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

2025-08-06

AI Summary

Most existing 3D line representations model individual lines in isolation, neglecting the prevalent parallelism constraints inherent in man-made environments. To address this, we propose a unified minimal representation grounded in Riemannian geometry: each line—and its parallel group—is jointly embedded onto the unit sphere (S^2) (encoding shared vanishing directions) and an orthogonal subspace (parameterizing scaled normal vectors), thereby decoupling global orientation from local geometry and enabling implicit structural regularization. We further introduce, for the first time, a manifold-based joint bundle adjustment within a factor graph framework, eliminating the need for explicit parallelism constraints. Evaluated on ICL-NUIM and TartanAir, our method achieves significant improvements in pose estimation and line reconstruction accuracy. Model parameters reduce from (4n) to (2n+2), yielding faster convergence and enhanced optimization stability. Our core contributions are a structure-aware, manifold-unified line representation and an efficient, constraint-free optimization paradigm.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 extbf{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 $mathcal{S}^2$, 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.

Problem

• Minimal parametrization of 3D lines for camera localization
• Representation of parallel lines in man-made environments
• Unified manifold-based bundle adjustment for optimization

Innovation

• Riemannian manifold representation for 3D lines
• Decouples lines into global and local components
• Integrates parameterization into factor graph framework