Abstract
The perspective three-point pose problem involves solving Grunert’s system of quadratic equations for the distances from the center of perspective to the three control points, typically resulting in multiple mathematical solutions. Relationships between the corresponding possible camera positions in space have only rarely been studied. Several efforts have been made though to understand the number of solution points using various assumptions. In this article, the number of solutions is determined in the limiting case, where the center of perspective is far from the plane containing the control points, as compared with its distance to the danger cylinder. Moreover, concise formulas are given for the other solutions based on a knowledge of one of the solutions. It turns out that the projection onto the control points plane of the various solution points lies at the intersection points of two rectangular hyperbolas. A certain deltoid curve also plays a crucial role.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Based on the author’s animated GIF image, which was inspired by Zachary Abel’s animated GIF images at http://blog.zacharyabel.com/2012/04/three-cornered-deltoids/.
References
Faugère, J.-C., Moroz, G., Rouillier, F., El-Din, M.S.: Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities. In: ISSAC’08, 21st International Symposium Symbolic and Algebraic Computation, pp. 79–86 (2008)
Gao, X.-S., Hou, X.-R., Tang, J., Cheng, H.-F.: Complete solution classification for the perspective-three-point problem. IEEE Trans. Pattern Anal. Mac. Intell. 25(8), 930–943 (2003)
Grunert, J.A.: Das pothenotische problem in erweiterter gestalt nebst über seine anwendungen in der geodäsie. Grunerts Arch. für Math. Phys. 1, 238–248 (1841)
Haralick, R.M., Lee, C.-N., Ottenberg, K., Nölle, N.: Review and analysis of solutions of the three point perspective pose estimation problem. J. Comput. Vis. 13(3), 331–356 (1994)
Nistér, D.: A minimal solution to the generalized 3-point pose problem. J. Math. Imaging Vis. 27(1), 67–79 (2007)
Rieck, M.Q.: A fundamentally new view of the perspective three-point pose problem. J. Math. Imaging Vis. 48(3), 499–516 (2014)
Sun, F.-M., Wang, B.: The solution distribution analysis of the P3P problem. In: SMC ’10, International Conference Systems, Man and Cybernetics, pp. 2033–2036 (2010)
Tang, J., Chen, W., Wang, J.: A study on the P3P problem. In: ICIC ’08, 4th International Conference Intelligent Computing 5226, 422–429 (2008)
Wolfe, W.J., Mathis, D., Sklair, C.W., Magee, M.: The perspective view of three points. IEEE Trans. Pattern Anal. Mach. Intell. 13(1), 66–73 (1991)
Xiaoshan, G., Hangfei, C.: New algorithms for the perspective-three-point problem. J. Comput. Sci. Tech. 16(3), 194–207 (2001)
Yang, L.: A simplified algorithm for solution classification of the perspective-three-point problem. In: em Tech. rep, MM-Preprints, MMRC, Academia Sinica (1998)
Zhang, C.-X., Hu, Z.-Y.: A general sufficient condition of four positive solutions of the P3P problem. J. Comput. Sci. Technol. 20(6), 836–842 (2005)
Zhang, C.-X., Hu, Z.-Y.: Why is the danger cylinder dangerous in the P3P problem? Acta Autom. Sin. 32(4), 504–511 (2006)
Zwikker, C.: The Advanced Geometry of Plane Curves and Their Applications. Dover, New York (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rieck, M.Q. Related Solutions to the Perspective Three-Point Pose Problem. J Math Imaging Vis 53, 225–232 (2015). https://doi.org/10.1007/s10851-015-0572-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-015-0572-1