  We present novel, convex relaxations for rotation and pose estimation
problems that can a posteriori guarantee global optimality for practical
measurement noise levels. Some such relaxations exist in the literature for
specific problem setups that assume the matrix von Mises-Fisher distribution
(a.k.a., matrix Langevin distribution or chordal distance)for isotropic
rotational uncertainty. However, another common way to represent uncertainty
for rotations and poses is to define anisotropic noise in the associated Lie
algebra. Starting from a noise model based on the Cayley map, we define our
estimation problems, convert them to Quadratically Constrained Quadratic
Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be
solved using standard interior-point optimization methods; global optimality
follows from Lagrangian strong duality. We first show how to carry out basic
rotation and pose averaging. We then turn to the more complex problem of
trajectory estimation, which involves many pose variables with both individual
and inter-pose measurements (or motion priors). Our contribution is to
formulate SDP relaxations for all these problems based on the Cayley map
(including the identification of redundant constraints) and to show them
working in practical settings. We hope our results can add to the catalogue of
useful estimation problems whose solutions can be a posteriori guaranteed to be
globally optimal.
