研究Wahba问题的算法对比以及新算法的提出

A Comparative Study of Algorithms for the Wahba Problem and the Proposal of a New Algorithm

摘要: Wahba问题是指在旋转矩阵估计中求解两组向量之间的最佳旋转矩阵，该问题常出现在航空航天工程、机器人应用和计算机视觉等领域。尤其在卫星姿态估计、图像拼接、3D重建等应用有着重要作用。在这些应用中，数据往往受到测量误差和异常值的影响，导致估计结果的精度下降，因此，如何在这些不理想条件下求解Wahba问题，成为了研究的核心问题之一。为了应对这些问题，要求算法具有更强的抗噪性和鲁棒性，以确保在不完全准确的数据条件下仍能获得可靠的结果。本文主要研究在三维空间中的Wahba问题，介绍了几种针对误差和异常值影响的算法，包括四元数解法(QUASAR)、快速全局配准算法(FGR)、RANSAC算法，此外，本文还在四元数解法的基础上提出了一种新的算法思路，旨在提高算法性能。文章简单阐述了算法的计算原理和推导过程，通过计算机进行数据仿真，对比了不同算法在存在误差和异常值情况下的精度、计算时间和鲁棒性。

Abstract: The Wahba problem refers to the estimation of the optimal rotation matrix between two sets of vectors in the context of rotation matrix estimation. This problem is commonly encountered in fields such as aerospace engineering, robotics, and computer vision. It plays an important role in applications like satellite attitude estimation, image stitching, and 3D reconstruction. In these applications, data are often affected by measurement errors and outliers, leading to a decrease in estimation accuracy. Therefore, solving the Wahba problem under these imperfect conditions has become a crucial research issue. To address these problems, algorithms are required to have stronger noise resistance and robustness, so as to ensure reliable results can still be obtained under conditions with inaccurate data. This paper primarily investigates the Wahba problem in three-dimensional space and introduces several algorithms designed to handle errors and outliers, including quaternion-based semidefinite relaxation for robust alignment (QUASAR), fast global registration (FGR), and RANSAC algorithm. Additionally, this paper proposes a new algorithmic approach based on the quaternion-based robust Wahba problem, aiming to improve algorithm performance. The computational principles and derivations of the proposed algorithms are briefly outlined, and through computer simulations, the paper compares the accuracy, computational time, and robustness of different algorithms under the influence of errors and outliers.