The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem since the ill-conditioning amplifies, in the solution, the noise present in the data. Recently introduced algorithms based on deep learning overwhelm the more traditional model-based approaches in performance, but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyze the trade-off between stability and accuracy of neural networks, when used to solve linear imaging inverse problems for not under-determined cases. Moreover, we propose different supervised and unsupervised solutions to increase the network stability and maintain a good accuracy, by means of regularization properties inherited from a model-based iterative scheme during the network training and pre-processing stabilizing operator in the neural networks. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed deep learning-based approaches to handle noise on the data.

翻译：在线性逆问题的求解中（例如信号与图像处理中出现的此类问题），由于病态性会放大数据中的噪声，因此这是一个具有挑战性的问题。近年来基于深度学习提出的算法在性能上超越了传统基于模型的方法，但它们通常对数据扰动存在不稳定性。本文从理论上分析了神经网络在解决非欠定线性成像逆问题时，其稳定性与准确性之间的权衡关系。此外，我们提出了不同的有监督和无监督解决方案，通过利用网络训练过程中基于模型的迭代方案所继承的正则化特性，以及在神经网络中引入预处理稳定算子，来提升网络稳定性并保持良好准确性。在图像去模糊问题上的大量数值实验证实了理论结果，并验证了所提出的基于深度学习的方法在处理数据噪声方面的有效性。