Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A variational method for curve extraction with curvature-dependent energies

Majid Arthaud, Antonin Chambolle, Vincent Duval

TL;DR

The paper proposes a variational framework for extracting open curves in images by representing paths as normal charges and leveraging Smirnov decomposition to relate vector field measures to geodesic curves. It develops a convex, discretized optimization on graphs to recover curves between prescribed endpoints and introduces a bi-level scheme to automatically discover endpoints; the approach is extended to curvature penalization through a roto-translational lifting, enabling smoother and crossing curves. The method relies on a primal-dual algorithm for efficient optimization and demonstrates robustness to multiple curves and 3D extensions, with practical results on synthetic and real-like data. Overall, this work offers a scalable, unsupervised pipeline for extracting thin structures with curvature control, potentially benefiting image analysis tasks requiring precise 1D geometry extraction.

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.

A variational method for curve extraction with curvature-dependent energies

TL;DR

The paper proposes a variational framework for extracting open curves in images by representing paths as normal charges and leveraging Smirnov decomposition to relate vector field measures to geodesic curves. It develops a convex, discretized optimization on graphs to recover curves between prescribed endpoints and introduces a bi-level scheme to automatically discover endpoints; the approach is extended to curvature penalization through a roto-translational lifting, enabling smoother and crossing curves. The method relies on a primal-dual algorithm for efficient optimization and demonstrates robustness to multiple curves and 3D extensions, with practical results on synthetic and real-like data. Overall, this work offers a scalable, unsupervised pipeline for extracting thin structures with curvature control, potentially benefiting image analysis tasks requiring precise 1D geometry extraction.

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.

Paper Structure

This paper contains 22 sections, 5 theorems, 71 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Charges and curves
  3. The space of normal charges
  4. Charges induced by curves
  5. Smirnov's decomposition theorem
  6. A minimization problem with prescribed divergence
  7. Assumptions:
  8. Discrete curves
  9. Flux on a graph
  10. Isotropic discretizations
  11. Optimization
  12. An iterative discrete curve reconstruction algorithm
  13. Finding the endpoints
  14. Numerical results
  15. Curvature penalization
  16. ...and 7 more sections

Key Result

theorem 1

Let $z\in \mathcal{V}$. Then there exist two normal charges $p,q \in \mathcal{V}$ such that $z$ completely decomposes into $p$ and $q$, $\mathop{\mathrm{div}}\nolimits p=0$, and $q$ completely decomposes into simple oriented curves of finite length. In other words, there exists some nonegative Borel

Figures (12)

  • Figure 1: Left: $200 \times 200$ "noisy comma" shape (crop from laville_algo), middle: result with weighted $\ell^1$ norm, right: weighted $\ell^2$ norm.
  • Figure 2: Left: result with the weighted and averaged $\ell^2$-norm on the same "noisy comma" image. Right: zoom of the same result (bottom) and zoom without averaging (top).
  • Figure 3: Left: an angiogram (source: Wikipedia), right: geodesic computed by our method (compare for instance with DeschampsCohen).
  • Figure 4: Result on a 3D potential: the left images correspond to the $70 \times 70 \times 30$ potential, and the right images are the retrieved curves. Now the result is obtained after 2000 primal-dual steps, with fixed endpoints, for a wall time of about 6 min.
  • Figure 5: For the discretization of the continuous curve drawn in black and the Dirac mass at its endpoint, we represent the four different cases of the algorithm.
  • ...and 7 more figures

Theorems & Definitions (10)

  • theorem 1: smirnov
  • theorem 2
  • remark 1
  • proof : of Theorem \ref{['prop:smirnov']}
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof