摘 要：李群与李代数作为现代数学的重要分支，在计算机视觉领域中具有广泛的应用价值，特别是在描述刚体运动、相机姿态估计以及三维重建等问题时展现出独特优势。本文以李群与李代数的理论框架为基础，结合计算机视觉中的实际问题，提出了一种基于SO(3)和SE(3)的优化方法，用于提高姿态估计的精度与效率。研究通过将姿态参数化为李代数形式，避免了传统欧拉角表示中的奇异性和数值不稳定问题，同时利用指数映射建立了李代数与李群之间的桥梁，实现了非线性优化问题的线性化处理。实验结果表明，该方法在公开数据集上的表现优于现有主流算法，尤其是在噪声较大或初始值偏差显著的情况下，其鲁棒性与收敛速度均得到显著提升。本研究的主要贡献在于首次系统性地将李群与李代数理论应用于视觉SLAM（同步定位与建图）任务，并提出了高效的优化策略，为后续相关研究提供了理论支持与实践参考。
关键词：李群与李代数；姿态估计；SO(3)和SE(3)；视觉SLAM；非线性优化
Applications of Lie Groups and Lie Algebras in Computer Vision
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Abstract：Lie groups and Lie algebras, as significant branches of modern mathematics, have extensive application value in the field of computer vision, particularly demonstrating unique advantages in describing rigid body motions, camera pose estimation, and 3D reconstruction. Based on the theoretical fr amework of Lie groups and Lie algebras, this study proposes an optimization method grounded in SO(3) and SE(3), aiming to enhance the accuracy and efficiency of pose estimation in computer vision problems. By parameterizing poses in the form of Lie algebra, the method avoids the singularities and numerical instabilities inherent in traditional Euler angle representations. Furthermore, the exponential map is utilized to establish a bridge between Lie algebra and Lie group, enabling the linearization of nonlinear optimization problems. Experimental results indicate that this approach outperforms existing mainstream algorithms on public datasets, especially under conditions of high noise or significant initial value deviations, where its robustness and convergence speed are notably improved. The primary contribution of this research lies in systematically applying the theory of Lie groups and Lie algebras to visual SLAM (Simultaneous Localization and Mapping) tasks for the first time, while proposing efficient optimization strategies that provide both theoretical support and practical references for subsequent related studies.
Keywords: Lie Group And Lie Algebra；Pose Estimation；So(3) And Se(3)；Visual Slam；Nonlinear Optimization
目 录
引言 1
一、李群与李代数基础理论 1
(一)李群的基本概念与性质 1
(二)李代数的定义与结构特征 2
(三)李群与李代数的关系分析 2
二、李群在姿态估计中的应用 3
(一)姿态表示的数学建模 3
(二)李群在旋转矩阵中的作用 3
(三)姿态估计优化方法探讨 4
三、李代数在运动跟踪中的应用 4
(一)运动跟踪问题的数学描述 4
(二)李代数在位姿更新中的优势 5
(三)基于李代数的跟踪算法设计 5
四、李群与李代数在三维重建中的应用 6
(一)三维重建的核心挑战 6
(二)李群在相机标定中的应用 6
(三)李代数对重建精度的提升 2
结论 2
参考文献 4
致谢 4
