Two-dimensional (2-D)/3-D pose-graph simultaneous localization and mapping (SLAM) is a problem of estimating a set of poses based on noisy measurements of relative rotations and translations. This article focuses on the relation between the graphical structure of pose-graph SLAM and Fisher information matrix (FIM), Cramér–Rao lower bounds (CRLB), and its optimal design metrics (T-optimality and D-optimality). As a main contribution, based on the assumption of isotropic Langevin noise for rotation and block-isotropic Gaussian noise for translation, the FIM and CRLB are derived and shown to be closely related to the graph structure, in particular, the weighted Laplacian matrix. We also prove that total node degree and weighted number of spanning trees, as two graph connectivity metrics, are, respectively, closely related to the trace and determinant of the FIM. The discussions show that, compared with the D-optimality metric, the T-optimality metric is more easily computed but less effective. We also present upper and lower bounds for the D-optimality metric, which can be efficiently computed and are almost independent of the estimation results. The results are verified with several well-known datasets, such as Intel, KITTI, sphere, and so on.

中文翻译：

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姿势图SLAM的Cramér-Rao界和最佳设计指标

二维（2-D）/ 3-D姿态图同时定位和映射（SLAM）是一个基于相对旋转和平移的噪声测量来估计一组姿态的问题。本文重点介绍姿势图SLAM的图形结构和Fisher信息矩阵（FIM），Cramér-Rao下界（CRLB）及其最佳设计指标（T优化和D优化）之间的关系。作为主要贡献，基于旋转的各向同性Langevin噪声和平移的块各向同性高斯噪声的假设，推导了FIM和CRLB并显示出它们与图结构特别是加权拉普拉斯矩阵密切相关。我们还证明了总节点度和生成树的加权数量分别作为两个图的连通性指标，与FIM的跟踪和决定因素密切相关。讨论表明，与D最优度量相比，T最优度量更容易计算，但效果较差。我们还提出了D最优性指标的上限和下限，可以有效地计算出该上限，而下限几乎与估计结果无关。结果已通过多个知名数据集进行了验证，例如Intel，KITTI，sphere等。