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Variational models with coupling terms are becoming increasingly popular inimage analysis. They involve auxiliary variables, such that their energyminimisation splits into multiple fractional steps that can be solved easierand more efficiently. In our paper we show that coupling models offer a numberof interesting properties that go far beyond their obvious numerical benefits.We demonstrate that discontinuity-preserving denoising can be achieved evenwith quadratic data and smoothness terms, provided that the coupling terminvolves the $L^1$ norm. We show that such an $L^1$ coupling term providesadditional information as a powerful edge detector that has remained unexploredso far. While coupling models in the literature approximate higher orderregularisation, we argue that already first order coupling models can beuseful. As a specific example, we present a first order coupling model thatoutperforms classical TV regularisation. It also establishes a theoreticalconnection between TV regularisation and the Mumford-Shah segmentationapproach. Unlike other Mumford-Shah algorithms, it is a strictly convexapproximation, for which we can guarantee convergence of a split Bregmanalgorithm.

Variational Coupling Revisited: Simpler Models, Theoretical Connections, and Novel Applications