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Modern depth sensors can generate a huge number of 3D points in few secondsto be latter processed by Localization and Mapping algorithms. Ideally, thesealgorithms should handle efficiently large sizes of Point Clouds under theassumption that using more points implies more information available. The EigenFactors (EF) is a new algorithm that solves SLAM by using planes as the maingeometric primitive. To do so, EF exhaustively calculates the error of allpoints at complexity $O(1)$, thanks to the {\em Summation matrix} $S$ ofhomogeneous points.  The solution of EF is highly efficient: i) the state variables are only thesensor poses -- trajectory, while the plane parameters are estimated previouslyin closed from and ii) EF alternating optimization uses a Newton-Raphson methodby a direct analytical calculation of the gradient and the Hessian, which turnsout to be a block diagonal matrix. Since we require to differentiate overeigenvalues and matrix elements, we have developed an intuitive methodology tocalculate partial derivatives in the manifold of rigid body transformations$SE(3)$, which could be applied to unrelated problems that require analyticalderivatives of certain complexity.  We evaluate EF and other state-of-the-art plane SLAM back-end algorithms in asynthetic environment. The evaluation is extended to ICL dataset (RGBD) andLiDAR KITTI dataset. Code is publicly available athttps://github.com/prime-slam/EF-plane-SLAM.

Eigen-Factors an Alternating Optimization for Back-end Plane SLAM of 3D Point Clouds