Geometric Interpretation of the Multi-solution Phenomenon in the P3P Problem
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Geometric Interpretation of the Multi-solution Phenomenon in the P3P Problem Bo Wang1 · Hao Hu2 · Caixia Zhang3 Received: 23 December 2019 / Accepted: 4 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its three control points and the position of the optical center. Since in any real applications, the knowledge on the exact number of possible solutions is a prerequisite for selecting the right one among all the possible solutions, and the study on the phenomenon of multiple solutions in the P3P problem has been an active topic since its very inception. In this work, we provide some new geometric interpretations on the multi-solution phenomenon in the P3P problem, and our main results include: (1) the necessary and sufficient condition for the P3P problem to have a pair of side-sharing solutions is the two optical centers of the solutions both lie on one of the three vertical planes to the base plane of control points; (2) the necessary and sufficient condition for the P3P problem to have a pair of point-sharing solutions is the two optical centers of the solutions both lie on one of the three so-called skewed danger cylinders;(3) if the P3P problem has other solutions in addition to a pair of side-sharing (point-sharing) solutions, these remaining solutions must be a point-sharing (side-sharing ) pair. In a sense, the side-sharing pair and the point-sharing pair are companion pairs; (4) there indeed exist such P3P problems that have four completely distinct solutions, i.e., the solutions sharing neither a side nor a point, closing a long guessing issue in the literature. In sum, our results provide some new insights into the nature of the multi-solution phenomenon in the P3P problem, and in addition to their academic value, they could also be used as some theoretical guidance for practitioners in real applications to avoid occurrence of multiple solutions by properly arranging the control points. Keywords P3P problem · Multiple solutions · Danger cylinder
1 Introduction and Related Works The Perspective-3-Point problem, or P3P problem, was first introduced by Grunert [1] in 1841 and popularized in computer vision community a century later by mainly the Fishler
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Caixia Zhang [email protected] Bo Wang [email protected] Hao Hu [email protected]
1
Institute of Automation, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
2
Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angels, CA, USA
3
Science College, North China University of Technology, Beijing 100041, People’s Republic of China
and Bolles’ work in 1981 [2]. Since it is the least number of points to have a finite number of solutions, it has been widely used in various fields ([3–12]), either for its minimal demand in restricted working environment, such as robotics and aeronautics, or for its computational efficiency of robust pose
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