Learned reconstruction methods with convergence guarantees

TL;DR

This paper surveys learned image reconstruction methods that incorporate convergence guarantees, emphasizing the importance of stability and reliability in critical applications like medical imaging.

Contribution

It introduces relevant convergence notions for data-driven reconstruction, reviews methods with mathematical guarantees, and highlights ICNN as a promising approach combining deep learning with convex regularization.

Findings

ICNN enables combining deep learning with convex regularization for provably convergent methods

The survey clarifies convergence concepts applicable to learned image reconstruction

Provides a mathematical foundation for empirical practices in data-driven reconstruction

Abstract

In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of data-driven methods in critical use-cases, for instance in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding their stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we will specify relevant notions of convergence for data-driven image reconstruction, which will form the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of ICNN, offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners, by providing an accessible description of useful convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation.