On The Primal and Dual Formulations Of The Discrete Mumford-Shah Functional

Abstract

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 DMS 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 behavior, especially due to KL properties and also most standard methods such as BFGS.