Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Optimal Algorithms in Multiview Geometry

  • Conference paper
Computer Vision – ACCV 2007 (ACCV 2007)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 4843))

Included in the following conference series:

Abstract

This is a survey paper summarizing recent research aimed at finding guaranteed optimal algorithms for solving problems in Multiview Geometry. Many of the traditional problems in Multiview Geometry now have optimal solutions in terms of minimizing residual imageplane error. Success has been achieved in minimizing L 2 (least-squares) or L  ∞  (smallest maximum error) norm. The main methods involve Second Order Cone Programming, or quasi-convex optimization, and Branch-and-bound. The paper gives an overview of the subject while avoiding as far as possible the mathematical details, which can be found in the original papers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, S., Chandraker, M.K., Kahl, F., Kriegman, D.J., Belongie, S.: Practical global optimization for multiview geometry. In: European Conf. Computer Vision, Graz, Austria, pp. 592–605 (2006)

    Google Scholar 

  2. 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 

  3. Boyd, S., Vanderberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  4. Byröd, M., Josephson, K., Åström, K.: Improving numerical accuracy in gröbner basis polynomial equation solvers. In: Int. Conf.Computer Vision, Rio de Janeiro, Brazil (2007)

    Google Scholar 

  5. Chandraker, M.K., Agarwal, S., Kriegman, D.J., Belongie, S.: Globally convergent algorithms for affine and metric upgrades in stratified autocalibration. In: Int. Conf. Computer Vision, Rio de Janeiro, Brazil (2007)

    Google Scholar 

  6. Farenzena, M., Fusiello, A., Dovier, A.: Reconstruction with interval constraints propagation. In: Proc. Conf. Computer Vision and Pattern Recognition, New York City, USA, pp. 1185–1190 (2006)

    Google Scholar 

  7. Faugeras, O.D., Maybank, S.J.: Motion from point matches: Multiplicity of solutions. Int. Journal Computer Vision 4, 225–246 (1990)

    Article  Google Scholar 

  8. Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Opt. 19(1), 83–102 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hartley, R., de Agapito, L., Hayman, E., Reid, I.: Camera calibration and the search for infinity. In: Proc. 7th International Conference on Computer Vision, Kerkyra, Greece, September 1999, pp. 510–517 (1999)

    Google Scholar 

  10. Hartley, R., Kahl, F.: Global optimization through searching rotation space and optimal estimation of the essential matrix. Int. Conf. Computer Vision  (2007)

    Google Scholar 

  11. Hartley, R., Schaffalitzky, F.: L ∞  minimization in geometric reconstruction problems. In: Conf. Computer Vision and Pattern Recognition, Washington DC, USA, vol. I, pp. 504–509 (2004)

    Google Scholar 

  12. Hartley, R., Sturm, P.: Triangulation. Computer Vision and Image Understanding 68(2), 146–157 (1997)

    Article  Google Scholar 

  13. Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  14. Horn, B.K.P.: Closed form solution of absolute orientation using unit quaternions. J. Opt. Soc. America 4(4), 629–642 (1987)

    Article  MathSciNet  Google Scholar 

  15. Horn, B.K.P.: Relative orientation. Int. Journal Computer Vision 4, 59–78 (1990)

    Article  Google Scholar 

  16. Horn, B.K.P.: Relative orientation revisited. J. Opt. Soc. America 8(10), 1630–1638 (1991)

    Google Scholar 

  17. Josephson, K., Kahl, F.: Triangulation of points, lines and conics. In: Scandinavian Conf. on Image Analysis, Aalborg, Denmark (2007)

    Google Scholar 

  18. Kahl, F.: Multiple view geometry and the L  ∞ -norm. In: Int. Conf. Computer Vision, Beijing, China, pp. 1002–1009 (2005)

    Google Scholar 

  19. Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. Int. Journal Computer Vision 74(1), 3–15 (2007)

    Article  Google Scholar 

  20. Ke, Q., Kanade, T.: Quasiconvex optimization for robust geometric reconstruction. In: Int. Conf. Computer Vision, Beijing, China, pp. 986–993 (2005)

    Google Scholar 

  21. Ke, Q., Kanade, T.: Uncertainty models in quasiconvex optimization for geometric reconstruction. In: Conf. Computer Vision and Pattern Recognition, New York City, USA, pp. 1199–1205 (2006)

    Google Scholar 

  22. Kim, J.H., Hartley, R., Frahm, J.M., Pollefeys, M.: Visual odometry for non-overlapping views using second-order cone programming. In: Asian Conf. Computer Vision (November 2007)

