School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, P.O. Box 14115-175, Iran.
Magn Reson Imaging. 2020 Sep;71:80-92. doi: 10.1016/j.mri.2020.03.008. Epub 2020 Apr 14.
Inspired by the first-order method of Malitsky and Pock, we propose a new variational framework for compressed MR image reconstruction which introduces the application of a rotation-invariant discretization of total variation functional into MR imaging while exploiting BM3D frame as a sparsifying transform. In the first step, we provide theoretical and numerical analysis establishing the exceptional rotation-invariance property of this total variation functional and observe its superiority over other well-known variational regularization terms in both upright and rotated imaging setups. Thereupon, the proposed MRI reconstruction model is presented as a constrained optimization problem, however, we do not use conventional ADMM-type algorithms designed for constrained problems to obtain a solution, but rather we tailor the linesearch-equipped method of Malitsky and Pock to our model, which was originally proposed for unconstrained problems. As attested by numerical experiments, this framework significantly outperforms various state-of-the-art algorithms from variational methods to adaptive and learning approaches and in particular, it eliminates the stagnating behavior of a previous work on BM3D-MRI which compromised the solution beyond a certain iteration.
受 Malitsky 和 Pock 的一阶方法的启发，我们提出了一种新的压缩磁共振图像重建变分框架，该框架将旋转不变离散全变分函数应用于磁共振成像中，同时利用 BM3D 框架作为稀疏变换。在第一步中，我们提供了理论和数值分析，确立了该全变分函数的特殊旋转不变性，并观察到它在直立和旋转成像设置中优于其他著名的变分正则化项。随后，提出的 MRI 重建模型被表示为一个约束优化问题，但我们没有使用专为约束问题设计的传统 ADMM 类型算法来获得解决方案，而是根据我们的模型对 Malitsky 和 Pock 的带线搜索的方法进行了调整，该方法最初是为无约束问题提出的。通过数值实验证明，该框架在各种从变分方法到自适应和学习方法的最新算法中表现出色，特别是它消除了之前关于 BM3D-MRI 的工作中的停滞行为，该行为在超过一定迭代次数后会影响解决方案。
