  Consider the problem of registering multiple point sets in some
$d$-dimensional space using rotations and translations. Assume that there are
sets with common points, and moreover the pairwise correspondences are known
for such sets. We consider a least-squares formulation of this problem, where
the variables are the transforms associated with the point sets. The present
novelty is that we reduce this nonconvex problem to an optimization over the
positive semidefinite cone, where the objective is linear but the constraints
are nevertheless nonconvex. We propose to solve this using variable splitting
and the alternating directions method of multipliers (ADMM). Due to the
linearity of the objective and the structure of constraints, the ADMM
subproblems are given by projections with closed-form solutions. In particular,
for $m$ point sets, the dominant cost per iteration is the partial
eigendecomposition of an $md \times md$ matrix, and $m-1$ singular value
decompositions of $d \times d$ matrices. We empirically show that for
appropriate parameter settings, the proposed solver has a large convergence
basin and is stable under perturbations. As applications, we use our method for
$2$D shape matching and $3$D multiview registration. In either application, we
model the shapes/scans as point sets and determine the pairwise correspondences
using ICP. In particular, our algorithm compares favorably with existing
methods for multiview reconstruction in terms of timing and accuracy.
