In this paper we consider inverse problems that are mathematically ill-posed. That is, given some (noisy) data, there is more than one solution that approximately fits the data. In recent years, deep neural techniques that find the most appropriate solution, in the sense that it contains a-priori information, were developed. However, they suffer from several shortcomings. First, most techniques cannot guarantee that the solution fits the data at inference. Second, while the derivation of the techniques is inspired by the existence of a valid scalar regularization function, such techniques do not in practice rely on such a function, and therefore veer away from classical variational techniques. In this work we introduce a new family of neural regularizers for the solution of inverse problems. These regularizers are based on a variational formulation and are guaranteed to fit the data. We demonstrate their use on a number of highly ill-posed problems, from image deblurring to limited angle tomography.

翻译：本文考虑数学上不适定的逆问题，即给定（含噪）数据时存在多个近似拟合数据的解。近年来，人们开发了能够找到最符合先验信息意义的解的深度神经网络方法。然而这些方法存在若干缺陷：首先，多数技术无法保证解在推理阶段与数据拟合；其次，虽然其推导灵感源于有效标量正则化函数的存在，但实际并未依赖此类函数，因而偏离了经典变分方法。本研究提出一类用于求解逆问题的新型神经网络正则化方法。这些正则化方法基于变分框架，并能保证与数据的拟合性。我们在多个高度不适定问题（从图像去模糊到有限角度断层成像）中验证了其有效性。