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Optimal Algorithms in Multiview Geometry

  • Richard Hartley
  • Fredrik Kahl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4843)

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.

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References

  1. 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. 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. 3.
    Boyd, S., Vanderberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  4. 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. 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. 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. 7.
    Faugeras, O.D., Maybank, S.J.: Motion from point matches: Multiplicity of solutions. Int. Journal Computer Vision 4, 225–246 (1990)CrossRefGoogle Scholar
  8. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 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. 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. 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. 12.
    Hartley, R., Sturm, P.: Triangulation. Computer Vision and Image Understanding 68(2), 146–157 (1997)CrossRefGoogle Scholar
  13. 13.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  14. 14.
    Horn, B.K.P.: Closed form solution of absolute orientation using unit quaternions. J. Opt. Soc. America 4(4), 629–642 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Horn, B.K.P.: Relative orientation. Int. Journal Computer Vision 4, 59–78 (1990)CrossRefGoogle Scholar
  16. 16.
    Horn, B.K.P.: Relative orientation revisited. J. Opt. Soc. America 8(10), 1630–1638 (1991)Google Scholar
  17. 17.
    Josephson, K., Kahl, F.: Triangulation of points, lines and conics. In: Scandinavian Conf. on Image Analysis, Aalborg, Denmark (2007)Google Scholar
  18. 18.
    Kahl, F.: Multiple view geometry and the L  ∞ -norm. In: Int. Conf. Computer Vision, Beijing, China, pp. 1002–1009 (2005)Google Scholar
  19. 19.
    Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. Int. Journal Computer Vision 74(1), 3–15 (2007)CrossRefGoogle Scholar
  20. 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. 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. 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. 23.
    Koenderink, J.J., van Doorn, A.J.: Affine structure from motion. J. Opt. Soc. America 8(2), 377–385 (1991)Google Scholar
  24. 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. 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. 26.
    Li, H., Hartley, R.: Five-point motion estimation made easy. In: Int. Conf. Pattern Recognition, pp. 630–633 (August 2006)Google Scholar
  27. 27.
    Li, H., Hartley, R.: The 3D – 3D registration problem revisited. In: Int. Conf. Computer Vision (October 2007)Google Scholar
  28. 28.
    Longuet-Higgins, H.C.: A computer algorithm for reconstructing a scene from two projections. Nature 293, 133–135 (1981)CrossRefGoogle Scholar
  29. 29.
    Lu, F., Hartley, R.: A fast optimal algorithm for l 2 triangulation. In: Asian Conf. Computer Vision (November 2007)Google Scholar
  30. 30.
    Marvell, A.: To his coy mistress. circa (1650)Google Scholar
  31. 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)CrossRefGoogle Scholar
  32. 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. 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. 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. 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. 36.
    Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs (1982)zbMATHGoogle Scholar
  37. 37.
    Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes in C. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  38. 38.
    Salzman, M., Hartley, R., Fua, P.: Convex optimization for deformable surface 3D tracking. In: Int. Conf. Computer Vision (October 2007)Google Scholar
  39. 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. 40.
    Seo, Y., Hartley, R.: Sequential L  ∞  norm minimization for triangulation. In: Asian Conf. Computer Vision (November 2007)Google Scholar
  41. 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. 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. 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. 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)CrossRefMathSciNetGoogle Scholar
  45. 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)CrossRefGoogle Scholar
  46. 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)CrossRefGoogle Scholar
  47. 47.
    Zhang, X.: Pose estimation using L ∞ . In: Image and Vision Computing New Zealand (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Richard Hartley
    • 1
  • Fredrik Kahl
    • 2
  1. 1.Research School of Information Sciences and Engineering, The Australian National University, National ICT Australia (NICTA) 
  2. 2.Centre for Mathematical Sciences, Lund UniversitySweden

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