变量投影框架下基于Wahba问题的多点透视问题求解算法

Solving Perspective-n-Point Problem in Variable Projection Framework Based on Wahba Problem

提出一种在变量投影框架下的多点透视问题求解算法。多点透视问题是确定相机旋转与平移参数的基础问题,广泛应用于计算机视觉、摄影测量、机器人、空间交会以及虚拟现实等场合。相机的旋转与平移参数在变量投影的框架下被分开估计。首先,将平移参数固定,根据Wahba问题的解法求解出最优旋转矩阵;然后,应用奇异值分解导数的计算公式计算当前最优旋转矩阵对平移参数的导数,并由此计算出代价函数对平移参数的导数;最后,针对平移参数,采用Levenberg-Marquardt算法迭代优化代价函数,直到精度满足要求。实验结果证明,本文算法不仅误差较小,计算效率较高,而且具有较大的收敛域。

We propose an algorithm that solves the perspective-n-point problem in the framework of variable projection. The perspective-n-point is the problem of determining the rotation and translation parameters of a camera, which is widely used in computer vision, photogrammetry, robotics, space rendezvous and virtual reality. We separately estimate the rotation and translation parameters in the variable projection framework. Firstly, an optimal rotation matrix is solved by using the Wahba problem with fixed translation parameters. By applying the formula for the derivative of the singular value decomposition, the derivative of the optimal rotation matrix is calculated with respect to the translation parameters, then the derivative of the objective function with respect to the translation parameters is obtained. Finally, the objective function is minimized by Levenberg-Marquardt algorithm over the translation parameters. The experimental results show that the proposed algorithm is one of the most accurate and effective algorithms and has a large convergence basin.