Advertisement

Finding the Exact Rotation between Two Images Independently of the Translation

  • Laurent Kneip
  • Roland Siegwart
  • Marc Pollefeys
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7577)

Abstract

In this paper, we present a new epipolar constraint for computing the rotation between two images independently of the translation. Against the common belief in the field of geometric vision that it is not possible to find one independently of the other, we show how this can be achieved by relatively simple two-view constraints. We use the fact that translation and rotation cause fundamentally different flow fields on the unit sphere centered around the camera. This allows to establish independent constraints on translation and rotation, and the latter is solved using the Gröbner basis method. The rotation computation is completed by a solution to the cheiriality problem that depends neither on translation, nor on feature triangulations. Notably, we show for the first time how the constraint on the rotation has the advantage of remaining exact even in the case of translations converging to zero. We use this fact in order to remove the error caused by model selection via a non-linear optimization of rotation hypotheses. We show that our method operates in real-time and compare it to a standard existing approach in terms of both speed and accuracy.

Keywords

Rotation Matrix Rotation Matrice Visual Odometry Pure Rotation Epipolar Constraint 
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.
    Nistér, D., Naroditsky, O., Bergen, J.: Visual odometry. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Washington, DC, USA, pp. 652–659 (2004)Google Scholar
  2. 2.
    Davison, A., Reid, D., Molton, D., Stasse, O.: MonoSLAM: Real-time single camera SLAM. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 26(6), 1052–1067 (2007)CrossRefGoogle Scholar
  3. 3.
    Klein, G., Murray, D.: Parallel tracking and mapping for small AR workspaces. In: Proceedings of the International Symposium on Mixed and Augmented Reality (ISMAR), Nara, Japan (2007)Google Scholar
  4. 4.
    Kneip, L., Chli, M., Siegwart, R.: Robust real-time visual odometry with a single camera and an IMU. In: Proceedings of the British Machine Vision Conference (BMVC), Dundee, Scotland (2011)Google Scholar
  5. 5.
    Fischler, M., Bolles, R.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM 24(6), 381–395 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Nistér, D.: An efficient solution to the five-point relative pose problem. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 26(6), 756–777 (2004)CrossRefGoogle Scholar
  7. 7.
    Stewénius, H., Engels, C., Nistér, D.: Recent developments on direct relative orientation. ISPRS Journal of Photogrammetry and Remote Sensing 60(4), 284–294 (2006)CrossRefGoogle Scholar
  8. 8.
    Kukelova, Z., Bujnak, M., Pajdla, T.: Polynomial Eigenvalue solutions to the 5-pt and 6-pt relative pose problems. In: Proceedings of the British Machine Vision Conference (BMVC), Leeds, UK (2008)Google Scholar
  9. 9.
    Longuet-Higgins, H.: Readings in computer vision: issues, problems, principles, and paradigms. Morgan Kaufmann Publishers Inc., San Francisco (1987)Google Scholar
  10. 10.
    Kruppa, E.: Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung. Sitzgsber. Akad. Wien, Math. Naturw. Abt., IIa 122, 1939–1948 (1913)zbMATHGoogle Scholar
  11. 11.
    Torr, P., Fitzgibbon, A., Zisserman, A.: Maintaining multiple motion model hypotheses over many views to recover matching and structure. In: Proceedings of the International Conference on Computer Vision (ICCV), Bombay, India, pp. 485–491 (1998)Google Scholar
  12. 12.
    Kalantari, M., Jung, F., Guedon, J.P., Paparoditis, N.: The five points pose problem: A new and accurate solution adapted to any geometric configuration. In: Proceedings of the Pacific Rim Symposium on Advances in Image and Video Technology, Tokyo, Japan, pp. 215–226 (2009)Google Scholar
  13. 13.
    Lim, J., Barnes, N., Li, H.: Estimating relative camera motion from the antipodal-epipolar constraint. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 32(10), 1907–1914 (2010)CrossRefGoogle Scholar
  14. 14.
    Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Undergraduate Texts in Mathematics. Springer-Verlag New York, Inc., Secaucus (2007)zbMATHGoogle Scholar
  15. 15.
    Cayley, A.: About the algebraic structure of the orthogonal group and the other classical groups in a field of characteristic zero or a prime characteristic. Reine Angewandte Mathematik 32 (1846)Google Scholar
  16. 16.
    Thompson, E.: A method for the construction of orthogonal matrices. The Photogrammetric Record 3(13), 55–59 (1958)CrossRefGoogle Scholar
  17. 17.
    Buchberger, B.: Multidimensional Systems Theory - Progress, Directions and Open Problems in Multidimensional Systems. Reidel Publishing Company, Dodrecht (1985)Google Scholar
  18. 18.
    Gebauer, R., Möller, H.M.: On an installation of Buchberger’s algorithm. Journal of Symbolic Computation 6(2-3), 275–286 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Giovini, A., Mora, T., Niesi, G., Robbiano, L., Traverso, C.: “One sugar cube, please” or selection strategies in the Buchberger algorithm. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation, Bonn, West Germany, pp. 49–54 (1991)Google Scholar
  20. 20.
    Faugère, J.: A new efficient algorithm for computing Gröbner bases (F4). Journal of Pure and Applied Algebra 139(1-3), 61–88 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kukelova, Z., Bujnak, M., Pajdla, T.: Automatic Generator of Minimal Problem Solvers. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 302–315. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Arun, K., Huang, T., Blostein, S.: Least-Squares Fitting of Two 3-D Point Sets. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 9(5), 698–700 (1987)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Laurent Kneip
    • 1
  • Roland Siegwart
    • 1
  • Marc Pollefeys
    • 2
  1. 1.Autonomous Systems LabETH ZurichSwitzerland
  2. 2.Computer Vision and Geometry GroupETH ZurichSwitzerland

Personalised recommendations