A note on the number of solutions of the coplanar P4P problem

We present a novel constrained system for the coplanar P4P problem. By converting perspective transformation to affine transformation and using invariance to 3D affine transformation, we explore the relationship between the image of the absolute conic (IAC) and the world coordinate of camera optical center from four coplanar correspondences and show how the coplanar P4P problem is cast into the problems of solving the common intersection points of a system of five spheres. In particular, we claim that, if the four control points are coplanar, the upper bound of the coplanar P4P problem under distance-based definition or orthogonal-transformation-based definition is 2 and also attainable. From the point of view of efficient numerical solution, we also propose a solving technique based on singular value decomposition (SVD) in order to estimating the camera pose under the effects of image noise. Finally, The advantages of our method are demonstrated by testing on synthetic data.