透视三点问题贝叶斯解法(BP3P)及其推广

透视三点问题（Perspective-Three-Point, P3P）是计算机视觉与摄影测量学领域的经典问题，在目标定位、视觉测量、虚拟现实及目标姿态计算等方面有重要的应用。提出基于贝叶斯的P3P问题新解法（BP3P）。从三控制点所确定的支撑平面出发，证明确定支撑平面是求解P3P问题的充要条件，并提出基于贝叶斯的支撑平面计算方法。利用从三控制点计算出的角度、长度比例等几何约束，通过归一化高斯函数来对其似然概率进行建模，并在高斯半球面上搜索最大似然概率求解支撑平面法向量。最后通过任意两控制点实际距离为参考计算平面的距离，确定支撑平面。对BP3P算法进行推广，能处理一般性平面几何约束，实现视觉定位。利用三组真实图像实验对算法进行验证。实验一的结果表明该算法不仅能解决P3P问题，还能对其多解现象进行分析。在实验二与实验三，算法利用一般性几何约束成功实现平面目标准确定位。

Bayesian Perspective-Three-Point (BP3P) and Its Extensions

The perspective-three-point (P3P) is a classic problem in both computer vision and photogrammetry fields, which has important applications in object localization, metrology, virtual reality, and pose estimation, etc. A novel algorithm is proposed, Bayesian P3P (BP3P), to solve the P3P problem. The determination of the support plane, which is uniquely defined by the three control points, is proven to be the necessary and sufficient condition to the P3P problem. A Bayesian approach to compute the support plane is given. Computation of the plane normal is formulated into a maximum likelihood problem by utilizing the geometric constraints of known angles and length ratios from the three control points. The likelihood for each constraint is modeled with normalized Gaussian function and the maximum joint likelihood is searched on Gaussian hemisphere to solve the plane normal. The plane distance is thus calculated readily from the actual distance between two arbitrary control points. Furthermore, the proposed BP3P algorithm can be extended to deal with more generalized planar constraints for localization rather than three control points. The proposed algorithm was validated with two real image experiments. In the first experiment, the algorithm was successfully applied to solve P3P problems. The multiple solution phenomenon of P3P was also illustrated and studied. In the second and third experiments, the algorithm was applied to localize planar object from generalized constraints with good results reported.