A Derivative-Free Approach to Total Variation Regularization

The goal of this paper is to present a novel approach for total variation regularization and Sobolev minimization, which are prominent tools for variational imaging. Thereby we use derivative free characterizations of the total variation semi-norm and Sobolev semi-norms of functions recently derived by Bourgain, Br\'ezis, Mironescu and D\'avila. Their analysis is to approximate the semi-norms of a function by singular integral operators. With this characterization we derive a series of novel regularization methods for total variation minimization which have as a novel feature a non-local double integral regularization term.