# Global Minimum for a Finsler Elastica Minimal Path Approach

arXiv: https://arxiv.org/abs/1612.00343 · HTML: https://mlsanity.com/paper/1612.00343/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2016.

## Built on by (most cited first, 11 of 11)

- [Finsler-Laplace-Beltrami Operators with Application to Shape Analysis](https://mlsanity.com/paper/2404.03999/) (2024)
- [CurvPnP: Plug-and-play Blind Image Restoration with Deep Curvature Denoiser](https://mlsanity.com/paper/2211.07286/) (2022)
- [Geodesic Models with Convexity Shape Prior](https://mlsanity.com/paper/2111.00794/) (2021)
- [Grouping Boundary Proposals for Fast Interactive Image Segmentation](https://mlsanity.com/paper/2309.04169/) (2023)
- [Confluent Vessel Trees with Accurate Bifurcations](https://mlsanity.com/paper/2103.14268/) (2021)
- [Surface Segmentation Using Implicit Divergence Constraint Between Adjacent Minimal Paths](https://mlsanity.com/paper/2111.13111/) (2021)
- [Morphology Edge Attention Network and Optimal Geometric Matching Connection model for vascular segmentation](https://mlsanity.com/paper/2306.01808/) (2023)
- [Dynamic Exploration on Segment-Proposal Graphs for Tubular Centerline Tracking](https://mlsanity.com/paper/2506.18930/) (2025)
- [Fracture interactive geodesic active contours for bone segmentation](https://mlsanity.com/paper/2509.14817/) (2025)
- [Geodesics with Unified Tangent-constrained Priors and Curvature Regularization](https://mlsanity.com/paper/2606.00139/) (2026)
- [Mask Proposal Voting Based on Geodesic Framework for Robust Image Segmentation](https://mlsanity.com/paper/2606.14912/) (2026)

## Abstract (arXiv)

In this paper, we propose a novel curvature-penalized minimal path model via an orientation-lifted Finsler metric and the Euler elastica curve. The original minimal path model computes the globally minimal geodesic by solving an Eikonal partial differential equation (PDE). Essentially, this first-order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models. To solve this problem, we present an Eikonal PDE-based Finsler elastica minimal path approach to address the curvature-penalized geodesic energy minimization problem. We were successful at adding the curvature penalization to the classical geodesic energy. The basic idea of this work is to interpret the Euler elastica bending energy via a novel Finsler elastica metric that embeds a curvature penalty. This metric is non-Riemannian, anisotropic and asymmetric, and is defined over an orientation-lifted space by adding to the image domain the orientation as an extra space dimension. Based on this orientation lifting, the proposed minimal path model can benefit from both the curvature and orientation of the paths. Thanks to the fast marching method, the global minimum of the curvature-penalized geodesic energy can be computed efficiently. We introduce two anisotropic image data-driven speed functions that are computed by steerable filters. Based on these orientation-dependent speed functions, we can apply the proposed Finsler elastica minimal path model to the applications of closed contour detection, perceptual grouping and tubular structure extraction. Numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results.