Bilevel parameter learning for higher-order total variation regularisation models

TL;DR

This paper introduces a bilevel optimization framework for learning parameters in higher-order total variation models, proposing a new cost functional and demonstrating improved image reconstruction performance through numerical experiments.

Contribution

It develops a bilevel learning approach with a novel Huber-regularized TV cost functional and compares TGV$^2$ and ICTV regularizers in image reconstruction.

Findings

The new cost functional improves reconstruction quality.

Bilevel optimization effectively learns model parameters.

Comparison reveals strengths and weaknesses of TGV$^2$ and ICTV.

Abstract

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost, based on a Huber regularised TV-seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a quasi-Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV$^2$ and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.