We present novel, tight, convex relaxations for rotation and pose estimation problems that can guarantee global optimality via strong Lagrangian duality. 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. 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, including the identification of sufficient redundant constraints to make them tight. We hope our results can add to the catalogue of useful estimation problems whose global optimality can be guaranteed.

翻译：我们提出新颖、紧致的旋转与位姿估计问题凸松弛方法，通过强拉格朗日对偶性保证全局最优性。现有文献中针对特定问题设定存在此类松弛方法，这些设定假设各向同性旋转不确定性服从矩阵von Mises-Fisher分布（亦称矩阵Langevin分布或弦距离）。然而，旋转与位姿不确定性的另一种常见表示方式是在相关李代数中定义各向异性噪声。从基于Cayley映射的噪声模型出发，我们定义估计问题，将其转化为二次约束二次规划（QCQP），再松弛为可通过标准内点优化方法求解的半定规划（SDP）。我们首先展示如何执行基本旋转与位姿平均，继而转向更复杂的轨迹估计问题——该问题涉及多个位姿变量及个体测量与位姿间测量（或运动先验）。我们的贡献在于为所有这些问题构建SDP松弛，包括识别保证松弛紧致性的充分冗余约束。希望我们的成果能丰富可保证全局最优性的实用估计问题集。