  We introduce a unified framework based on bi-level optimization schemes to
deal with parameter learning in the context of image processing. The goal is to
identify the optimal regularizer within a family depending on a parameter in a
general topological space. Our focus lies on the situation with non-compact
parameter domains, which is, for example, relevant when the commonly used box
constraints are disposed of. To overcome this lack of compactness, we propose a
natural extension of the upper-level functional to the closure of the parameter
domain via Gamma-convergence, which captures possible structural changes in the
reconstruction model at the edge of the domain. Under two main assumptions,
namely, Mosco-convergence of the regularizers and uniqueness of minimizers of
the lower-level problem, we prove that the extension coincides with the
relaxation, thus admitting minimizers that relate to the parameter optimization
problem of interest. We apply our abstract framework to investigate a quartet
of practically relevant models in image denoising, all featuring nonlocality.
The associated families of regularizers exhibit qualitatively different
parameter dependence, describing a weight factor, an amount of nonlocality, an
integrability exponent, and a fractional order, respectively. After the
asymptotic analysis that determines the relaxation in each of the four
settings, we finally establish theoretical conditions on the data that
guarantee structural stability of the models and give examples of when
stability is lost.
