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On the Probability of the Number of Solutions for the P4P Problem

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Abstract

This paper studies the multi-solution phenomenon for the perspective four point (P4P) problem from geometric and algebraic aspects. We give a pure geometric proof that the P4P problem could have up to five solutions. We also give a clear picture on how these five solutions could be realized. We prove that with probability one, the P4P problem has a unique solution which can be represented by a set of rational functions in the parameters. The simulant experiments show that to solve the P4P problem with the rational functions is stable and accurate.

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Institute of Systems Science, Academia Sinica, Beijing, 100080, China

    Xiao-Shan Gao

  2. College of Science, Shenzhen University, Shenzhen, 518060, China

    Jianliang Tang

Authors
  1. Xiao-Shan Gao
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  2. Jianliang Tang
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Xiao-Shan Gao is a professor in the Institute of Systems Science, Chinese Academy of Sciences. His research interests include: automated reasoning, symbolic computation, intelligent CAD, CAGD, and robotics. He was published over one hundred research papers, two monographs and edited four books or conference proceedings. Webpage: http://www.mmrc.iss.ac.cn/~xgao.

Jianliang Tang received his PhD degree from Chinese Academy of Sciences in 2004. Currently, he is an assistant professor in SheZhen University. His research interests include camera calibration and symbolic computation.

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Gao, XS., Tang, J. On the Probability of the Number of Solutions for the P4P Problem. J Math Imaging Vis 25, 79–86 (2006). https://doi.org/10.1007/s10851-006-5149-6

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  • DOI: https://doi.org/10.1007/s10851-006-5149-6

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