Foundations of Supervised Deep Learning for Inverse Problems

逆问题的监督深度学习基础

• 批准号: 464101190
• 负责人: Professor Dr. Martin Burger
• 金额: --
• 依托单位: Projektträger DESY
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/464101190?language=en
• 关键词: Foundations; Supervised; Deep; Learning; Inverse

Over the last decade, deep learning methods have excelled at various data processing tasks including the solution of ill-posed inverse problems. While many works have demonstrated the superiority of such deep networks over classical (e.g. variational) regularization methods in image reconstruction, the theoretical foundation for truly understanding deep networks as regularization techniques, which can reestablish a continuous dependence of the solution on the data, is largely missing. The goal of this proposal is to close this gap in three step: First we will study deep network architectures that map a discrete (finite dimensional) representation of the data to a discrete representation of the solution in such a way, that we establish data consistency in a similar way as discretizations of classical regularization methods do. Secondly, we will study how to design, interpret and train deep networks as mappings between infinite-dimensional function spaces. Finally, we will investigate how to transfer finite-dimensional error estimates to the infinite-dimensional setting, by making suitable assumptions on the data to reconstruct as well as the data to train the network with, and utilizing suitable regularization schemes for the supervised training of the networks themselves. We will evaluate our networks and test our finding numerically on linear inverse problems in imaging using image deconvolution and computerized tomography as common test settings.

在过去的十年中，深度学习方法在各种数据处理任务中表现出色，包括解决不适定逆问题。虽然许多工作已经证明了这种深度网络在图像重建中优于经典（例如变分）正则化方法，但真正理解深度网络作为正则化技术的理论基础，可以重新建立解决方案对数据的连续依赖性，在很大程度上是缺失的。本提案的目标是通过三个步骤缩小这一差距：首先，我们将研究深度网络架构，该架构将数据的离散（有限维）表示映射到解决方案的离散表示，以便我们建立数据一致性。与经典正则化方法的离散化类似的方式。其次，我们将研究如何将深度网络设计、解释和训练为无限维函数空间之间的映射。最后，我们将研究如何将有限维的误差估计转移到无限维的设置，通过对重建数据以及训练网络的数据进行适当的假设，并利用适当的正则化方案进行网络本身的监督训练。我们将评估我们的网络，并测试我们的发现数字上的线性逆问题成像使用图像反卷积和计算机断层扫描作为常见的测试设置。