  Variational models with coupling terms are becoming increasingly popular in
image analysis. They involve auxiliary variables, such that their energy
minimisation splits into multiple fractional steps that can be solved easier
and more efficiently. In our paper we show that coupling models offer a number
of interesting properties that go far beyond their obvious numerical benefits.
We demonstrate that discontinuity-preserving denoising can be achieved even
with quadratic data and smoothness terms, provided that the coupling term
involves the $L^1$ norm. We show that such an $L^1$ coupling term provides
additional information as a powerful edge detector that has remained unexplored
so far. While coupling models in the literature approximate higher order
regularisation, we argue that already first order coupling models can be
useful. As a specific example, we present a first order coupling model that
outperforms classical TV regularisation. It also establishes a theoretical
connection between TV regularisation and the Mumford-Shah segmentation
approach. Unlike other Mumford-Shah algorithms, it is a strictly convex
approximation, for which we can guarantee convergence of a split Bregman
algorithm.
