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Triangulation of Points, Lines and Conics

  • Klas Josephson
  • Fredrik Kahl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4522)

Abstract

The problem of reconstructing 3D scene features from multiple views with known camera motion and given image correspondences is considered. This is a classical and one of the most basic geometric problems in computer vision and photogrammetry. Yet, previous methods fail to guarantee optimal reconstructions - they are either plagued by local minima or rely on a non-optimal cost-function.

A common framework for the triangulation problem of points, lines and conics is presented. We define what is meant by an optimal triangulation based on statistical principles and then derive an algorithm for computing the globally optimal solution. The method for achieving the global minimum is based on convex and concave relaxations for both fractionals and monomials. The performance of the method is evaluated on real image data.

Keywords

Fractional Programming Bundle Adjustment Convex Envelope Second Order Cone Program Point Case 
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.

References

  1. 1.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  2. 2.
    Slama, C.C.: Manual of Photogrammetry. American Society of Photogrammetry, Falls Church (1980)Google Scholar
  3. 3.
    Hartley, R., Sturm, P.: Triangulation. Computer Vision and Image Understanding 68(2), 146–157 (1997)CrossRefGoogle Scholar
  4. 4.
    Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier, Amsterdam (1996)zbMATHCrossRefGoogle Scholar
  5. 5.
    Stewénius, H., Schaffalitzky, F., Nistér, D.: How hard is three-view triangulation really? In: Int. Conf. Computer Vision, Beijing, China (2005)Google Scholar
  6. 6.
    Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. In: Int. Conf. Computer Vision, Beijing, China (2005)Google Scholar
  7. 7.
    Agarwal, S., Chandraker, M.K., Kahl, F., Kriegman, D.J., Belongie, S.: Practical Global Optimization for Multiview Geometry. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3951, pp. 592–605. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Harris, C., Stephens, M.: A combined corner and edge detector. In: Alvey Vision Conference, pp. 147–151 (1988)Google Scholar
  9. 9.
    Triggs, B.: Detecting Keypoints with Stable Position, Orientation, and Scale under Illumination Changes. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 100–113. Springer, Heidelberg (2004)Google Scholar
  10. 10.
    Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. Journal of Global Optimization 19, 83–102 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. Journal of Global Optimization 20, 133–154 (2001)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software 11/12, 625–653 (1999)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. Journal of Global Optimization 19(4), 403–424 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22, 343–364 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optimization Methods and Software 18, 219–229 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Klas Josephson
    • 1
  • Fredrik Kahl
    • 1
  1. 1.Centre for Mathematical Sciences, Lund University, LundSweden

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