Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus
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.
Cite
Text
Mollenhoff and Cremers. "Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019. doi:10.1109/CVPR.2019.01137Markdown
[Mollenhoff and Cremers. "Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019.](https://mlanthology.org/cvpr/2019/mollenhoff2019cvpr-lifting/) doi:10.1109/CVPR.2019.01137BibTeX
@inproceedings{mollenhoff2019cvpr-lifting,
title = {{Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus}},
author = {Mollenhoff, Thomas and Cremers, Daniel},
booktitle = {Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition},
year = {2019},
doi = {10.1109/CVPR.2019.01137},
url = {https://mlanthology.org/cvpr/2019/mollenhoff2019cvpr-lifting/}
}