# Lifting Vectorial Variational Problems: A Natural Formulation based on Geometric Measure Theory and Discrete Exterior Calculus

## TL;DR

This paper introduces a novel convex formulation for vectorial variational problems in imaging by lifting them to the space of currents and discretizing with Whitney forms, enabling more effective shape optimization.

## Contribution

It presents a new convex relaxation approach for vector-valued variational problems using geometric measure theory and discrete exterior calculus, extending multilabeling methods.

## Key findings

- Convex relaxation via currents improves problem tractability.
- Discretization with Whitney forms generalizes multilabeling approaches.
- The method facilitates shape optimization in imaging tasks.

## Abstract

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.