Abstract: The noncoplanar P4P problem with an uncalibrated two-parameter pinhole camera is systematically originated in this paper from the mathematical point of view.It is proved algebraically that if the 4 space control points are noncoplanar，the P4P problem can have the following 2 possible cases：It has at most 8 real solutions，and this upper bound is also attainable;It has an infinite number of solutions.In addition，algebraic conditions are provided for the above two cases.Some practical algorithms are also presented to compute admissible solutions.The obtained results are of practical importance in theory and applications such as object pose estimation，hand-eye coordination and robot navigation.

Key words: camera calibration, noncoplanar P4P problem, planar perspective transformation, elimination by resultant

摘要： 针对二参数模型的针孔摄像机，从数学上比较系统地研究了未标定情况下的非共面P4P解数问题。主要结论是当空间4个控制点不共面时，未标定的P4P问题的解仅有两种可能：至多有8个实解，而且解的上限可以达到;有无穷多解。给出了至多有8个实解和有无穷多解的代数条件以及求解未标定非共面P4P问题的具体算法。所得结果在物体定位、手眼定标、机器人导航等领域具有比较重要的理论及应用价值。

关键词: 摄像机标定, 非共面P4P问题, 平面透视变换, 结式消去法