基于非凸全变差正则的核磁共振图像重构算法

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。

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

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.