Lifting vectorial variational problems: A natural formulation based on geometric measure theory and discrete exterior calculus

Mollenhoff T, Cremers D (2019)

Publication Type: Conference contribution

Publication year: 2019

Journal

Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition IEEE Computer Society; 1999

Publisher: IEEE Computer Society

Book Volume: 2019-June

Pages Range: 11109-11118

Conference Proceedings Title: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition

Event location: Long Beach, CA, USA

ISBN: 9781728132938

DOI: 10.1109/CVPR.2019.01137

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.

Involved external institutions

Technische Universität München (TUM)

Germany (DE)

How to cite

APA:

Mollenhoff, T., & Cremers, D. (2019). Lifting vectorial variational problems: A natural formulation based on geometric measure theory and discrete exterior calculus. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (pp. 11109-11118). Long Beach, CA, USA: IEEE Computer Society.

MLA:

Mollenhoff, Thomas, and Daniel Cremers. "Lifting vectorial variational problems: A natural formulation based on geometric measure theory and discrete exterior calculus." Proceedings of the 32nd IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA IEEE Computer Society, 2019. 11109-11118.

BibTeX: Download