Department of Artificial Intelligence, Korea University, Seoul 02841 South Korea.
Department of Artificial Intelligence, Korea University, Seoul 02841 South Korea.
Comput Methods Programs Biomed. 2022 Oct;225:107090. doi: 10.1016/j.cmpb.2022.107090. Epub 2022 Aug 29.
Recent unfolding based compressed sensing magnetic resonance imaging (CS-MRI) methods only reinterpret conventional CS-MRI optimization algorithms and, consequently, inherit the weaknesses of the alternating optimization strategy. In order to avoid the structural complexity of the alternating optimization strategy and achieve better reconstruction performance, we propose to directly optimize the ℓ regularized convex optimization problem using a deep learning approach.
In order to achieve direct optimization, a system of equations solving the ℓ regularized optimization problem is constructed from the optimality conditions of a novel primal-dual form proposed for the effective training of the sparsifying transform. The optimal solution is obtained by a cascade of unfolding networks of the preconditioned conjugate gradient (PCG) algorithm trained to minimize the mean element-wise absolute difference (ℓ loss) between the terminal output and ground truth image in an end-to-end manner. The performance of the proposed method was compared with that of U-Net, PD-Net, ISTA-Net+, and the recently proposed projection-based cascaded U-Net, using single-coil knee MR images of the fastMRI dataset.
In our experiment, the proposed network outperformed existing unfolding-based networks and the complex version of U-Net in several subsampling scenarios. In particular, when using the random Cartesian subsampling mask with 25 % sampling rate, the proposed model outperformed PD-Net by 0.76 dB, ISTA-Net+ by 0.43 dB, and U-Net by 1.21 dB on the positron density without suppression (PD) dataset in term of peak signal to noise ratio. In comparison with the projection-based cascade U-Net, the proposed algorithm achieved approximately the same performance when the sampling rate was 25% with only 1.62% number of network parameters at the cost of a longer reconstruction time (approximately twice).
A cascade of unfolding networks of the PCG algorithm was proposed to directly optimize the ℓ regularized CS-MRI optimization problem. The proposed network achieved improved reconstruction performance compared with U-Net, PD-Net, and ISTA-Net+, and achieved approximately the same performance as the projection-based cascaded U-Net while using significantly fewer network parameters.
最近出现的基于展开的压缩感知磁共振成像（CS-MRI）方法仅重新解释了传统的 CS-MRI 优化算法，因此继承了交替优化策略的弱点。为了避免交替优化策略的结构复杂性并实现更好的重建性能，我们建议使用深度学习方法直接优化ℓ正则化凸优化问题。
为了实现直接优化，从为稀疏变换的有效训练提出的新的原始对偶形式的最优条件构建了一个求解ℓ正则化优化问题的方程组。通过级联预处理共轭梯度（PCG）算法的展开网络来获得最优解，该算法经过训练可以最小化终端输出和地面真实图像之间的逐元素绝对差（ℓ损失）的平均值，以端到端的方式。将所提出的方法的性能与 U-Net、PD-Net、ISTA-Net+以及最近提出的基于投影的级联 U-Net进行了比较，使用了 fastMRI 数据集的单线圈膝关节 MR 图像。
在我们的实验中，所提出的网络在几个欠采样场景中优于现有的基于展开的网络和复杂版本的 U-Net。特别是，在使用具有 25%采样率的随机笛卡尔欠采样掩模时，在所提出的 PD 数据集上，与 PD-Net 相比，所提出的模型在正电子密度（PD）没有抑制（PD）数据集上的峰值信噪比提高了 0.76 dB，与 ISTA-Net+相比提高了 0.43 dB，与 U-Net 相比提高了 1.21 dB。与基于投影的级联 U-Net 相比，当采样率为 25%时，所提出的算法的性能大致相同，但其网络参数数量仅增加了 1.62%，重建时间增加了约两倍。
提出了一种 PCG 算法的展开网络级联，以直接优化ℓ正则化 CS-MRI 优化问题。与 U-Net、PD-Net 和 ISTA-Net+相比，所提出的网络实现了改进的重建性能，并且在使用明显更少的网络参数的情况下，其性能与基于投影的级联 U-Net 大致相同。
