  Numerous tasks in imaging and vision can be formulated as variational
problems over vector-valued maps. We approach the relaxation and
convexification of such vectorial variational problems via a lifting to the
space of currents. To that end, we recall that functionals with polyconvex
Lagrangians can be reparametrized as convex one-homogeneous functionals on the
graph of the function. This leads to an equivalent shape optimization problem
over oriented surfaces in the product space of domain and codomain. A convex
formulation is then obtained by relaxing the search space from oriented
surfaces to more general currents. We propose a discretization of the resulting
infinite-dimensional optimization problem using Whitney forms, which also
generalizes recent "sublabel-accurate" multilabeling approaches.
