求解位姿估计问题的对偶方法

A dual method for the pose estimation problem

位姿估计是计算机图形学、机器视觉、摄影测量学等研究领域中的核心问题之一,利用给定的3D-2D参考点来估计相机与对象间的旋转和平移.针对该问题的四元数模型,人们最近开发应用半定规划松弛(SDR)和平方和松弛(SOS)得到了很好的计算效果.在原始模型的基础上,通过添加冗余约束,提出了Lagrangian对偶松弛方法(Dual).这三种方法的核心是各自求解一个常数维度的半定规划问题,调用SeDuMi求解的系数矩阵规模分别为SDR:117×32,SOS:266×70和Dual:81×12,大量的数值实验表明Lagrangian对偶松弛方法在进一步缩短了计算时间的同时计算效果也十分卓越.

The pose estimation problem is one of the key problems in computer graphics, machine vision, and photogrammetry. It is to estimate the rotation and translation between a camera and an object based on given 3D-to-2D reference points. Recently, with the help of a quaternion model, semidefinite programming relaxation （SDR） and sum-of-square relaxation （SOS） are proposed in literature. In this paper, by adding redundant constraints to the original problem, we develop a Lagrangian dual model for pose estimation, which can be reformulated as a semidefinite program. We use SeDuMi to solve these three models. They are captured in matrices of 117 × 32 （SDR）, 266 × 70 （SOS） and 81 × 12 （Dual）, respectively. Numerical results show that our method is not only fast but also very efficient.