Computational imaging has been revolutionized by compressed sensing algorithms, which offer guaranteed uniqueness, convergence, and stability properties. Model-based deep learning methods that combine imaging physics with learned regularization priors have emerged as more powerful alternatives for image recovery. The main focus of this paper is to introduce a memory efficient model-based algorithm with similar theoretical guarantees as CS methods. The proposed iterative algorithm alternates between a gradient descent involving the score function and a conjugate gradient algorithm to encourage data consistency. The score function is modeled as a monotone convolutional neural network. Our analysis shows that the monotone constraint is necessary and sufficient to enforce the uniqueness of the fixed point in arbitrary inverse problems. In addition, it also guarantees the convergence to a fixed point, which is robust to input perturbations. We introduce two implementations of the proposed MOL framework, which differ in the way the monotone property is imposed. The first approach enforces a strict monotone constraint, while the second one relies on an approximation. The guarantees are not valid for the second approach in the strict sense. However, our empirical studies show that the convergence and robustness of both approaches are comparable, while the less constrained approximate implementation offers better performance. The proposed deep equilibrium formulation is significantly more memory efficient than unrolled methods, which allows us to apply it to 3D or 2D+time problems that current unrolled algorithms cannot handle.

翻译：计算成像领域已被压缩感知算法彻底改变，这类算法提供了唯一性、收敛性和稳定性的理论保证。融合成像物理与学习型正则化先验的模型驱动深度学习方法，已发展为更强大的图像重建方案。本文主要提出一种在记忆效率上与压缩感知方法具有相同理论保证的模型驱动算法。该迭代算法交替执行涉及得分函数的梯度下降和共轭梯度算法，以促进数据一致性。得分函数被建模为单调卷积神经网络。理论分析表明，单调约束是保证任意逆问题中不动点唯一性的充分必要条件。该约束同时确保了算法收敛至不动点，且对输入扰动具有鲁棒性。我们提出两种单调约束施加方式的MOL框架实现：第一种方法施加严格单调约束，第二种方法依赖近似约束——虽在严格意义上不保证理论性质，但实证研究表明两种方法的收敛性与鲁棒性相当，而约束较弱的近似实现具有更优性能。相较于展开方法，所提出的深度均衡公式显著提升内存效率，可应用于当前展开算法无法处理的3D或2D+时间问题。