Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization

TL;DR

This paper develops optimality conditions and a numerical algorithm for bilevel total variation image denoising problems, enabling better parameter learning and solution stability in image processing tasks.

Contribution

It derives M- and B-stationarity conditions for bilevel TV denoising, characterizes the solution mapping's subdifferential, and proposes a non-smooth trust-region algorithm for solving these problems.

Findings

Derived M-stationarity conditions for bilevel TV problems.

Characterized the Bouligand subdifferential of the solution mapping.

Implemented a two-phase algorithm tested on experimental settings.

Abstract

We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.