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PnP Problem Revisited

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Abstract

Perspective-n-Point camera pose determination, or the PnP problem, has attracted much attention in the literature. This paper gives a systematic investigation on the PnP problem from both geometric and algebraic standpoints, and has the following contributions: Firstly, we rigorously prove that the PnP problem under distance-based definition is equivalent to the PnP problem under orthogonal-transformation-based definition when n > 3, and equivalent to the PnP problem under rotation-transformation-based definition when n = 3. Secondly, we obtain the upper bounds of the number of solutions for the PnP problem under different definitions. In particular, we show that for any three non-collinear control points, we can always find out a location of optical center such that the P3P problem formed by these three control points and the optical center can have 4 solutions, its upper bound. Additionally a geometric way is provided to construct these 4 solutions. Thirdly, we introduce a depth-ratio based approach to represent the solutions of the whole PnP problem. This approach is shown to be advantageous over the traditional elimination techniques. Lastly, degenerated cases for coplanar or collinear control points are also discussed. Surprisingly enough, it is shown that if all the control points are collinear, the PnP problem under distance-based definition has a unique solution, but the PnP problem under transformation-based definition is only determined up to one free parameter.

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Authors and Affiliations

  1. National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing, 100080, People's Republic of China

    Yihong Wu & Zhanyi Hu

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  1. Yihong Wu
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  2. Zhanyi Hu
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Yihong Wu received her Bachelor of Science degree in Mathematics from Shanxi Yanbei Normal College in 1995; a Master of Science degree in Computational Algebra from Shaanxi Normal University in 1998; a Doctor of Science degree in Geometric Invariants and Applications from MMRC, Institute of Systems Science, Chinese Academy of Sciences, in 2001. From June 2001 to July 2003, she did her postdoctoral research in NLPR, Institute of Automation, Chinese Academy of Sciences. After then, she joined NLPR as an associate professor. Her research interests include polynomial elimination and applications, geometric invariant and applications, automated geometric theorem proving, camera calibration, camera pose determination, and 3D reconstruction. She has published more than 15 papers on major international journals and major international conferences.

Zhanyi Hu was born in Shanxi province, P. R. China in 1961. He received the B.S. Degree in Automation from the North China University of Technology in 1985, the Ph.D. Degree (Docteur d’Etat) in Computer Science from the University of Liege, Belgium, in Jan. 1993. Since 1993, he has been with the Institute of Automation, Chinese Academy of Sciences. From May 1997 to May 1998, he also acted as a visiting scholar of Chinese University of Hong Kong on invitation. Dr. Hu now is a Research Professor of Computer Vision, a member of the Executive Expert Committee of the Chinese National High Technology R&D Program, a deputy editor-in-chief for Chinese Journal of CAD and CG, and an associate editor for Journal of Computer Science and Technology. His current research interests include Camera Calibration, 3D Reconstruction, Feature Extraction, Vision Guided Robot Navigation etc. Dr. Hu has published more than 70 peer-reviewed papers on major national and international journals.

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Wu, Y., Hu, Z. PnP Problem Revisited. J Math Imaging Vis 24, 131–141 (2006). https://doi.org/10.1007/s10851-005-3617-z

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