The recovery of magnetic resonance (MR) images from undersampled measurements is a key problem that has seen extensive research in recent years. Unrolled approaches, which rely on end-to-end training of convolutional neural network (CNN) blocks within iterative reconstruction algorithms, offer state-of-the-art performance. These algorithms require a large amount of memory during training, making them difficult to employ in high-dimensional applications. Deep equilibrium (DEQ) models and the recent monotone operator learning (MOL) approach were introduced to eliminate the need for unrolling, thus reducing the memory demand during training. Both approaches require a Lipschitz constraint on the network to ensure that the forward and backpropagation iterations converge. Unfortunately, the constraint often results in reduced performance compared to unrolled methods. The main focus of this work is to relax the constraint on the CNN block in two different ways. Inspired by convex-non-convex regularization strategies, we now impose the monotone constraint on the sum of the gradient of the data term and the CNN block, rather than constrain the CNN itself to be a monotone operator. This approach enables the CNN to learn possibly non-monotone score functions, which can translate to improved performance. In addition, we only restrict the operator to be monotone in a local neighborhood around the image manifold. Our theoretical results show that the proposed algorithm is guaranteed to converge to the fixed point and that the solution is robust to input perturbations, provided that it is initialized close to the true solution. Our empirical results show that the relaxed constraints translate to improved performance and that the approach enjoys robustness to input perturbations similar to MOL.

翻译：从欠采样测量中恢复磁共振（MR）图像是一个关键问题，近年来得到了广泛研究。展开方法（unrolled approaches）通过在迭代重建算法中对卷积神经网络（CNN）模块进行端到端训练，展现出最先进的性能。然而，这些算法在训练过程中需要大量内存，使其难以应用于高维任务。深度均衡（DEQ）模型和近期提出的单调算子学习（MOL）方法无需展开处理，从而降低了训练时的内存需求。这两种方法均要求网络满足Lipschitz约束，以确保前向传播和反向传播迭代的收敛性。但遗憾的是，与展开方法相比，该约束往往导致性能下降。本文主要致力于以两种不同方式放宽对CNN模块的约束。受凸-非凸正则化策略的启发，我们不再将CNN本身约束为单调算子，而是对数据项梯度与CNN模块之和施加单调性约束。这种方法使CNN能够学习可能非单调的评分函数，从而提升性能。此外，我们仅要求算子在图像流形周围的局部邻域内保持单调性。理论结果表明，若算法初始化接近真实解，则保证能收敛至不动点，且解对输入扰动具有鲁棒性。实验结果显示，放宽约束可转化为性能提升，且该方法与MOL类似，对输入扰动具有鲁棒性。