Globally Optimal Least Squares Solutions for Quasiconvex 1D Vision Problems

  • Carl Olsson
  • Martin Byröd
  • Fredrik Kahl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5575)


Solutions to non-linear least squares problems play an essential role in structure and motion problems in computer vision. The predominant approach for solving these problems is a Newton like scheme which uses the hessian of the function to iteratively find a local solution. Although fast, this strategy inevitably leeds to issues with poor local minima and missed global minima.

In this paper rather than trying to develop an algorithm that is guaranteed to always work, we show that it is often possible to verify that a local solution is in fact also global. We present a simple test that verifies optimality of a solution using only a few linear programs. We show on both synthetic and real data that for the vast majority of cases we are able to verify optimality. Further more we show even if the above test fails it is still often possible to verify that the local solution is global with high probability.


  1. 1.
    Stewénius, H., Schaffalitzky, F., Nistér, D.: How hard is three-view triangulation really? In: Int. Conf. Computer Vision, Beijing, China, pp. 686–693 (2005)Google Scholar
  2. 2.
    Hartley, R., Kahl, F.: Optimal algorithms in multiview geometry. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds.) ACCV 2007, Part I. LNCS, vol. 4843, pp. 13–34. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Triggs, B., McLauchlan, P.F., Hartley, R.I., Fitzgibbon, A.W.: Bundle adjustment – A modern synthesis. In: Triggs, B., Zisserman, A., Szeliski, R. (eds.) ICCV-WS 1999. LNCS, vol. 1883, pp. 298–372. Springer, Heidelberg (2000); in conjunction with ICCV 1999CrossRefGoogle Scholar
  4. 4.
    Engels, C., Stewénius, H., Nistér, D.: Bundle adjustment rules. In: Photogrammetric Computer Vision (PCV) (2006)Google Scholar
  5. 5.
    Kai, N., Steedly, D., Dellaert, F.: Out-of-core bundle adjustment for large-scale 3D reconstruction. In: Conf. Computer Vision and Pattern Recognition, Minneapolis, USA (2007)Google Scholar
  6. 6.
    Hartley, R., Kahl, F.: Critical configurations for projective reconstruction from multiple views. Int. Journal Computer Vision 71, 5–47 (2007)CrossRefGoogle Scholar
  7. 7.
    Åström, K., Enqvist, O., Olsson, C., Kahl, F., Hartley, R.: An L  ∞  approach to structure and motion problems in 1d-vision. In: Int. Conf. Computer Vision, Rio de Janeiro, Brazil (2007)Google Scholar
  8. 8.
    Hartley, R., Seo, Y.: Verifying global minima for L 2 minimization problems. In: Conf. Computer Vision and Pattern Recognition, Anchorage, USA (2008)Google Scholar
  9. 9.
    Olsson, C., Kahl, F., Hartley, R.: Projective Least Squares: Global Solutions with Local Optimization. In: Proc. Int. Conf. Computer Vision and Pattern Recognition (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Carl Olsson
    • 1
  • Martin Byröd
    • 1
  • Fredrik Kahl
    • 1
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

Personalised recommendations