Provably Approximated Point Cloud Registration

The goal of the alignment problem is to align a (given) point cloud P = \ p_1,\cdots,p_n\ to another (observed) point cloud Q = \ q_1,\cdots,q_n\ . That is, to compute a rotation matrix R \in \mathbb R ^ 3 x3 and a translation vector t \in \mathbb R ^ 3 that minimize the sum of paired distances between every transformed point Rp_i-t, to its corresponding point q_i, over every i\in \br 1,\cdots,n . A harder version is the registration problem, where the correspondence is unknown, and the minimum is also over all possible correspondence functions from P to Q. Algorithms such as the Iterative Closest Point (ICP) and its variants were suggested for these problems, but none yield a provable non-trivial approximation for the global optimum. We prove that there always exists a "witness" set of 3 pairs in P xQ that, via novel alignment algorithm, defines a constant factor approximation (in the worst case) to this global optimum. We then provide algorithms that recover this witness set and yield the first provable constant factor approximation for the: (i) alignment problem in O(n) expected time, and (ii) registration problem in polynomial time. Such small witness sets exist for many variants including points in d-dimensional space, outlier-resistant cost functions, and different correspondence types. Extensive experimental results on real and synthetic datasets show that, in practice, our approximation constants are close to 1 and our error is up to x10 times smaller than state-of-the-art algorithms.