A variational method for curve extraction with curvature-dependent energies

2025-12-01

AI Summary

This paper addresses the unsupervised extraction of curves and one-dimensional structures in images with known endpoints. The proposed method integrates geometric modeling and variational principles: curves are lifted to a position-orientation space, where sub-Riemannian or Finsler metrics naturally encode curvature-dependent energy; a differentiable energy functional is constructed based on Smirnov’s vector field decomposition theorem, and a bilevel optimization framework is designed to compute globally optimal paths under endpoint constraints. By jointly discretizing the energy and enforcing geometric regularization, the approach significantly improves connectivity robustness in weak-boundary and low signal-to-noise ratio scenarios. Experiments demonstrate superior accuracy and generalizability compared to state-of-the-art variational and learning-based methods—without requiring manual parameter tuning or ground-truth annotations—making it particularly effective for challenging 1D structure extraction tasks such as biological image analysis and vascular segmentation.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Extracts curves between endpoints using variational methods
• Designs unsupervised approach for curve and 1D structure extraction
• Extends method to curvature-dependent energies with geometric lifting

Innovation

• Variational approach for curve extraction
• Bi-level minimization for unsupervised structure detection
• Curvature-dependent energies with sub-Riemannian lifting