# Geodesics with Unified Tangent-constrained Priors and Curvature Regularization

arXiv: https://arxiv.org/abs/2606.00139 · HTML: https://mlsanity.com/paper/2606.00139/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 56 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 18 of 18)

- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [Image Segmentation Using Deep Learning: A Survey](https://mlsanity.com/paper/2001.05566/) (2020)
- [The Medical Segmentation Decathlon](https://mlsanity.com/paper/2106.05735/) (2021)
- [Conditional Random Fields as Recurrent Neural Networks](https://mlsanity.com/paper/1502.03240/) (2015)
- [Medical Image Segmentation Review: The success of U-Net](https://mlsanity.com/paper/2211.14830/) (2022)
- [Salient Object Detection: A Discriminative Regional Feature Integration Approach](https://mlsanity.com/paper/1410.5926/) (2014)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [PolarMask++: Enhanced Polar Representation for Single-Shot Instance Segmentation and Beyond](https://mlsanity.com/paper/2105.02184/) (2021)
- [Global Minimum for a Finsler Elastica Minimal Path Approach](https://mlsanity.com/paper/1612.00343/) (2016)
- [Minimal Paths for Tubular Structure Segmentation with Coherence Penalty and Adaptive Anisotropy](https://mlsanity.com/paper/1809.07987/) (2018)
- [A Region-based Randers Geodesic Approach for Image Segmentation](https://mlsanity.com/paper/1912.10122/) (2019)
- [A Generalized Asymmetric Dual-front Model for Active Contours and Image Segmentation](https://mlsanity.com/paper/2006.07839/) (2020)
- [Geodesic Paths for Image Segmentation with Implicit Region-based Homogeneity Enhancement](https://mlsanity.com/paper/2008.06909/) (2020)
- [Geodesic Models with Convexity Shape Prior](https://mlsanity.com/paper/2111.00794/) (2021)
- [Progressive Minimal Path Method with Embedded CNN](https://mlsanity.com/paper/2204.00944/) (2022)
- [Grouping Boundary Proposals for Fast Interactive Image Segmentation](https://mlsanity.com/paper/2309.04169/) (2023)
- [Computing Geodesic Paths Encoding a Curvature Prior](https://mlsanity.com/paper/2308.15729/) (2023)

## Abstract (arXiv)

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.