    Google Scholar 

  23. Koenderink, J.J., van Doorn, A.J.: Affine structure from motion. J. Opt. Soc. America 8(2), 377–385 (1991)

    Google Scholar 

  24. Kumar, P., Torr, P.H.S., Zisserman, A.: Solving markov random fields using second order cone programming relaxations. In: Conf. Computer Vision and Pattern Recognition, pp. 1045–1052 (2006)

    Google Scholar 

  25. Li, H.: A practical algorithm for L-infinity triangulation with outliers. In: CVPR, vol. 1, pp. 1–8. IEEE Computer Society, Los Alamitos (2007)

    Google Scholar 

  26. Li, H., Hartley, R.: Five-point motion estimation made easy. In: Int. Conf. Pattern Recognition, pp. 630–633 (August 2006)

    Google Scholar 

  27. Li, H., Hartley, R.: The 3D – 3D registration problem revisited. In: Int. Conf. Computer Vision (October 2007)

    Google Scholar 

  28. Longuet-Higgins, H.C.: A computer algorithm for reconstructing a scene from two projections. Nature 293, 133–135 (1981)

    Article  Google Scholar 

  29. Lu, F., Hartley, R.: A fast optimal algorithm for l 2 triangulation. In: Asian Conf. Computer Vision (November 2007)

    Google Scholar 

  30. Marvell, A.: To his coy mistress. circa (1650)

    Google Scholar 

  31. Nistér, D.: An efficient solution to the five-point relative pose problem. IEEE Trans. Pattern Analysis and Machine Intelligence 26(6), 756–770 (2004)

    Article  Google Scholar 

  32. Nistér, D., Hartley, R., Stewénius, H.: Using Galois theory to prove that structure from motion algorithms are optimal. In: Conf. Computer Vision and Pattern Recognition (June 2007)

    Google Scholar 

  33. Nistér, D., Kahl, F., Stewénius, H.: Structure from motion with missing data is NP-hard. In: Int. Conf. Computer Vision, Rio de Janeiro, Brazil (2007)

    Google Scholar 

  34. Olsson, C., Eriksson, A., Kahl, F.: Efficient optimization of L  ∞ -problems using pseudoconvexity. In: Int. Conf. Computer Vision, Rio de Janeiro, Brazil (2007)

    Google Scholar 

  35. Olsson, C., Kahl, F., Oskarsson, M.: Optimal estimation of perspective camera pose. In: Int. Conf. Pattern Recognition, Hong Kong, China, vol. II, pp. 5–8 (2006)

    Google Scholar 

  36. Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs (1982)

    MATH  Google Scholar 

  37. Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes in C. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  38. Salzman, M., Hartley, R., Fua, P.: Convex optimization for deformable surface 3D tracking. In: Int. Conf. Computer Vision (October 2007)

    Google Scholar 

  39. Seo, Y., Hartley, R.: A fast method to minimize L  ∞  error norm for geometric vision problems. In: Int. Conf. Computer Vision (October 2007)

    Google Scholar 

  40. Seo, Y., Hartley, R.: Sequential L  ∞  norm minimization for triangulation. In: Asian Conf. Computer Vision (November 2007)

    Google Scholar 

  41. Sim, K., Hartley, R.: Recovering camera motion using the L  ∞ -norm. In: Conf. Computer Vision and Pattern Recognition, New York City, USA, pp. 1230–1237 (2006)

    Google Scholar 

  42. Sim, K., Hartley, R.: Removing outliers using the L  ∞ -norm. In: Conf. Computer Vision and Pattern Recognition, New York City, USA, pp. 485–492 (2006)

    Google Scholar 

  43. 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 

  44. Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software 11(12), 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  45. Tomasi, C., Kanade, T.: Shape and motion from image streams under orthography: A factorization approach. Int. Journal Computer Vision 9(2), 137–154 (1992)

    Article  Google Scholar 

  46. Triggs, W., McLauchlan, P.F., Hartley, R.I., Fitzgibbon, A.: Bundle adjustment for structure from motion. In: Vision Algorithms: Theory and Practice, pp. 298–372. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  47. Zhang, X.: Pose estimation using L ∞ . In: Image and Vision Computing New Zealand (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research School of Information Sciences and Engineering, The Australian National University, National ICT Australia (NICTA),  

    Richard Hartley

  2. Centre for Mathematical Sciences, Lund University, Sweden

    Fredrik Kahl

Authors
  1. Richard Hartley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fredrik Kahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Yasushi Yagi Sing Bing Kang In So Kweon Hongbin Zha

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hartley, R., Kahl, F. (2007). Optimal Algorithms in Multiview Geometry. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds) Computer Vision – ACCV 2007. ACCV 2007. Lecture Notes in Computer Science, vol 4843. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76386-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-76386-4_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-76385-7

  • Online ISBN: 978-3-540-76386-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation