The structure of optimal parameters for image restoration problems

TL;DR

This paper investigates the properties and existence of optimal regularisation parameters in variational image restoration models, analyzing their behavior through bilevel optimization and convergence of smoothed approximations.

Contribution

It proves the existence and boundedness of optimal parameters in bilevel models for various regularisers, including TV, TGV, and ICTV, in function space.

Findings

Optimal parameters exist under certain data conditions.

Optimal parameters are bounded away from zero.

Smoothed bilevel problems converge to the original as smoothing vanishes.

Abstract

We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from $0$ which we prove in this paper. The analysis is done on the original -- in image restoration typically non-smooth variational problem -- as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it $\Gamma$ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.