Abstract
This paper presents a practical method for obtaining the global minimum to the least-squares (L 2) triangulation problem. Although optimal algorithms for the triangulation problem under L ∞ -norm have been given, finding an optimal solution to the L 2 triangulation problem is difficult. This is because the cost function under L 2-norm is not convex. Since there are no ideal techniques for initialization, traditional iterative methods that are sensitive to initialization may be trapped in local minima. A branch-and-bound algorithm was introduced in [1] for finding the optimal solution and it theoretically guarantees the global optimality within a chosen tolerance. However, this algorithm is complicated and too slow for large-scale use. In this paper, we propose a simpler branch-and-bound algorithm to approach the global estimate. Linear programming algorithms plus iterative techniques are all we need in implementing our method. Experiments on a large data set of 277,887 points show that it only takes on average 0.02s for each triangulation problem.
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References
Agarval, S., Chandraker, M., Kahl, F., Belongie, S., Kriegman, D.: Practical global optimization for multiview geometry. In: Proc. European Conference on Computer Vision (2005)
Hartley, R., Schaffalitzky, F.: L ∞ minimization in geometric reconstruction problems. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition, Washington DC, pp. I–504–509 (2004)
Triggs, W., McLauchlan, P.F., Hartley, R.I., Fitzgibbon, A.: Bundle adjustment for structure from motion. In: Triggs, B., Zisserman, A., Szeliski, R. (eds.) Vision Algorithms: Theory and Practice. LNCS, vol. 1883, pp. 298–372. Springer, Heidelberg (2000)
Kahl, F.: Multiple view geometry and the L ∞ -norm. In: Proc. International Conference on Computer Vision, pp. 1002–1009 (2005)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003)
Hartley, R.I., Sturm, P.: Triangulation. Computer Vision and Image Understanding 68(2), 146–157 (1997)
Stewenius, H., Schaffalitzky, F., Nister, D.: How hard is 3-view triangulation really. In: Proc. International Conference on Computer Vision, pp. 686–693 (2005)
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Oxford University Press, Oxford (2006)
Boyd, S., Vanderberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Snavely, N., Seitz, S., Szeliski, R.: Photo tourism: Exploring photo collections in 3d. ACM Trans on Graphics 25(3), 835–846 (2006)
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Lu, F., Hartley, R. (2007). A Fast Optimal Algorithm for L 2 Triangulation. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds) Computer Vision – ACCV 2007. ACCV 2007. Lecture Notes in Computer Science, vol 4844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76390-1_28
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DOI: https://doi.org/10.1007/978-3-540-76390-1_28
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