Closed-Form Solution to the Noncoplanar P4P Problem for Uncalibrated Camera
This paper presents a closed-form solution to determination of the position and orientation of a perspective camera with two unknown effective focal lengths for the noncoplanar perspective four point (P4P) problem. Given four noncoplanar 3D points and their correspondences in image coordinate, we convert perspective transformation to affine transformation, and formulate the problem using invariance to 3D affine transformation and arrive to a closed-form solution. We show how the noncoplanar P4P problem is cast into the problem of solving an eighth degree polynomial equation in one unknown. This result shows the noncoplanar P4P problem with two unknown effective focal lengths has at most 8 solutions. Last, we confirm the conclusion by an example. Although developed as part of landmark-guided navigation, the solution might well be used for landmark-based tracking problem, hand-eye coordination, and for fast determination of interior and exterior camera parameters. Because our method is based on closed-form solution, its speed makes it a potential candidate for solving above problems.
Y. Guo, "Closed-Form Solution to the Noncoplanar P4P Problem for Uncalibrated Camera", Applied Mechanics and Materials, Vol. 145, pp. 6-10, 2012