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Solving the perspective-three-point problem using comprehensive Gröbner systems

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Abstract

A complete solution classification of the perspective-three-point (P3P) problem is given by using the Gröbner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be obtained by computing a comprehensive Gröbner system. Combining with properties of the generalized discriminant sequences, the authors give the explicit conditions to determine the number of distinct real positive solutions of the P3P problem. Several examples are provided to illustrate the effectiveness of the proposed conditions.

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References

  1. Fischler M A and Bolles R C, Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography, Communications of the ACM, 1981, 24(6): 381–395.

    Article  MathSciNet  Google Scholar 

  2. Ganapathy S, Decomposition of transformation matrices for robot vision, Pattern Recognition Letters, 1984, 2(6): 401–412.

    Article  Google Scholar 

  3. Hu Z Y and Wu F C, A note on the number of solutions of the noncoplanar P4P problem, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24(4): 550–555.

    Article  MathSciNet  Google Scholar 

  4. Wu Y H and Hu Z Y, PnP problem revisited, Journal of Mathematical Imaging and Vision, 2006, 24(1): 131–141.

    Article  MathSciNet  Google Scholar 

  5. Horaud R, Conio B, Leboulleux O, et al., An analytic solution for the perspective 4-point problem, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE, 1989, 500–507.

    Google Scholar 

  6. Moreno-Noguer F, Lepetit V, and Fua P, Accurate non-iterative O(n) solution to the PnP problem, IEEE 11th International Conference on Computer Vision, 2007, 1–8.

    Google Scholar 

  7. Schweighofer G and Pinz A, Globally optimal O(n) solution to the PnP problem for general camera models, BMVC, 2008, 1–10.

    Google Scholar 

  8. Tang J, Chen W S, and Wang J, A novel linear algorithm for P5P problem, Applied Mathematics and Computation, 2008, 205(2): 628–634.

    Article  MathSciNet  MATH  Google Scholar 

  9. Lepetit V, Moreno-Noguer F, and Fua P, EPnP: An accurate O(n) solution to the PnP problem, International journal of computer vision, 2009, 81(2): 155–166.

    Article  Google Scholar 

  10. Kneip L, Scaramuzza D, and Siegwart R, A novel parametrization of the perspective-three-point problem for a direct computation of absolute camera position and orientation, IEEE Conference on Computer Vision and Pattern Recognition, 2011, 2969–2976.

    Google Scholar 

  11. Haralick R M, Lee D, Ottenburg K, and Nolle M, Analysis and solutions of the three point perspective pose estimation problem, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1991, 592–598.

    Google Scholar 

  12. Yang L, A simplified algorithm for solution classification of the perspective-three-point problem, MM Research Preprints, 1998, 17: 17–135.

    Google Scholar 

  13. Gao X S, Hou X R, Tang J L, and Cheng H F, Complete solution classification for the perspectivethree-point problem, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2003, 28(8): 930–943.

    Google Scholar 

  14. Reid G, Tang J L, and Zhi L H, A complete symbolic-numeric linear method for camera pose determination, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ACM Press, 2003, 215–223.

    Google Scholar 

  15. Faugère J C, Moroz G, Rouillier F, and El Din M S, Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities, Proceedings of the 2008 International Symposium on Symbolic and Algebraic Computation, ACM Press, 2008, 79–86.

    Google Scholar 

  16. Yang L, Recent advances on determining the number of real roots of parametric polynomials, Journal of Symbolic Computation, 1999, 28(1): 225–242.

    Article  MathSciNet  MATH  Google Scholar 

  17. Weispfenning V, Comprehensive Gröbner bases, Journal of Symbolic Computation, 1992, 14(1): 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang L, Zhang J Z, and Hou X R, Nonlinear Algebraic Equation System and Automated Theorem Proving, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1996 (in Chinease).

    Google Scholar 

  19. Sylvester J, On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure, Philosophical Transactions of the Royal Society of London, 1853, 407–548.

    Google Scholar 

  20. Kapur D, Sun Y, and Wang D K, A new algorithm for computing comprehensive Gröbner systems, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM Press, 2010, 29–36.

    Chapter  Google Scholar 

  21. Yang L and Xia B C, Real solution classification for parametric semi-algebraic systems, Proceeding of the A3L, 2005, 281–289.

    Google Scholar 

  22. Cox D A, Little J B, and O’shea D, Using Algebraic Geometry, Second Edition, Springer, New York, 1998.

    Book  MATH  Google Scholar 

  23. Suzuki A and Sato Y, An alternative approach to comprehensive Gröbner bases, Journal of Symbolic Computation, 2003, 36(3): 649–667.

    Article  MathSciNet  MATH  Google Scholar 

  24. Montes A, A new algorithm for discussing Gröbner bases with parameters, Journal of Symbolic Computation, 2002, 33(2): 183–208.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Jie Zhou & Dingkang Wang

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  1. Jie Zhou
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  2. Dingkang Wang
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Correspondence to Jie Zhou.

Additional information

This research was supported by the National Nature Science Foundation of China under Grant Nos. 11371356 and 61121062.

This paper was recommended for publication by Editor LI Hongbo.

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Zhou, J., Wang, D. Solving the perspective-three-point problem using comprehensive Gröbner systems. J Syst Sci Complex 29, 1446–1471 (2016). https://doi.org/10.1007/s11424-015-4310-y

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  • DOI: https://doi.org/10.1007/s11424-015-4310-y

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