Precise Relaxation of the Mumford-Shah Functional

Jumps, edges and cutoffs are prevalent in our world across many modalities. The Mumford-Shah functional is a classical and elegant approach for modeling such discontinuities but global optimization of this non-convex functional remains challenging. The state of the art are convex representations based on the theory of calibrations. The major drawback of these approaches is the ultimate discretization of the co-domain into labels. For the case of total variation regularization, this issue has been partially resolved by recent sublabel-accurate relaxations, a generalization of which to other regularizers is not straightforward. In this work, we show that sublabel-accurate lifting approaches can be derived by discretizing a continuous relaxation of the Mumford-Shah functional by means of finite elements. We thereby unify and generalize existing functional lifting approaches. We show the efficiency of the proposed discretizations on discontinuity-preserving denoising tasks.