Le Mai, Fessler Jeffrey A
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109 USA.
IEEE Trans Comput Imaging. 2017 Mar;3(1):11-21. doi: 10.1109/TCI.2016.2626999. Epub 2016 Nov 8.
Undersampling is an effective method for reducing scan acquisition time for MRI. Strategies for accelerated MRI such as parallel MRI and Compressed Sensing MRI present challenging image reconstruction problems with non-differentiable cost functions and computationally demanding operations. Variable splitting (VS) can simplify implementation of difficult image reconstruction problems, such as the combination of parallel MRI and Compressed Sensing, CS-SENSE-MRI. Combined with augmented Lagrangian (AL) and alternating minimization strategies, variable splitting can yield iterative minimization algorithms with simpler auxiliary variable updates. However, arbitrary variable splitting schemes are not guaranteed to converge. Many variable splitting strategies are combined with periodic boundary conditions. The resultant circulant Hessians enable 𝒪( log ) computation but may compromise image accuracy at the spatial boundaries. We propose two methods for CS-SENSE-MRI that use regularization with non-periodic boundary conditions to prevent wrap-around artifacts. Each algorithm computes one of the resulting variable updates efficiently in 𝒪() time using a parallelizable tridiagonal solver. AL-tridiag is a VS method designed to enable efficient computation for non-periodic boundary conditions. Another proposed algorithm, ADMM-tridiag, uses a similar VS scheme but also ensures convergence to a minimizer of the proposed cost function using the Alternating Direction Method of Multipliers (ADMM). AL-tridiag and ADMM-tridiag show speeds competitive with previous VS CS-SENSE-MRI reconstruction algorithm AL-P2. We also apply the tridiagonal VS approach to a simple image inpainting problem.
欠采样是一种减少磁共振成像（MRI）扫描采集时间的有效方法。诸如并行MRI和压缩感知MRI等加速MRI策略带来了具有不可微代价函数和计算量大的操作的具有挑战性的图像重建问题。变量分裂（VS）可以简化诸如并行MRI和压缩感知（CS-SENSE-MRI）组合等困难图像重建问题的实现。结合增广拉格朗日（AL）和交替最小化策略，变量分裂可以产生具有更简单辅助变量更新的迭代最小化算法。然而，任意的变量分裂方案不能保证收敛。许多变量分裂策略与周期性边界条件相结合。由此产生的循环海森矩阵实现了𝒪(log)计算，但可能会在空间边界处损害图像精度。我们提出了两种用于CS-SENSE-MRI的方法，它们使用非周期性边界条件的正则化来防止卷绕伪影。每种算法都使用可并行化的三对角求解器在𝒪()时间内有效地计算其中一个结果变量更新。AL-tridiag是一种VS方法，旨在实现非周期性边界条件的高效计算。另一种提出的算法ADMM-tridiag使用类似的VS方案，但还使用乘子交替方向法（ADMM）确保收敛到所提出代价函数的极小值。AL-tridiag和ADMM-tridiag显示出与先前的VS CS-SENSE-MRI重建算法AL-P2具有竞争力的速度。我们还将三对角VS方法应用于一个简单的图像修复问题。
