Abstract
Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We introduce a new polynomial equation system for 4-point pose estimation and apply our symbolic-numeric method to solve it stably and efficiently. In particular, our algorithm can also recognize the points near critical configurations and deal with these near critical cases carefully. Numerical experiments are given to show the performance of the hybrid algorithm.
Supported by NKBRPC 2004CB318000 and Chinese National Science Foundation under Grant 10401035 and Reids Canadian NSERC Grant.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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 17(5), 534–538 (1995)
Ameller, M.A., Triggs, B., Quan, L.: Camera Pose Revisited - New Linear Algorithms. Private Communication
Bonasia, J., Reid, G.J., Zhi, L.H.: Determination of approximate symmetries of differential equations. In: Gomez-Ullate, Winternitz (eds.) CRM Proceedings and Lecture Notes; Amer. Math. Soc. 39, 233–249 (2004)
Buchberger, B.: An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal, PhD. Thesis, Univ. of Innsbruck Math. Inst (1965)
Faugére, J.C.: A New Efficient Algorithm for Computing Gröbner Bases without Reduction to Zero(F5). In: Mora, T. (ed.) Proc. ISSAC, New York, pp. 75–83. ACM Press, New York (2002)
Gao, X.S., Hou, X.R., Tang, J.L., Cheng, H.: Complete Solution Classification for the Perspective-Three-Point Problem. IEEE Tran. on Pattern Analysis and Machine Intelligence 25(8), 534–538 (2003)
Gao, X.-S., Tang, J.L.: On the Solution Number of Solutions for the P4P Problem. Mathematics-Mechanization Research Center Preprints Preprint 21, 64–76 (2002)
Horaud, R., Conio, B., Leboulleux, O.: An Analytic Solution for the Perspective 4-Point Problem. CVGIP 47, 33–44 (1989)
Horn, B.K.P.: Closed Form Solution of Absolute Orientation Using Unit Quaternions. Journal of the Optical Society of America 5(7), 1127–1135 (1987)
Kuranishi, M.: On E. Cartan’s Prolongation Theorem of Exterior Differential Systems. Amer. J. Math 79, 1–47 (1957)
Hu, Z.Y., Wu, F.C.: A Note on the Number Solution of the Non-coplanar P4P Problem. IEEE Transaction on Pattern Analysis and Machine Intelligence 24(4), 550–555 (2002)
Lazard, D.: Gaussian Elimination and Resolution of Systems of Algebraic Equations. In: Proc. EUROCAL 1983, pp. 146–157 (1993)
Macaulay, F.S.: The Algebraic Theory of Modular Systems. In: Cambridge tracts in Math. and Math. Physics, vol. 19. Cambridge Univ. Press, Cambridge (1916)
Mourrain, B.: Computing the Isolated Roots by Matrix Methods. J. Symb. Comput. 26, 715–738 (1998)
Mourrain, B., Trébuchet, P.: Solving Projective Complete Intersection Faster. In: Traverso, C. (ed.) Proc. ISSAC, pp. 430–443. ACM Press, New York (2000)
Möller, H.M., Sauer, T.: H-bases for polynomial interpolation and system solving. Advances Comput. Math. (to appear)
Mourrain, B.: A New Criterion for Normal Form Algorithms. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 430–443. Springer, Heidelberg (1999)
P. Trébuchet, Vers une Résolution Stable et Rapide des Équations Algébriques. Ph.D. Thesis, Université Pierre et Marie Curie (2002)
Pommaret, J.F.: Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach / Science Publishers (1978)
Quan, L., Lan, Z.: Linear N-Point Camera Pose Determination. IEEE Transaction on PAMI 21(8), 774–780 (1999)
Rives, P., Bouthémy, P., Prasada, B., Dubois, E.: Recovering the Orientation and the Position of a Rigid Body in Space from a Single View, Technical Report, INRS-Télécommunications, Quebec, Canada (1981)
Reid, G.J., Lin, P., Wittkopf, A.D.: Differential elimination-completion algorithms for DAE and PDAE. Studies in Applied Mathematics 106(1), 1–45 (2001)
Reid, G.J., Tang, J., Zhi, L.: A complete symbolic-numeric linear method for camera pose determination. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, Scotland, pp. 215–223. ACM Press, New York (2003)
Reid, G.J., Smith, C., Verschelde, J.: Geometric Completion of Differential Systems using Numeric-Symbolic Continuation. SIGSAM Bulletin 36(2), 1–17 (2002)
Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004)
Auzinger, W., Stetter, H.: An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations. In: Numerical Mathematics Proceedings of the International Conference, Singapore, pp. 11–30 (1988)
Seiler, W.M.: Analysis and Application of the Formal Theory of Partial Differential Equations, PhD. Thesis, Lancaster University (1994)
Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6), 2022–2046 (2001)
Wang, D.: Characteristic Sets and Zero Structures of Polynomial Sets, Preprint RISC-LINZ (1989)
Wittkopf, A.D., Reid, G.J.: Fast Differential Elimination in C: The CDiffElim Environment. Comp. Phys. Comm. 139(2), 192–217 (2001)
Wu, W.T.: Basic principles of mechanical theorem proving in geometries Volume I: Part of Elementary Geometries. Science Press, Beijing (1984) (in Chinese) (English Version Springer-Verlag 1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Reid, G., Tang, J., Yu, J., Zhi, L. (2005). Hybrid Method for Solving New Pose Estimation Equation System. In: Li, H., Olver, P.J., Sommer, G. (eds) Computer Algebra and Geometric Algebra with Applications. IWMM GIAE 2004 2004. Lecture Notes in Computer Science, vol 3519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11499251_5
Download citation
DOI: https://doi.org/10.1007/11499251_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26296-1
Online ISBN: 978-3-540-32119-4
eBook Packages: Computer ScienceComputer Science (R0)