Global registration of multiple point clouds using semidefinite programming

Consider N points in d-dimensions and M >= 2 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. We study the problem of estimating the global coordinates (up to a global rigid transform) of the points from such measurements. This problem comes up in distributed approaches to molecular conformation (determination of molecular structures using NMR) and sensor network localization, and also in computer vision and graphics applications. We formulate a least-squares-based program, in which the variables are the global coordinates and the rigid transforms associated with each local coordinate system. While there exists a well-known closed form SVD-based solution when M=2, for M > 2 the least-squares problem is non-convex and no closed-form solution is known. We show that the non-convex least-squares problem can be relaxed into a standard semidefinite program (SDP), and demonstrate how the global coordinates (and rigid transforms) can be efficiently computed from this SDP. Under specific measurement and noise models, we prove that the proposed convex relaxation exactly and stably recovers the global coordinates from the noisy local coordinates.