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运筹学学报 ›› 2013, Vol. 17 ›› Issue (3): 86-92.

• 运筹学 • 上一篇    下一篇

求解位姿估计问题的对偶方法

韩颖薇1, 夏勇1,*   

  1. 1. 软件开发环境国家重点实验室，“数学、信息与行为”教育部重点实验室，北京航空航天大学数学与系统科学学院，北京 100191
  • 出版日期:2013-09-15 发布日期:2013-09-15
  • 通讯作者: 夏勇 E-mail:dearyxia@gmail.com
  • 基金资助:

    国家自然科学基金项目(Nos. 11001006, 91130019/A011702), 软件开发环境国家重点实验室开放课题(No. SKLSDE-2013ZX-13)

A dual method for the pose estimation problem

HAN Yingwei1, XIA Yong1,*   

  1. 1. State Key Laboratory of Software Development Environment, Key Laboratory of Mathematics, Informatics and Behavioral Semantics of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China
  • Online:2013-09-15 Published:2013-09-15

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摘要： 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\times32, SOS: 266\times70和Dual: 81\times 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.

关键词: 位姿问题, Lagrangian对偶, 半定规划

Abstract: 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\times32 (SDR), 266\times70 (SOS) and  81\times 12 (Dual), respectively. Numerical results show that our method is not only fast but also very efficient.

Key words: pose estimation, Lagrangian dual, semidefinite programming

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