  We present an approach for variational regularization of inverse and imaging
problems for recovering functions with values in a set of vectors. We introduce
regularization functionals, which are derivative-free double integrals of such
functions. These regularization functionals are motivated from double
integrals, which approximate Sobolev semi-norms of intensity functions. These
were introduced in Bourgain, Br\'ezis and Mironescu, "Another Look at Sobolev
Spaces". In: Optimal Control and Partial Differential Equations-Innovations and
Applications, IOS press, Amsterdam, 2001. For the proposed regularization
functionals we prove existence of minimizers as well as a stability and
convergence result for functions with values in a set of vectors.
