Neumann Networks for Linear Inverse Problems in Imaging
Neumann Networks for Linear Inverse Problems in Imaging
复制标题
DOI：
10.1109/tci.2019.2948732
复制
发表时间：
2020-01-01
影响因子：
5.4
通讯作者：
Willett, Rebecca
中科院分区：
文献类型：
--
作者：
Gilton, Davis;Ongie, Greg;Willett, Rebecca
许多具有挑战性的图像处理任务可以通过不适定的线性逆问题来描述：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成本函数，该成本函数由数据拟合项和正则化器组成，数据拟合项测量图像与观测值的匹配程度，正则化器反映先验知识并促进图像具有所需的属性，如平滑度。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正则化器，它可以优于更传统的正则化器。我们提出了一个端到端的，数据驱动的方法来解决反问题的灵感来自诺依曼级数，我们称之为诺依曼网络。而不是展开一个迭代优化算法，我们截断的诺依曼级数直接解决线性逆问题的数据驱动的非线性正则化。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，当图像属于一个联盟的子空间，并在适当的假设下的前向模型，我们证明存在一个Neumann网络配置，以及近似的最佳预言估计的反问题，并证明经验，训练的Neumann网络的形式预测的理论。
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.