Consider @math points in @math and @math 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 @math 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 of this problem, though non-convex, has a well known closed-form solution when @math (based on the singular value decomposition). However, no closed form solution is known for @math . In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results from rigidity theory, we prove conditions for exact and stable recovery for the SDP relaxation. In particular, we prove that the SDP relaxation can guarantee recovery under more adversarial conditions compared to earlier proposed spectral relaxations, and derive error bounds for the registration error incurred by the SDP relaxation. We also present results of 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 least-squares 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.
We propose a novel algorithm to register multiple 3D point sets within a common reference frame using a manifold optimization approach. The point sets are obtained with multiple laser scanners or a mobile scanner. Unlike most prior algorithms, our approach performs an explicit optimization on the manifold of rotations, allowing us to formulate the registration problem as an unconstrained minimization on a constrained manifold. This approach exploits the Lie group structure of SO 3 and the simple representation of its associated Lie algebra so 3 in terms of R3. Our contributions are threefold. We present a new analytic method based on singular value decompositions that yields a closed-form solution for simultaneous multiview registration in the noise-free scenario. Secondly, we use this method to derive a good initial estimate of a solution in the noise-free case. This initialization step may be of use in any general iterative scheme. Finally, we present an iterative scheme based on Newton's method on SO 3 that has locally quadratic convergence. We demonstrate the efficacy of our scheme on scan data taken both from the Digital Michelangelo project and from scans extracted from models, and compare it to some of the other well known schemes for multiview registration. In all cases, our algorithm converges much faster than the other approaches, (in some cases orders of magnitude faster), and generates consistently higher quality registrations.