  In this note we present some results that were already conjectured in the
work [9] by Bildhauer, Fuchs and Weickert, where they have investigated
analytical aspects of coupled variational models with applications to
mathematical imaging. Here we focus on variants of linear growth, which require
a treatment in the framework of relaxation theory and convex analysis.
Following basic ideas from [6] and [7], we establish existence and regularity
of (dual-)solutions.
