Chinese Journal of Computers   Full Text
Title	An Analytic Solution for the P5P Problem with an Uncalibrated Camera
Authors	GUO Yang1) XU Xin-He2)
Address	1)(Department of Mathematics, Northeastern University, Shenyang 110004) 2)(Institute of Artificial Intelligence and Robotics, Northeastern University, Shenyang 110004)
Year	2007
Issue	No.7(1195—1200)
Abstract &   Background	Abstract Classical PnP problem is based on an important precondition that the intrinsic parameters of a camera are known. But it is more valuable in real applications to investigate the PnP problem with an uncalibrated camera. In this paper, an analytic solution for the P5P problem with an uncalibrated four-parameter pinhole camera is proposed. The position and orientation of a camera with respect to world coordinate frame and intrinsic parameter of the camera can be obtained together by solving this problem. According to projection equations of five control points and properties of rotation matrix, the analysis is performed first by writing a suitable sixteen equations in sixteen unknowns. Then, by a specifically-developed elimination scheme, the equation set that is consisted of the sixteen equations is reduced to a biquadratic polynomial equation with only one unknown. Last, we confirm the P5P problem with an uncalibrated four-parameter pinhole camera has at most 4 solutions, generally. A large number of synthetic experiments show that on the one hand the robustness of this method is very strong, on the other hand the results are very accurate for fast determination of extrinsic parameters. The algorithm might well be used for object pose estimation, hand-eye coordination and landmark-guided navigation. keywords P5P problem; camera pose; camera instrinsic paraineter; dialytic elimination method; analytic solution background The work in this paper belongs to the PnP problem of in machine vision basically. Classical PnP problem is based on an important precondition that the intrinsic parameters of a camera are known. The problem has been solved well. For example, P3P has at most 4 solutions, and this upper bound is also attainable; P4P has unique solution when 4 control points is coplanar, and has at most 4 solutions when 4 control points are not coplanar; P5P has at most 2 solutions and this upper bound is also attainable. But few investigate the PnP problem with an uncalibrated camera. Professor Wu Fu-Chao et al first propose this problem, and solve the P5P problem with an uncalibrated camera by using SVD decomposition. It is proved that if no 4 control points among the 5 control points are coplanar,or if no 3 image points are collinear in case the 4 control points are coplanar, the P5P problem can have the following two possible cases: (1) It has at most 4 solutions, and this upper bound is also attainable; (2) It has an infinite number of solutions. Because SVD decomposition is sensitive to numerical value, the authors propose analytic algorithm for solving the P5P problem with an uncalibrated four-parameter pinhole camera. The conclusion is the same as that of Professor Wu, but the authors’ method is more robust and accurate. The work in this paper is a part of vision-based navigation of autonomous mobile robot. The navigation is based on artificial landmarks, so it is very easy to recognize control points and obtain their world coordinates and image coordinates. So the algorithm in this paper can be used easily for camera calibration and camera localization in navigation. The authors adopt camera self-calibration once in navigation, and propose an analytic solution of a linear camera self-calibration based on Kruppa equations therefore, but on the one hand this self-calibration is time-consuming, on the other hand it is not accurate, especially, it is very sensitive to the accuracy of fundamental matrix and epipole, however, at presents it is very difficult to solve accurate fundamental matrix and epipole, so it is impossible in real applications. For zoom-changing camera in navigation, we need a fast, robust and accurate method of camera calibration and camera localization. The work in this paper can achieve these functions.