Recently there has been a surge of interest to use branch-and-bound (bnb) optimisation for 3D point cloud registration. While bnb guarantees globally optimal solutions, it is usually too slow to be practical. A fundamental source of difficulty is the search for the rotation parameters in the 3D rigid transform. In this work, assuming that the translation parameters are known, we focus on constructing a fast rotation search algorithm. With respect to an inherently robust geometric matching criterion, we propose a novel bounding function for bnb that allows rapid evaluation. Underpinning our bounding function is the usage of stereographic projections to precompute and spatially index all possible point matches. This yields a robust and global algorithm that is significantly faster than previous methods. To conduct full 3D registration, the translation can be supplied by 3D feature matching, or by another optimisation framework that provides the translation. On various challenging point clouds, including those taken out of lab settings, our approach demonstrates superior efficiency.
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