杨青海 杨敏

摘  要：通过低秩加稀疏矩阵分解模型重建欠采样动态磁共振图像时，常采用变量分裂算法来求解。针对共轭梯度法在二次项更新中迭代计算较为复杂的问题，为了加快重建速度，提出一种考虑数据采集算子形式的高效变量分裂方案，将数据采集算子根据欠采样掩码矩阵、傅里叶变换算子和线圈灵敏度矩阵进行拆分，简化算法子问题中二次项更新所涉及的矩阵逆运算，达到加快算法收敛速度的目的。仿真实验结果表明：与迭代软阈值法和共轭梯度法相比，所提算法在心电影数据集中收敛速度分别提高了57.9%和83.0%，结构相似性分别提升了3.3%和1.4%;在心脏灌注数据集中收敛速度分别提高了55.5%和79.6%，结构相似性分别提升了1.5%和0.4%。

关键词：动态磁共振成像;压缩感知;低秩稀疏分解;变量分裂

中图分类号：TP391     文献标识码：A

Dynamic MRI Reconstruction based on Low-rank

Sparse Decomposition Fast Algorithm

YANG Qinghai， YANG Min

（School of Automation， Nanjing University of Posts and Telecommunications， Nanjing 210023， China）

1789321617@qq.com; yangm@njupt.edu.cn

Abstract： Variable splitting algorithm is often used to solve the problem of under-sampled dynamic MRI （Magnetic Resonance Imaging） reconstruction by low rank and sparse matrix decomposition model. Aiming at the complex iterative calculation of conjugate gradient method in quadratic term updating， in order to speed up the reconstruction， this paper proposes an efficient variable splitting scheme considering the form of the data acquisition operator. The data acquisition operator is divided according to the under-sampled mask code matrix， the Fourier transform operator and the coil sensitivity matrix， which simplifies the matrix inverse operation involved in the update of the quadratic term in the algorithm sub-problem， and achieves the purpose of accelerating the convergence speed the algorithm. Simulation results show that compared with the iterative soft threshold method and the conjugate gradient method， the convergence speed of the proposed algorithm in the cardiac cine dataset has increased by 57.9% and 83.0% respectively， and the structural similarity has increased by 3.3% and 1.4% respectively. In the cardiac perfusion dataset， the convergence speed has increased by 55.5% and 79.6% respectively， and the structural similarity has increased by 1.5% and 0.4% respectively.

Keywords： dynamic magnetic resonance imaging; compressed sensing; low rank sparse decomposition; variable

splitting

1   引  言（Introduction）

對于动态磁共振图像，可以把同一层空间不同时间帧的图像看作一个列向量，作为时空矩阵的一列，由多个时间帧构建成一个低秩矩阵，从而把动态磁共振成像（MRI）重建问题转化为低秩矩阵恢复问题。OTAZO等[1]和GAO等[2]基于L+S模型，通过IST算法对临床医学图像进行重建，并取得了可观的重建效果。此前的大量研究表明，基于L+S模型对动态MRI进行重建整体上可以取得较好的重建精度，但重建速度仍有提升空间。

本文考虑到在常规的变量分解优化方案中，二次项的更新一般需要共轭梯度法来迭代求解，提出基于L+S框架下结合数据采集算子的快速变量分裂法。……