Regularization of Inverse Problems by Neural Networks

TL;DR

This paper reviews how neural networks can be used as regularization tools to solve inverse problems in imaging, highlighting their advantages over classical methods and extending theoretical understanding.

Contribution

It provides a comprehensive overview of neural network-based regularization for inverse problems and extends existing theoretical results in this area.

Findings

Neural networks significantly outperform classical methods in inverse problems.

Deep learning methods can be viewed through the lens of regularization theory.

Theoretical results on neural network regularization are extended.

Abstract

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.