# RiemanLine: Riemannian Manifold Representation of 3D Lines for Factor Graph Optimization

arXiv: https://arxiv.org/abs/2508.04335 · HTML: https://mlsanity.com/paper/2508.04335/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 33 references that are in the map.
- First posted on arXiv 6 Aug 2025.

## Builds on (most cited first, 13 of 13)

- [ORB-SLAM: a Versatile and Accurate Monocular SLAM System](https://mlsanity.com/paper/1502.00956/) (2015)
- [ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual-Inertial and Multi-Map SLAM](https://mlsanity.com/paper/2007.11898/) (2020)
- [ORB-SLAM2: an Open-Source SLAM System for Monocular, Stereo and RGB-D Cameras](https://mlsanity.com/paper/1610.06475/) (2016)
- [TartanAir: A Dataset to Push the Limits of Visual SLAM](https://mlsanity.com/paper/2003.14338/) (2020)
- [Direct Sparse Odometry](https://mlsanity.com/paper/1607.02565/) (2016)
- [Kimera: an Open-Source Library for Real-Time Metric-Semantic Localization and Mapping](https://mlsanity.com/paper/1910.02490/) (2019)
- [RTAB-Map as an Open-Source Lidar and Visual SLAM Library for Large-Scale and Long-Term Online Operation](https://mlsanity.com/paper/2403.06341/) (2024)
- [Robust Visual SLAM with Point and Line Features](https://mlsanity.com/paper/1711.08654/) (2017)
- [Structure-SLAM: Low-Drift Monocular SLAM in Indoor Environments](https://mlsanity.com/paper/2008.01963/) (2020)
- [Convex Relaxations for Pose Graph Optimization with Outliers](https://mlsanity.com/paper/1801.02112/) (2018)
- [ManhattanSLAM: Robust Planar Tracking and Mapping Leveraging Mixture of Manhattan Frames](https://mlsanity.com/paper/2103.15068/) (2021)
- [StructVIO : Visual-inertial Odometry with Structural Regularity of Man-made Environments](https://mlsanity.com/paper/1810.06796/) (2018)
- [Open-Structure: Structural Benchmark Dataset for SLAM Algorithms](https://mlsanity.com/paper/2310.10931/) (2023)

## Abstract (arXiv)

Minimal parametrization of 3D lines plays a critical role in camera localization and structural mapping. Existing representations in robotics and computer vision predominantly handle independent lines, overlooking structural regularities such as sets of parallel lines that are pervasive in man-made environments. This paper introduces RiemanLine, a unified minimal representation for 3D lines formulated on Riemannian manifolds that jointly accommodates both individual lines and parallel-line groups. Our key idea is to decouple each line landmark into global and local components: a shared vanishing direction optimized on the unit sphere S2, and scaled normal vectors constrained on orthogonal subspaces, enabling compact encoding of structural regularities. For n parallel lines, the proposed representation reduces the parameter space from 4n (orthonormal form) to 2n+2, naturally embedding parallelism without explicit constraints. We further integrate this parameterization into a factor graph framework, allowing global direction alignment and local reprojection optimization within a unified manifold-based bundle adjustment. Extensive experiments on ICL-NUIM, TartanAir, and synthetic benchmarks demonstrate that our method achieves significantly more accurate pose estimation and line reconstruction, while reducing parameter dimensionality and improving convergence stability.