Geodesics with Unified Tangent-constrained Priors and Curvature Regularization

2026-05-28

AI Summary

This work addresses the susceptibility of existing geodesic active contour models to topological shortcuts when segmenting objects with complex shapes or weak boundaries, primarily due to insufficient constraints on path tangents. To overcome this limitation, the authors propose a unified geodesic framework that integrates spatially varying tangent priors—such as skeleton-based intrinsic shape representations—with curvature regularization in a lifted orientation space. This approach yields, for the first time, a family of Finsler metrics that explicitly enforce hard tangent constraints. The resulting energy minimization problem is efficiently solved via the Hamilton–Jacobi–Bellman equation using an enhanced fast marching method. Experiments demonstrate that the proposed method significantly suppresses topological shortcuts, enhances robustness to weak boundaries, and achieves superior shape fidelity across synthetic, natural, and medical images.

Abstract

Curvature-penalized geodesic models have proven their effectiveness in image segmentation by computing globally optimal curves. Unfortunately, these models remain susceptible to shortcuts when delineating objects with complex shapes and image intensity distributions, as they lack mechanisms to enforce shape-aware tangent constraints. To address this limitation, we propose a unified geodesic framework that integrates tangent-constrained priors with curvature penalization. The key idea is to formulate tangent admissibility directly within the orientation-lifted space, where path tangents are restricted to spatially varying angular sectors derived from intrinsic shape representatives (ISR) such as skeletons or interior landmarks. This formulation gives rise to a family of tangent-constrained Finslerian metrics, extending the classical curvature-penalized geodesic models while enforcing mandatory tangent constraints. The resulting Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) admit efficient numerical solutions via variants of the fast marching method, preserving the single-pass computational complexity. Experiments on synthetic, natural, and medical images demonstrate that the proposed geodesic framework indeed improves robustness against weak boundaries and topological shortcuts, yielding segmentation results with enhanced shape fidelity compared to existing geodesic models.

Problem

• geodesic
• curvature regularization
• tangent constraints
• image segmentation
• shortcuts

Innovation

• tangent-constrained geodesics
• curvature regularization
• orientation-lifted space
• Finslerian metrics
• shape-aware segmentation