  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.
