On the primal and dual formulations of the Discrete Mumford-Shah functional

This work focuses on the discrete Mumford-Shah (D-MS) functional which aims to perform jointly image reconstruction and contour detection but at the price of minimizing a non-convex objective function. This functional was of main interest during the 90's but was then forsaken in order to focus on the unique restoration task relying on non-smooth convex minimization. Recent advances about D-MS were dedicated to alternative objective functions for which efficient numerical solution based on proximal iterations can be designed. In the 90's literature about D-MS, equivalences between primal and dual formulations were derived. However, in the framework obtained by more recent developments dedicated to D-MS such an equivalence was not yet derived and it is the goal of this work. By providing both a primal and dual formulation, a large panel of algorithms can be employed including recent proximal-based algorithms benefiting of good convergence behaviour, especially due to KL properties and also most standard methods such as BFGS.