Analysis and optimisation of a variational model for mixed Gaussian and Salt & Pepper noise removal

TL;DR

This paper analyzes a variational model for removing mixed Gaussian and Salt & Pepper noise, focusing on parameter effects, asymptotic behavior, and automatic parameter selection, supported by theoretical insights and numerical experiments.

Contribution

It provides a detailed analysis of the model's parameter effects, asymptotic behavior, and introduces a bilevel optimization approach for automatic parameter selection.

Findings

Asymptotic analysis clarifies parameter influence on noise removal.

Exact solutions for simple cases validate the theoretical model.

Numerical results support the effectiveness of the parameter selection strategy.

Abstract

We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines $L^1$ and $L^2$ terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with $L^1$ and $L^2$ data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind" model selection.