A QCQP Approach to Triangulation

  • Chris Aholt
  • Sameer Agarwal
  • Rekha Thomas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7572)


Triangulation of a three-dimensional point from n ≥ 2 two-dimensional images can be formulated as a quadratically constrained quadratic program. We propose an algorithm to extract candidate solutions to this problem from its semidefinite programming relaxations. We then describe a sufficient condition and a polynomial time test for certifying when such a solution is optimal. This test has no false positives. Experiments indicate that false negatives are rare, and the algorithm has excellent performance in practice. We explain this phenomenon in terms of the geometry of the triangulation problem.


Singular Value Decomposition Camera Center Global Optimality Condition Epipolar Constraint Trifocal Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Opt. 19(1), 83–102 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hartley, R., Seo, Y.: Verifying global minima for l 2 minimization problems. In: CVPR (2008)Google Scholar
  3. 3.
    Hartley, R., Sturm, P.: Triangulation. CVIU 68(2), 146–157 (1997)Google Scholar
  4. 4.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press (2003)Google Scholar
  5. 5.
    Heyden, A., Åström, K.: Algebraic properties of multilinear constraints. Math. Methods Appl. Sci. 20(13), 1135–1162 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Jeyakumar, V., Rubinov, A., Wu, Z.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Mathematical Programming 110(3), 521–541 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kahl, F., Agarwal, S., Chandraker, M.K., Kriegman, D.J., Belongie, S.: Practical global optimization for multiview geometry. IJCV 79(3), 271–284 (2008)CrossRefGoogle Scholar
  8. 8.
    Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. IJCV 74(1), 3–15 (2007)CrossRefGoogle Scholar
  9. 9.
    Kanatani, K., Niitsuma, H., Sugaya, Y.: Optimization without search: Constraint satisfaction by orthogonal projection with applications to multiview triangulation. Memoirs of the Faculty of Engineering 44, 32–41 (2010)Google Scholar
  10. 10.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. IMA Vol. Math. Appl., vol. 149, pp. 157–270. Springer (2009)Google Scholar
  11. 11.
    Li, G.: Global quadratic minimization over bivalent constraints: Necessary and sufficient global optimality condition. Journal of Optimization Theory and Applications, 1–17 (2012)Google Scholar
  12. 12.
    Lofberg, J.: YALMIP: A toolbox for modeling and optimization in matlab. In: Int. Symp. on Computer Aided Control Systems Design, pp. 284–289 (2004)Google Scholar
  13. 13.
    Lu, F., Hartley, R.I.: A Fast Optimal Algorithm for L 2 Triangulation. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds.) ACCV 2007, Part II. LNCS, vol. 4844, pp. 279–288. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Moré, J.: Generalizations of the trust region problem. Optimization methods and Software 2(3-4), 189–209 (1993)CrossRefGoogle Scholar
  15. 15.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer (1999)Google Scholar
  16. 16.
    Olsson, C., Kahl, F., Hartley, R.: Projective least-squares: Global solutions with local optimization. In: CVPR, pp. 1216–1223 (2009)Google Scholar
  17. 17.
    Pinar, M.: Sufficient global optimality conditions for bivalent quadratic optimization. Journal of Optimization Theory and Applications 122(2), 433–440 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Shafarevich, I.: Basic Algebraic Geometry I: Varieties in Projective Space. Springer (1998)Google Scholar
  19. 19.
    Snavely, N., Seitz, S.M., Szeliski, R.: Photo tourism: Exploring photo collections in 3d. In: SIGGRAPH, pp. 835–846 (2006)Google Scholar
  20. 20.
    Stewenius, H., Schaffalitzky, F., Nister, D.: How hard is 3-view triangulation really? In: ICCV, pp. 686–693 (2005)Google Scholar
  21. 21.
    Sturm, J.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Opt. Meth. and Soft. 11-12, 625–653 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Zheng, X., Sun, X., Li, D., Xu, Y.: On zero duality gap in nonconvex quadratic programming problems. Journal of Global Optimization 52(2), 229–242 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Chris Aholt
    • 1
  • Sameer Agarwal
    • 2
  • Rekha Thomas
    • 1
  1. 1.University of WashingtonUSA
  2. 2.Google Inc.USA

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