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：压缩感知算法通过提供唯一性、收敛性和稳定性保障，彻底革新了计算成像领域。结合成像物理与学习型正则化先验的基于模型深度学习方法，已成为图像恢复领域更为强大的替代方案。本文主要目标是在保持与压缩感知方法类似理论保证的前提下，提出一种记忆高效的基于模型算法。所提出的迭代算法在涉及分数函数的梯度下降与共轭梯度算法之间交替进行，以促进数据一致性。该分数函数被建模为单调卷积神经网络。我们的分析表明，单调约束是保证任意逆问题中固定点唯一性的必要且充分条件。同时，该约束还确保了向固定点的收敛性，且该收敛过程对输入扰动具有鲁棒性。我们提出了所提出的MOL框架的两种实现方式，二者在施加单调性质的方式上有所区别：第一种方法强制严格单调约束，而第二种方法依赖近似处理。严格意义上，第二种方法的保障并不成立。然而，我们的实证研究表明，两种方法的收敛性与鲁棒性相当，而约束较少的近似实现可提供更优性能。所提出的深度平衡公式相比展开方法具有显著更高的记忆效率，这使其能够应用于当前展开算法无法处理的3D或2D+时间问题。