Regularization of Inverse Problems by Neural Networks

Markus Haltmeier () and Linh Nguyen ()
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Markus Haltmeier: University of Innsbruck, Department of Mathematics
Linh Nguyen: University of Idaho, Department of Mathematics

Chapter 29 in Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, 2023, pp 1065-1093 from Springer

Abstract: Abstract Inverse problems arise in a variety of imaging applications, including computed tomography, non-destructive testing, and remote sensing. 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 showed 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.

Keywords: Inverse problems; Deep learning; Neural networks; Regularization theory; Ill-posedness; Stability; Theoretical foundation (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/978-3-030-98661-2_81

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