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.