透视三点问题的一种快速且稳定的代数解法

A Fast and Stable Algebraic Solution to Perspective-Three-Point Problem

对于经典的透视三点(P3P)问题,当三维控制点的Z轴坐标在较大范围内随机分布时,仍然存在数值稳定性差、图像噪声增加导致的退化、计算效率低的问题。为此,文中提出了一种快速且稳定的代数求解方法。首先,在根据3个三维到二维的对应点对估计已校准摄像机的旋转角度和相对位置时,在世界坐标系与相机坐标系之间引入中间坐标系,以减少未知参数的数量,并对旋转矩阵进行归一化,以简化计算过程,提高计算效率;然后,选取两个控制点的中心作为中间坐标系的坐标原点,以提高P3P问题在退化配置中的抗噪性能;最后,通过使用Gröbner基将P3P问题转化为求解只有一个未知参数的四次方程,求出P3P问题的封闭解。实验结果表明,文中算法的数值稳定性以及在退化配置中的抗噪性能,均优于其他3种P3P问题的经典算法。

For the classic perspective-three-point(P3P)problem,when the Z-axis coordinates of the three-dimensional control points are randomly distributed in a large range,there are still problems of poor numerical stability,degradation caused by increased image noise,and low computational efficiency.Therefore,a fast and stable algebraic solution method was proposed in this article.Firstly,when the proposed solution directly estimates the rotation and position of a calibrated camera from three 3D to 2D point correspondences,an intermediate coordinate frame is introduced between the world coordinate frame and the camera coordinate frame to reduce the number of unknown parameters,and the rotation matrix is normalized to simplify the calculation process and improve the calculation efficiency.Secondly,the midpoint between the two control points was chosen as the origin point of the intermediate coordinate frame,so as to improve the anti-noise performance of the P3P problem in degenerate configurations.Finally,the P3P problem was transformed into a biquadratic equation with one unknown parameter by using a Gröbner basis,then a closed solution to the P3P problem was obtained.Experimental results show that the proposed algorithm can achieve better numerical stability and anti-noise performance compared with other three classic algorithm of the P3P problem.