Inverse problems are mathematically ill-posed. Thus, given some (noisy) data, there is more than one solution that 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：逆问题在数学上是不适定的。因此，给定某些含噪数据时，存在多个解能拟合该数据。近年来，人们开发了能够找到最合适解（即在意义上包含先验信息）的深度神经技术。然而，这些技术存在若干缺陷。首先，多数技术无法保证解在推理阶段能拟合数据。其次，尽管这些技术的推导灵感来源于有效标量正则化函数的存在，但在实际应用中并未依赖此类函数，因此偏离了经典变分方法。本研究针对逆问题提出了一种新的神经正则化函数族。这些正则化函数基于变分框架，并能确保数据拟合。我们将其应用于多个高度不适定问题（如图像去模糊和有限角度层析成像）中，展示其性能。