Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Interpretation of conic motion and its applications

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

The indeterminacy of conic motion is analyzed in terms of Lie group theory. It is shown that an image motion of a conic is associated with a group ofinvisible motions that do not cause a visible change of the conic. All such groups are isomorphic to the group associated with a special conic called thestandard circle, for which the group of invisible motions is the (three-dimensional)Lorentz group. Similar results are obtained forinvisible optical flows. Finally, our analysis is extended toconic stereo: the 3-D position and orientation of a conic in the scene are computed from two projections. This algorithm also works with one camera if a circular pattern is projected from a light source.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. F. Bergholm, Motion from flow along contours: A note on robustness and ambiguous cases,Intern. J. Comput. Vis. 4: 395–415, 1989.

    Google Scholar 

  2. F. Bergholm and S. Carlsson. A “theory” of optical flow,Comput. Vis. Graph. Image Process.: Image Understanding 53: 171–188, 1991.

    Google Scholar 

  3. F.L. Bookstein, Fitting conic sections to scattered data,Comput. Graph. Image Process. 9: 56–71, 1979.

    Google Scholar 

  4. L.S. Davis, Z. Wu, and H. Sun, Contour-based motion estimation,Comput. Vis. Graph. Image Process. 33: 313–326, 1983.

    Google Scholar 

  5. T. Ellis, A. Abboot, and B. Brillault, Ellipse detection and matching with uncertainty,Image Vis. Comput. 10: 271–276, 1992.

    Google Scholar 

  6. D. Forsyth, J.L. Mundy, A. Zisserman, C. Coelho, A. Heller, and C. Rothwell, Invariant descriptors for 3-D object recognition and pose,IEEE Trans. Patt. Anal. Mach. Intell. 13: 971–991, 1991.

    Google Scholar 

  7. A. Griewank and G.F. Corliss (eds.),Automatic Differentiation of Algorithms: Theory, Implementation, and Application. Soc. for Indust. and Applied Mathematics: Philadelphia, PA, 1991.

    Google Scholar 

  8. K. Kanatani, Tracing planar surface motion from a projection without knowing correspondence,Comput. Vis. Graph. Image Process. 29: 1–12, 1985.

    Google Scholar 

  9. K. Kanatani, Detecting the motion of a planar surface by line and surface integrals,Comput. Vis. Graph. Image Process. 29: 13–22, 1985.

    Google Scholar 

  10. K. Kanatani,Group-Theoretical Methods in Image Understanding. Springer: Berlin, 1990.

    Google Scholar 

  11. K. Kanatani, Computational projective geometry,Comput. Vis. Graph. Image Process.: Image Understanding 54: 333–348, 1991.

    Google Scholar 

  12. K. Kanatani, Statistical analysis of focal-length calibration using vanishing points,IEEE Trans. Robot. Autom. 8(6): 1992 (to appear).

  13. K. Kanatani,Geometric Computation for Machine Vision. Oxford Univ. Press: Oxford, 1993.

    Google Scholar 

  14. K. Kanatani, Statistical bias of conic fitting and renormalization,IEEE Trans. Patt. Anal. Mach. Intell. 15(1): 1993 (to appear).

  15. K. Kanatani and W. Liu, 3-D interpretation of conics and orthogonality,Comput. Vis. Graph. Image Process.: Image Understanding (to appear).

  16. K. Kanatani and Y. Onodera, Anatomy of camera calibration using vanishing points,IEICE Trans. Infor. Sys. 74(10): 3369–3378, 1991.

    Google Scholar 

  17. H.C. Longuet-Higgins, The reconstruction of a plane surface from two perspective projections,Proc. Roy. Soc. London B-227: 399–410, 1986.

    Google Scholar 

  18. S.D. Ma, S.H. Si, and Z.Y. Chen, Quadric curve based stereo,Proc. 11th Intern. Conf. Patt. Recog., August–September, the Hague, the Netherlands, vol. 1, pp. 1–4, 1992.

  19. J. Porrill, Fitting ellipses and predicting confidence envelopes using a bias corrected Kalman filter,Image Vis. Comput. 8: 37–51, 1990.

    Google Scholar 

  20. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,Numerical Recipes in C. Cambridge Univ. Press: Cambridge, 1988.

    Google Scholar 

  21. C.A. Rothwell, A. Zisserman, C.I. Marinos, D.A. Forsyth, and J.L. Mundy, Relative motion and pose from arbitrary plane curves.Image Vis. Comput. 10(4): 250–262, 1992.

    Google Scholar 

  22. R. Safaee-Rad, I. Tchoukanov, B. Benhabib, and K.C. Smith, Accurate parameter estimation of quadratic curves from grey-level images.Comput. Vis. Graph. Image Process.: Image Understanding 54: 259–274, 1991.

    Google Scholar 

  23. R. Safaee-Rad, I. Tchoukanov, K.C. Smith, and B. Benhabib, Constraints on quadratic-curved features under perspective projection,Image Vis. Comput. 19(8): 532–548, 1992.

    Google Scholar 

  24. R. Safaee-Rad, I. Tchoukanov, K.C. Smith, and B. Benhabib, Three-dimensional location estimation of circular features for machine vision,IEEE Trans. Robot. Autom. 8(5): 624–640, 1992.

    Google Scholar 

  25. P.D. Sampson, Fitting conic sections to “very scattered” data: An iterative refinement of the Bookstein algorithm,Comput. Graph. Image Process. 18: 97–108, 1982.

    Google Scholar 

  26. J.G. Semple and G.T. Kneebone,Algebraic Projective Geometry. Clarendon Press: Oxford, 1952 (reprinted 1979).

    Google Scholar 

  27. R.Y. Tsai and T.S. Huang, Estimating three-dimensional motion parameters of rigid planar patch,IEEE Trans. Acoust. Speech Sig. Process. 29: 1147–1152, 1981.

    Google Scholar 

  28. T.Y. Tsai and T.S. Huang, Estimating 3-D motion parameters of a rigid planar patch III. Finite point correspondences and the three-view problem,IEEE Trans. Acoust. Speech Sig. Process. 32: 213–220, 1984.

    Google Scholar 

  29. R.Y. Tsai, T.S. Huang, and W.-L. Zhu, Estimating three-dimensional motion parameters of a rigid planar patch, II: Singular value decomposition,IEEE Trans. Acoust. Speech Sig. Process. 30: 525–534, 1982.

    Google Scholar 

  30. A.M. Waxman and K. Wohn, Contour evolution, neighborhood deformation, and global image flow: Planar surfaces in motion,Intern. J. Robot. Res. 4: 95–108, 1985.

    Google Scholar 

  31. K. Wohn and A.M. Waxman, The analytic structure of image flows: Deformation and segmentation,Comput. Vis. Graph. Image Process. 49: 127–151, 1990.

    Google Scholar 

  32. K.Y. Wohn, J. Wu, and R.W. Brocket, A contour-based recovery of image flow: Iterative transformation method,IEEE Trans. Patt. Anal. Mach. Intell. 13: 746–760, 1991.

    Google Scholar 

  33. T. Yoshida, Automatic derivative derivation system (in Japanese),Trans. Inform. Process. Soc. Japan 30(7): 799–806, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Gunma University, 376, Kiryu, Gunma, Japan

    Wu Liu & Kenichi Kanatani

Authors
  1. Wu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenichi Kanatani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, W., Kanatani, K. Interpretation of conic motion and its applications. Int J Comput Vision 10, 67–84 (1993). https://doi.org/10.1007/BF01440848

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01440848

Keywords

Search

Navigation