Regularization of Inverse Problems by Neural Networks
This provides a theoretical foundation for applying deep learning to stabilize inverse problems in fields like computed tomography, though it is largely a review with some extensions.
The paper tackles the instability and non-uniqueness of solutions in inverse problems, such as in imaging applications, by reviewing how neural networks can be used for data-driven regularization, showing they significantly outperform classical methods.
Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their solutions. Therefore, any reasonable solution method requires the use of regularization tools that select specific solutions and at the same time stabilize the inversion process. Recently, data-driven methods using deep learning techniques and neural networks demonstrated to significantly outperform classical solution methods for inverse problems. In this chapter, we give an overview of inverse problems and demonstrate the necessity of regularization concepts for their solution. We show that neural networks can be used for the data-driven solution of inverse problems and review existing deep learning methods for inverse problems. In particular, we view these deep learning methods from the perspective of regularization theory, the mathematical foundation of stable solution methods for inverse problems. This chapter is more than just a review as many of the presented theoretical results extend existing ones.