Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method of solving inverse problems inspired by the Neumann series, which we call a Neumann network. Rather than unroll an iterative optimization algorithm, we truncate a Neumann series which directly solves the linear inverse problem with a data-driven nonlinear regularizer. The Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. Finally, when the images belong to a union of subspaces and under appropriate assumptions on the forward model, we prove there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem and demonstrate empirically that the trained Neumann network has the form predicted by theory.

中文翻译：

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用于成像线性逆问题的 Neumann 网络

许多具有挑战性的图像处理任务可以用不适定线性逆问题来描述：去模糊、去卷积、修复、压缩感知和超分辨率都在这个框架中。传统的逆问题求解器最小化由数据拟合项和正则化器组成的成本函数，该值函数衡量图像与观测值的匹配程度，正则化器反映先验知识并提升具有所需特性（如平滑度）的图像。机器学习和图像处理的最新进展表明，通常可以从训练数据中学习到一个正则化器，其性能优于传统的正则化器。我们提出了一种端到端的、数据驱动的方法来解决受诺依曼级数启发的逆问题，我们称之为诺依曼网络。与其展开迭代优化算法，我们截断了一个 Neumann 级数，它使用数据驱动的非线性正则化器直接解决了线性逆问题。Neumann 网络架构在标准数据集上优于传统的逆问题解决方法、无模型深度学习方法和最先进的展开迭代方法。最后，当图像属于子空间的并集并且在前向模型的适当假设下，我们证明存在一个 Neumann 网络配置，它很好地逼近了逆问题的最优预言机估计器，并凭经验证明了经过训练的 Neumann 网络具有理论预测的形式。无模型深度学习方法，以及标准数据集上最先进的展开迭代方法。最后，当图像属于子空间的并集并且在前向模型的适当假设下，我们证明存在一个 Neumann 网络配置，它很好地逼近了逆问题的最优预言机估计器，并凭经验证明了经过训练的 Neumann 网络具有理论预测的形式。无模型深度学习方法，以及标准数据集上最先进的展开迭代方法。最后，当图像属于子空间的并集并且在前向模型的适当假设下，我们证明存在一个 Neumann 网络配置，它很好地逼近了逆问题的最优预言机估计器，并凭经验证明了经过训练的 Neumann 网络具有理论预测的形式。