# A variational method for curve extraction with curvature-dependent energies

arXiv: https://arxiv.org/abs/2512.01494 · HTML: https://mlsanity.com/paper/2512.01494/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 41 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Builds on (most cited first, 1 of 1)

- [Formulation of branched transport as geometry optimization](https://mlsanity.com/paper/2109.07820/) (2021)

## Abstract (arXiv)

We introduce a variational approach for extracting curves between a list of possible endpoints, based on the discretization of an energy and Smirnov's decomposition theorem for vector fields. It is used to design a bi-level minimization approach to automatically extract curves and 1D structures from an image, which is mostly unsupervised. We extend then the method to curvature-dependent energies, using a now classical lifting of the curves in the space of positions and orientations equipped with an appropriate sub-Riemanian or Finslerian metric.