Simplified Jacobians in 6-DoF Camera Tracking

Aligning point clouds forms the front end of many visual odometry, 3D reconstruction and SLAM systems. Such alignment of point clouds from two or more different views allows to obtain an overall change in the translation and rotation relative to a given reference view. In this report, we focus on such problem of aligning point clouds particularly looking at the Iterative Closest Point (ICP) [2] which is one of the most popular and simplest algorithms for point cloud alignment. The algorithm alternates between finding the best possible correspondences and optimising over the 6-DoF robot pose given these correspondences. Over the years many different variations [4] have emerged that yield better and robust performance. We use the recommended point-plane variant of ICP in the following. Denoting R = {ri}i=1 as the set of 3D points in the reference view and L = {lj}j=1, the set in the new incoming live view. ICP then seeks to obtain the SE(3) transformation Trl (read it as live to ref transformation) that aligns the point in the live view to the reference view. The point-plane variant of ICP measures the perpendicular distance of the point from the plane and allows to slide the planar regions on top of each other. However, unlike point-point ICP which requires only the 3D positions of the points, this variant comes with an additional computational expense requiring 3D point surface normals to compute the point-plane distance. Next, we formulate the cost functions that measure the discrepency error which is optimised until a standard convergence criteria is satisfied that confirms that points are best possibly aligned. We assume in the following that