Bilevel approaches for learning of variational imaging models

TL;DR

This paper reviews bilevel optimization methods in variational imaging, focusing on function space treatment, analytical properties, and numerical solutions, with extensive computational validation across diverse imaging scenarios.

Contribution

It provides a comprehensive overview of recent bilevel learning approaches in variational imaging, including analytical results and advanced numerical techniques.

Findings

Existence and structure of minimisers established

Newton methods effectively solve bilevel problems

Validated techniques on large, diverse image datasets

Abstract

We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.