Advertisement

On the Non-linear Optimization of Projective Motion Using Minimal Parameters

  • Adrien Bartoli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)

Abstract

I address the problem of optimizing projective motion over a minimal set of parameters. Most of the existing works overparameterize the problem. While this can simplify the estimation process and may ensure well-conditioning of the parameters, this also increases the computational cost since more unknowns than necessary are involved.

I propose a method whose key feature is that the number of parameters employed is minimal. The method requires singular value decomposition and minor algebraic manipulations and is therefore straightforward to implement. It can be plugged into most of the optimization algorithms such as Levenberg-Marquardt as well as the corresponding sparse versions. The method relies on the orthonormal camera motion representation that I introduce here. This representation can be locally updated using minimal parameters.

I give a detailled description for the implementation of the two-view case within a bundle adjustment framework, which corresponds to the maximum likelihood estimation of the fundamental matrix and scene structure. Extending the algorithm to the multiple-view case is straightforward. Experimental results using simulated and real data show that algorithms based on minimal parameters perform better than the others in terms of the computational cost, i.e. their convergence is faster, while achieving comparable results in terms of convergence to a local optimum. An implementation of the method will be made available.

Keywords

Projection Matrix Fundamental Matrix Normalization Constraint Bundle Adjustment Epipolar Geometry 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Bartoli and P. Sturm. Three new algorithms for projective bundle adjustment with minimum parameters. Research Report 4236, inria, Grenoble, France, August 2001.Google Scholar
  2. 2.
    A. Bartoli, P. Sturm, and R. Horaud. Projective structure and motion from two views of a piecewise planar scene. In Proceedings of the 8th International Conference on Computer Vision, Vancouver, Canada, volume 1, pages 593–598, July 2001.Google Scholar
  3. 3.
    P. Beardsley, P. Torr, and A. Zisserman. 3D model acquisition from extended image sequences. In B. Buxton and R. Cipolla, editors, Proceedings of the 4th European Conference on Computer Vision, Cambridge, England, volume 1065 of Lecture Notes in Computer Science, pages 683–695. Springer-Verlag, April 1996.Google Scholar
  4. 4.
    R. Hartley. In defence of the 8-point algorithm. In Proceedings of the 5th International Conference on Computer Vision, Cambridge, Massachusetts, USA, pages 1064–1070, June 1995.Google Scholar
  5. 5.
    R. Hartley and P. Sturm. Triangulation. Computer Vision and Image Understanding, 68(2): 146–157, 1997.CrossRefGoogle Scholar
  6. 6.
    R.I. Hartley. Euclidean reconstruction from uncalibrated views. In Proceeding of the darpaesprit workshop on Applications of Invariants in Computer Vision, Azores, Portugal, pages 187–202, October 1993.Google Scholar
  7. 7.
    R.I. Hartley. Projective reconstruction and invariants from multiple images. ieee Transactions on Pattern Analysis and Machine Intelligence, 16(10):1036–1041, October 1994.Google Scholar
  8. 8.
    R.I. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, June 2000.Google Scholar
  9. 9.
    M. Irani and P. Anadan. Parallax geometry of pairs of points for 3d scene analysis. In Proceedings of the 4th European Conference on Computer Vision, Cambridge, England, pages 17–30. Springer-Verlag, 1996.Google Scholar
  10. 10.
    Q.T. Luong and O. Faugeras. The fundamental matrix: Theory, algorithms and stability analysis. International Journal of Computer Vision, 17(1):43–76, 1996.CrossRefGoogle Scholar
  11. 11.
    Q.T. Luong and T. Vieville. Canonic representations for the geometries of multiple projective views. Computer Vision and Image Understanding, 64(2): 193–229, 1996.CrossRefGoogle Scholar
  12. 12.
    P. F. McLauchlan. Gauge invariance in projective 3D reconstruction. In Proceedings of the Multi-View Workshop, Fort Collins, Colorado, USA, 1999.Google Scholar
  13. 13.
    J. Oliensis. The error surface for structure and motion. Technical report, NEC, 2001.Google Scholar
  14. 14.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes in C-The Art of Scientific Computing. Cambridge University Press, 2nd edition, 1992.Google Scholar
  15. 15.
    C.C. Slama, editor. Manual of Photogrammetry, Fourth Edition. American Society of Photogrammetry and Remote Sensing, Falls Church, Virginia, USA, 1980.Google Scholar
  16. 16.
    P. Sturm and B. Triggs. A factorization based algorithm for multi-image projective structure and motion. In B. Buxton and R. Cipolla, editors, Proceedings of the 4th European Conference on Computer Vision, Cambridge, England, volume 1065 of Lecture Notes in Computer Science, pages 709–720. Springer-Verlag, April 1996.Google Scholar
  17. 17.
    B. Triggs, P.F. McLauchlan, R.I. Hartley, and A. Fitzgibbon. Bundle ajustment — a modern synthesis. In B. Triggs, A. Zisserman, and R. Szeliski, editors, Vision Algorithms: Theory and Practice, volume 1883 of Lecture Notes in Computer Science, pages 298–372. Springer-Verlag, 2000.CrossRefGoogle Scholar
  18. 18.
    Z. Zhang. Determining the epipolar geometry and its uncertainty: A review. International Journal of Computer Vision, 27(2):161–195, March 1998.Google Scholar
  19. 19.
    Z. Zhang and C. Loop. Estimating the fundamental matrix by transforming image points in projective space. Computer Vision and Image Understanding, 82(2): 174–180, May 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Adrien Bartoli
    • 1
  1. 1.INRIA Rhône-AlpesSt. Ismier cedexFrance

Personalised recommendations