Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal

We analyse a variational regularisation problem for mixed noise removal that has been recently proposed in Calatroni et al (2017 SIAM J. Imaging Sci. 10 1196–233). 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. Here, we perform a fine analysis of the model that emphasises 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 its solutions to the ones of the corresponding variational models with L^1 and L^2 data fidelity. Extensions to the general linear inverse problems setting are also discussed. Furthermore, we compute exact solutions to the denoising problem, for simple data functions in the case of total variation regularisation. Using these theoretical results, we then analytically study a bilevel optimisation strategy for the automatic selection of model parameters by means of a training set. Finally, we report some numerical results which confirm the validity of our analysis and the use of popular data models in the case of 'blind' optimal noise model selection