Journal of Mathematical Imaging and Vision

, Volume 51, Issue 3, pp 378–384 | Cite as

A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

Article

Abstract

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.

Keywords

Rotation Lie group Exponential map Derivative of rotation Cross-product matrix Rodrigues parameters Rotation vector 

References

  1. 1.
    Altmann, S.: Rotations, Quaternions, And Double Groups. Clarendon, Oxford (1986)MATHGoogle Scholar
  2. 2.
    Bauchau, O., Choi, J.Y.: The vector parameterization of motion. Nonlinear Dyn. 33(2), 165–188 (2003)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cheng, H., Gupta, K.: An historical note on finite rotations. J. Appl. Mech. 56(1), 139–145 (1989)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Dambreville, S., Sandhu, R., Yezzi, A., Tannenbaum, A.: A ge‘ometric approach to joint 2D region-based segmentation and 3D pose estimation using a 3D shape prior. SIAM J. Img. Sci. 3(1), 110–132 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Enqvist, O., Kahl, F.: Robust optimal pose estimation. Proceedings of 10th European Conference on Computer Vision, ECCV’08, pp. 141–153. Springer, Berlin, Heidelberg (2008)Google Scholar
  6. 6.
    Gurwitz, C., Overton, M.: A globally convergent algorithm for minimizing over the rotation group of quadratic forms. IEEE Trans. Pattern Anal. Mach. Intell. 11(11), 1228–1232 (1989)CrossRefGoogle Scholar
  7. 7.
    Hall, B.: Lie Groups, Lie Algebras and Representations. Graduate Text in Mathematics. Springer, Berlin (2004)Google Scholar
  8. 8.
    Hartley, R., Kahl, F.: Global optimization through searching rotation space and optimal estimation of the essential matrix. In: Proceedings of IEEE 11th International Conference on Computer Vision, ICCV 2007, pp. 1–8 (2007)Google Scholar
  9. 9.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003)Google Scholar
  10. 10.
    Helgason, S.: Differential geometry and symmetric spaces. Elsevier Science, Pure and Applied Mathematics, Amsterdam (1962)MATHGoogle Scholar
  11. 11.
    Horn, B.K.P.: Relative Orientation. Int. J. Comput. Vis. 4(1), 59–78 (1990)CrossRefGoogle Scholar
  12. 12.
    Kuehnel, F.O.: On the minimization over SO(3) manifolds. Technical Report 03.12, Research Institute for Advanced Computer Science (RIACS). NASA Ames Research Center, USA (2003)Google Scholar
  13. 13.
    Ma, Y., Soatto, S., Kosecká, J., Sastry, S.: An Invitation to 3-D Vision: From Images to Geometric Models. Springer, Berlin (2003)Google Scholar
  14. 14.
    Murray, R., Li, Z., Sastry, S.: A Mathematical Introduction to Robotic Manipulation. CRC, Boca Raton (1994)MATHGoogle Scholar
  15. 15.
    Olsson, C., Kahl, F., Oskarsson, M.: Optimal estimation of perspective camera pose. In: Proceedings of 18th International Conference on Pattern Recognition, ICPR 2006, vol. 2, pp. 5–8 (2006)Google Scholar
  16. 16.
    Open Source Computer Vision (OpenCV): http://opencv.org
  17. 17.
    Ritto-Corrêa, M., Camotim, D.: On the differentiation of the Rodrigues formula and its significance for the vector-like parameterization of Reissner–Simo beam theory. Int. J. Num. Methods Eng. 55(9), 1005–1032 (2002)Google Scholar
  18. 18.
    Sandhu, R., Dambreville, S., Tannenbaum, A.: Point set registration via particle filtering and stochastic dynamics. IEEE Trans. Pattern Anal. Mach. Intell. 32(8), 1459–1473 (2010) Google Scholar
  19. 19.
    Stuelpnagel, J.: On the parametrization of the three-dimensional rotation group. SIAM Rev. 6(4), 422–430 (1964)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Unal, G.B., Yezzi, A.J., Soatto, S., Slabaugh, G.G.: A variational approach to problems in calibration of multiple cameras. IEEE Trans. Pattern Anal. Mach. Intell. 29(8), 1322–1338 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Grupo de Tratamiento de ImágenesUniversidad Politécnica de MadridMadridSpain
  2. 2.Robotics and Perception Group, AI LaboratoryUniversity of ZürichZürichSwitzerland
  3. 3.School of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations