Lifting Vectorial Variational Problems: A Natural Formulation based on Geometric Measure Theory and Discrete Exterior Calculus
This work addresses variational problems in imaging and vision, offering a novel formulation that could improve optimization methods, though it appears incremental as it builds on existing geometric measure theory and discrete exterior calculus.
The paper tackles the relaxation and convexification of vectorial variational problems in imaging and vision by lifting them to the space of currents, leading to an equivalent shape optimization problem over oriented surfaces. It proposes a discretization using Whitney forms, generalizing recent multilabeling approaches.
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.