Global registration of multiple point clouds using semidefinite programming

Consider N points in d-dimension and M local coordinate systems that are related through unknown rigid transforms. For each point we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The least-squares formulation, though non-convex, has a well known closed-form solution for the case M=2 (based on the singular value decomposition). However, no closed form solution is known for M>2. In this paper, we propose a semidefinite relaxation of the least-squares formulation, and prove conditions for exact and stable recovery for both this relaxation and for a previously proposed spectral relaxation. In particular, using results from rigidity theory and the theory of semidefinite programming, we prove that the semidefinite relaxation can guarantee recovery under more adversarial measurements compared to the spectral counterpart. We perform numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the semidefinite program (i.e., we are able to solve the original non-convex problem) up to a certain noise threshold, and (b) the semidefinite program performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.