  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.

Optimization and Control