algorithms[J]. IEEE Signal Processing Magazine, 2020, 37(1):33-40.
[20] ZHANG Z, XU Y, YANG J, et al. A survey of sparse representation:algorithms and applications[J]. IEEE Access, 2015, 3:490-530.
[21] BOYD S, PARIKH N, CHU E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers[J]. Foundations and Trends in Machine Learning, 2011, 3(1):1-122.
[22] ZOU J, SHEN M, ZHANG Y, et al. Total variation denoising with non-convex regularizers[J]. IEEE Access, 2019, 7:4422-4431.
[23] 王斌,胡辽林,曹京京,等. 基于Moreau包络平滑l1/全变差范数模型的图像脉冲噪声去除方法[J]. 光学学报, 2014, 34(12):No.1211002. (WANG B, HU L L, CAO J J,et al. Impulse noise removal method based on Moreau envelope smoothing l1/TV norm model[J]. Acta Optica Sinica, 2014, 34(12):No.1211002.)
[24] LANZA A, MORIGI S, SGALLARI F. Convex image denoising via non-convex regularization with parameter selection[J]. Journal of Mathematical Imaging and Vision, 2016, 56(2):195-220.