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.

翻译：本文针对旋转与姿态估计问题提出了新颖的凸松弛方法，能够在实际测量噪声水平下后验保证全局最优性。现有文献中针对特定问题设置存在此类松弛方法，通常假设各向同性旋转不确定性服从矩阵冯·米塞斯-费希尔分布（亦称矩阵朗之万分布或弦距离）。然而，另一种表示旋转与姿态不确定性的常用方法是在相关李代数中定义各向异性噪声。基于凯莱映射的噪声模型，我们定义了估计问题，将其转化为二次约束二次规划问题，进而松弛为半定规划问题——该问题可采用标准内点优化方法求解，其全局最优性源自拉格朗日强对偶性。我们首先展示了基础旋转与姿态平均的实现方法，进而转向更复杂的轨迹估计问题，该问题涉及多个姿态变量及个体与姿态间测量（或运动先验）。本文的核心贡献在于：基于凯莱映射为所有这些问题构建了半定规划松弛模型（包括冗余约束的识别），并验证了其在实践场景中的有效性。我们希望本研究能够扩充可后验保证全局最优解的有效估计问题类型库。