A variational method for curve extraction with curvature-dependent energies

TL;DR

This paper presents a variational method for extracting curves and 1D structures from images, incorporating curvature-dependent energies and leveraging advanced mathematical tools like Smirnov's theorem and sub-Riemannian geometry.

Contribution

It introduces a novel variational framework that automatically extracts curves using a bi-level minimization approach with curvature considerations, extending previous methods with a new energy formulation.

Findings

Effective extraction of curves from images with minimal supervision

Incorporation of curvature-dependent energies improves accuracy

Extension to sub-Riemannian and Finslerian metrics enhances flexibility

Abstract

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.