Modern depth sensors can generate a huge number of 3D points in few seconds to be latter processed by Localization and Mapping algorithms. Ideally, these algorithms should handle efficiently large sizes of Point Clouds under the assumption that using more points implies more information available. The Eigen Factors (EF) is a new algorithm that solves SLAM by using planes as the main geometric primitive. To do so, EF exhaustively calculates the error of all points at complexity $O(1)$, thanks to the {\em Summation matrix} $S$ of homogeneous points. The solution of EF is highly efficient: i) the state variables are only the sensor poses -- trajectory, while the plane parameters are estimated previously in closed from and ii) EF alternating optimization uses a Newton-Raphson method by a direct analytical calculation of the gradient and the Hessian, which turns out to be a block diagonal matrix. Since we require to differentiate over eigenvalues and matrix elements, we have developed an intuitive methodology to calculate partial derivatives in the manifold of rigid body transformations $SE(3)$, which could be applied to unrelated problems that require analytical derivatives of certain complexity. We evaluate EF and other state-of-the-art plane SLAM back-end algorithms in a synthetic environment. The evaluation is extended to ICL dataset (RGBD) and LiDAR KITTI dataset. Code is publicly available at https://github.com/prime-slam/EF-plane-SLAM.

翻译：摘要：现代深度传感器可在数秒内生成海量三维点云，这些数据随后需由定位与地图构建算法处理。理想情况下，这些算法应能高效处理大规模点云，其基本假设是使用更多点意味着可获取更多信息。Eigen-Factors（EF）是一种新型算法，以平面作为主要几何基元解决SLAM问题。通过利用齐次点构成的{求和矩阵}$S$，EF能以$O(1)$复杂度穷举计算所有点的误差。EF的解决方案具有极高效率：（i）状态变量仅为传感器位姿（即轨迹），而平面参数则通过闭式解预先估计；（ii）EF交替优化采用牛顿-拉弗森方法，通过直接解析计算梯度与海森矩阵（最终呈现为块对角矩阵）实现。由于需要处理特征值与矩阵元素的微分，我们开发了一种直观的方法来计算刚体变换流形$SE(3)$上的偏导数，该方法可推广至需要特定复杂度解析导数的其他问题。我们在合成环境中评估了EF及其他最新平面SLAM后端算法，并将评估扩展至ICL数据集（RGBD）与LiDAR KITTI数据集。代码已开源：https://github.com/prime-slam/EF-plane-SLAM。