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

TL;DR

RiemanLine introduces a Riemannian manifold-based minimal representation for 3D lines that captures both individual lines and parallel groups, improving accuracy and efficiency in camera localization and mapping.

Contribution

It proposes a unified minimal parametrization for 3D lines on Riemannian manifolds that models structural regularities like parallel lines, reducing parameters and enhancing optimization.

Findings

Achieves more accurate pose estimation and line reconstruction.

Reduces parameter space for parallel lines from 4n to 2n+2.

Improves convergence stability in bundle adjustment.

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 \textbf{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.