Journal of Mathematical Imaging and Vision

, Volume 51, Issue 2, pp 326–337 | Cite as

Mathematical Analysis of the Multisolution Phenomenon in the P3P Problem

  • M. VynnyckyEmail author
  • K. Kanev


The perspective 3-point problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis and robotics. One line of activity involves formulating it mathematically in terms of finding the solution to a quartic equation. However, in general, the equation does not have a unique solution, and in some situations there are no solutions at all. Here, we present a new approach to the solution of the problem; this involves closer scrutiny of the coefficients of the polynomial, in order to understand how many solutions there will be for a given set of problem parameters. We find that, if the control points are equally spaced, there are four positive solutions to the problem at 25 % of all available spatial locations for the control-point combinations, and two positive solutions at the remaining 75 %.


P3P Quartic polynomial Multiple solutions 



M. Vynnycky acknowledges the support of the Mathematics Applications Consortium for Science and Industry, funded by the Science Foundation Ireland (SFI) Mathematics Initiative Grant 06/MI/005 and SFI Grant 12/IA/1683, as well as funding support from Research Institute of Electronics, Shizuoka University, Japan, for visiting and cooperative research. The authors also acknowledge useful discussions with Prof. J. A. Cuminato, and the comments of an anonymous referee.


  1. 1.
    Finsterwalder, S., Scheufele, W.: Das Rückwärtseinschneiden im Raum. Sebastian Finsterwalder zum 75. Geburtstage. Verlag Herbert Wichmann, Berlin (1937)Google Scholar
  2. 2.
    Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM 24, 381–396 (1981)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Gao, X.S., Hou, X.R., Tang, J., Cheng, H.F.: Complete solution classification for the perspective-threepoint problem. IEEE Trans. Pattern Anal. Mach. Intell. 25, 930–943 (2003)CrossRefGoogle Scholar
  4. 4.
    Grafarend, E.W., Lohse, P., Schaffrin, B.: Dreidimensionaler rückwärtsschnitt, Teil I: Die projektiven Gleichungen, pp. 1–37. Zeitschrift für Vermessungswesen, Geodätisches Institut, Universität, Stuttgart (1989)Google Scholar
  5. 5.
    Grunert, J.A.: Das Pothenotische Problem in erweiterter Gestalt nebst über seine Anwendungen in der Geodäsie. Grunerts Archiv für Mathematik und Physik 1, 238–248 (1841)Google Scholar
  6. 6.
    Haralick, R.M., Lee, C., Ottenberg, K., Nolle, M.: Review and analysis of solutions of the 3-point perspective pose estimation problem. Int. J. Comput. Vis. 13, 331–356 (1994)CrossRefGoogle Scholar
  7. 7.
    Lazard, D.: Quantifier elimination: optimal solution for two classical examples. J. Symb. Comput. 5, 261–266 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Linnainmaa, S., Harwood, D., Davis, L.S.: Pose determination of a three-dimensional object using triangle pairs. IEEE Trans. Pattern Anal. Mach. Intell. 10, 634–647 (1988)CrossRefGoogle Scholar
  9. 9.
    Merritt, E.L.: Explicit three-point resection in space. Photogr. Eng. XV, 649–655 (1949)Google Scholar
  10. 10.
    Quan, L., Lan, Z.: Linear N-point camera pose determination. IEEE Trans. Pattern Anal. Mach. Intell. 21, 774–780 (1999)CrossRefGoogle Scholar
  11. 11.
    Rees, E.L.: Graphical discussion of the roots of a quartic equation. Am. Math. Mon. 29, 51–55 (1922)CrossRefzbMATHGoogle Scholar
  12. 12.
    Rieck, M.Q.: Handling repeated solutions to the perspective three-point pose problem. In: Richard, P., Braz, J. (eds.) VISAPP (1), pp. 395–399. INSTICC Press, Madeira (2010)Google Scholar
  13. 13.
    Rieck, M.Q.: An algorithm for finding repeated solutions to the general perspective three-point pose problem. J. Math. Imaging Vis. 42, 92–100 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Rieck, M.Q.: A fundamentally new view of the perspective three-point pose problem. J. Math. Imaging Vis. 48, 499–516 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Su, C., Xu, Y., Li, H., Liu, S.: Application of Wu’s method in computer animation. Proc. Fifth Int. Conf. CAD/CG 1, 211–215 (1997)Google Scholar
  16. 16.
    Su, C., Xu, Y., Li, H., Liu, S.: Necessary and sufficient condition of positive root mumber of P3P problem. Chin. J. Comput. Sci. 21, 1084–1095 (1998)Google Scholar
  17. 17.
    Wolfe, W.J., Mathis, D., Sklair, C.W., Magee, M.: The perspective view of three points. IEEE Trans. Pattern Anal. Mach. Intell. 13, 66–73 (1991)Google Scholar
  18. 18.
    Yang, L.: A simplified algorithm for solution classification of the perspective-three-point problem. Tech. rep, MM-Preprints, MMRC, Academia Sinica (1998)Google Scholar
  19. 19.
    Yuan, J.S.C.: A general photogrammetric method for determining object position and orientation. IEEE Trans. Robot. Autom. 5, 129–142 (1989)CrossRefGoogle Scholar
  20. 20.
    Zhang, C.X., Hu, Z.Y.: A general sufficient condition of four positive solutions of the P3P problem. J. Comput. Sci. Technol. 20, 836–843 (2005)CrossRefGoogle Scholar
  21. 21.
    Zhang, C.X., Hu, Z.Y.: Why is the danger cylinder dangerous in the P3P problem? Acta Autom. Sin. 32, 504–511 (2006)Google Scholar
  22. 22.
    Zhang, C.X., Hu, Z.Y.: A probabilistic study on the multiple solutions of the P3P problem. J. Softw. 18, 2100–2104 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematics Applications Consortium for Science and Industry (MACSI), Department of Mathematics and StatisticsUniversity of LimerickLimerickRepublic of Ireland
  2. 2.Division of Casting of Metals, Department of Materials Science and EngineeringRoyal Institute of TechnologyStockholmSweden
  3. 3.Research Institute of ElectronicsShizuoka UniversityHamamatsuJapan

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