A convex relaxation of the Ambrosio-Tortorelli elliptic functionals for the mumford-shah functional

In this paper, we revisit the phase-field approximation of Ambrosio and Tortorelli for the Mumford - Shah functional. We then propose a convex relaxation for it to attempt to compute globally optimal solutions rather than solving the nonconvex functional directly, which is the main contribution of this paper. Inspired by McCormick's seminal work on factorable nonconvex problems, we split a nonconvex product term that appears in the Ambrosio - Tortorelli elliptic functionals in a way that a typical alternating gradient method guarantees a globally optimal solution without completely removing coupling effects. Furthermore, not only do we provide a fruitful analysis of the proposed relaxation but also demonstrate the capacity of our relaxation in numerous experiments that show convincing results compared to a naive extension of the McCormick relaxation and its quadratic variant. Indeed, we believe the proposed relaxation and the idea behind would open up a possibility for convexifying a new class of functions in the context of energy minimization for computer vision.