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.

通过对梯度数据和卷积神经网络块的总和施加单调性约束，本研究旨在放松对卷积神经网络块的约束以提高性能，并仅将算子在图像流形附近的局部邻域内保持单调性约束，从而实现从稀疏测量恢复磁共振图像的创新方法。

使用非单调算子的本地单调算子学习：MnM-MOL

Local monotone operator learning using non-monotone operators: MnM-MOL

Model-based deep learning methods that combine imaging physics with learned
regularization priors have been emerging as powerful tools for parallel MRI
acceleration. The main focus of this paper is to determine the utility of the
monotone operator learning (MOL) framework in the parallel MRI setting. The MOL
algorithm alternates between a gradient descent step using a monotone
convolutional neural network (CNN) and a conjugate gradient algorithm to
encourage data consistency. The benefits of this approach include similar
guarantees as compressive sensing algorithms including uniqueness, convergence,
and stability, while being significantly more memory efficient than unrolled
methods. We validate the proposed scheme by comparing it with different
unrolled algorithms in the context of accelerated parallel MRI for static and
dynamic settings.

通过探究单调算子学习 (MOL) 框架在并行 MRI 领域的适用性，本文提出了一种利用单调卷积神经网络 (CNN) 和共轭梯度算法交替迭代来保持数据一致性的并行 MRI 加速方法，并验证了该方法在静态和动态设置下的有效性。