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.

翻译：三维直线的最小参数化在相机定位与结构重建中具有关键作用。机器人学与计算机视觉领域的现有表示方法主要处理独立直线，忽略了人造环境中普遍存在的结构规律性（如平行线集）。本文提出 \textbf{RiemanLine}，一种建立在黎曼流形上的统一最小化三维直线表示方法，可同时处理独立直线与平行线组。我们的核心思想是将每个直线路标分解为全局与局部分量：在单位球面 $\mathcal{S}^2$ 上优化的共享消失方向，以及约束在正交子空间上的缩放法向量，从而实现对结构规律性的紧凑编码。对于 $n$ 条平行线，所提表示将参数空间从 $4n$（标准正交形式）缩减至 $2n+2$，无需显式约束即可自然嵌入平行性。我们进一步将该参数化集成至因子图框架中，使得全局方向对齐与局部重投影优化能够在基于流形的统一光束法平差中实现。在 ICL-NUIM、TartanAir 及合成基准数据集上的大量实验表明，本方法在显著降低参数维度并提升收敛稳定性的同时，实现了更精确的位姿估计与直线重建。