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New algorithms for the perspective-three-point problem. (English) Zbl 0983.65020

Two approaches are used to solve the perspective-three-point problem (P3P): the symbolic computation approach and the geometric approach. In the symbolic case it is used a zero decomposition algorithm to find a complete solution decomposition for the equation system derived from the P3P. The decomposition provides an analytical solution to the problem. In the geometrical case, there are given some new pure geometrical criteria for the number of solutions of the P3P problem. Finally, certain special cases of the P3P are solved completely.

MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65H10 Numerical computation of solutions to systems of equations
68W30 Symbolic computation and algebraic computation
68W05 Nonnumerical algorithms

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[1] Abidi M A, Chandra T. A new efficient and direct solution for pose estimation using quadrangular targets: Algorithm and evaluation.IEEE Transaction on Pattern Analysis and Machine Intelligence, May 1995, 17(5): 543–538. · Zbl 05111447 · doi:10.1109/34.391388
[2] Fischler M A, Bolles R C. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartomated cartography.Communications of the ACM, June 1981, 24(6): 381–395. · doi:10.1145/358669.358692
[3] Horaud R, Conio B, Leboulleux O. An analytic solution for the perspective 4-point problem.Computer Vision, Graphics, and Image Processing, 1989, 47: 33–44. · doi:10.1016/0734-189X(89)90052-2
[4] Su C, Xu Y, Li H, Liu S. Wu’s methods application in computer animation. InThe Fifth Int. Conf. CAD/CG, China, Vol.1, 1997, pp.211–215.
[5] Yuan J S C. A general photogrammetric method for determining object position and orientation.IEEE Transactions on Robotics and Automation, April 1989, 5(2): 129–142. · doi:10.1109/70.88034
[6] Wolfe W J, Mathis D, Weber C, Magee M. The perspective view of three points.IEEE Transactions on Pattern Analysis and Machine Intelligence, Jan. 1991, 13(1): 66–73. · Zbl 05110473 · doi:10.1109/34.67632
[7] Hung Y, Yeh P, Harwood D. Passive ranging to known planar points sets. InProc. IEEE Int. Conf. Rob. and Auto., St. Louis, Vol.1, 1985, pp.80–85.
[8] Haralick R M, Lee C, Ottenberg K, Nolle M. Analysis and solutions of the three-point perspective pose estimation problem. InProc of the Int. Conf. Computer Vision and Pattern Recognition, 1991.
[9] DeMenthon D, Davis L S. Exact and approximate solutions of the perspective-three-point.IEEE Transaction on Pattern Analysis and Machine Intelligence, Nov. 1992, 14(11): 1100–1105. · Zbl 05112718 · doi:10.1109/34.166625
[10] Wolfe W J, Jones K. Camera calibration using the perspective view of a triangle. InProc. SPIE Conf. Auto. Inspection Measurement, Vol.730, Cambridge, 1986.
[11] Su C, Xu C Y, Li H, Liu S. Necessary and sufficient condition of positive root number of P3P problem.Chinese Journal of Computers, Dec. 1998, 21(12): 1084–1095. (in Chinese)
[12] Yang L. A simplified algorithm for solution classification of the perspective-three-point problem.MM-Preprints, 1998, 17: 135–145.
[13] Wu W T. Basic Principles of Mechanical Theorem Proving in Geometries, Volume I: Part of Elementary Geometrics. Beijing: Science Press, 1984 (in Chinese), English Version, Springer-Verlag, 1995.
[14] Gao X S, Cheng H F. On the solution classification of the ”P3P” problem. InProceedings of the Third Asian Symposium on Computer Mathematics, Li Z B (ed), LanZhou University Press, 1998, pp.185–200.
[15] Collins G E. Quantifier elimination for real closed fields by cylindrical algebraic decomposition.LNCS, Vol. 33, Springer-Verlag, 1975, pp.134–183. · Zbl 0318.02051
[16] Mishra B. Algorithmic Algebra. New York: Springer-Verlag, 1993. · Zbl 0804.13009
[17] Buchberger B, Collins G E, Loos R. Computer Algebra Symbolic and algebraic Computation. Berlin: Springer-Verlag, 1988. · Zbl 0491.00019
[18] Gao X S, Chou S C. On the theory of resolvents and its applications.Sys. Sci. and Math. Sci., 1999, 12(Suppl.): 170. · Zbl 1073.13505
[19] Arnon D S. Geometric reasoning with logic and algebra.Artificial Intelligence, 1988, 37(1–3): 37–60. · Zbl 0705.68086 · doi:10.1016/0004-3702(88)90049-5
[20] Gao X S, Zhang J Z, Chou S C. Geometry Expert. Taipai: Nine Chapter Pub., Taiwan, 1998 (in Chinese).
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