  In image processing, classical methods minimize a suitable functional that
balances between computational feasibility (convexity of the functional is
ideal) and suitable penalties reflecting the desired image decomposition. The
fact that algorithms derived from such minimization problems can be used to
construct (deep) learning architectures has spurred the development of
algorithms that can be trained for a specifically desired image decomposition,
e.g. into cartoon and texture. While many such methods are very successful,
theoretical guarantees are only scarcely available. To this end, in this
contribution, we formalize a general class of intersection point problems
encompassing a wide range of (learned) image decomposition models, and we give
an existence result for a large subclass of such problems, i.e. giving the
existence of a fixpoint of the corresponding algorithm. This class generalizes
classical model-based variational problems, such as the TV-l2 -model or the
more general TV-Hilbert model. To illustrate the potential for learned
algorithms, novel (non learned) choices within our class show comparable
results in denoising and texture removal.
