Skip to main content
Book cover

European Conference on Computer Vision

ECCV 1998: Computer Vision — ECCV'98 pp 72–88Cite as

Recognition of planar point configurations using the density of affine shape

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1406)

Abstract

In this paper, we study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Such studies have been made for affine invariants in e.g. [GHJ92], [Wer93], where in the former case a bound on errors are used instead of errors described by density functions, and in the latter case a first order approximation gives an ellipsis as uncertainty region. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. Similar results have been obtained for the special case of the density of the cross ratio, see [May95,åst96].

In particular, we will concentrate on the affine shape, where often analytical computations are possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, similarity, projective etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations and error analysis of reconstruction algorithms. Another approach towards this problem, in the case of similarity transformations, can be found in [Ken89]. For the special case of normally distributed feature points in a plane and similarity transformations, see [BOO86], [MD89].

Finally, an example will be given, illustrating an application of the theory for the problem of recognising planar point configurations from images taken by an affine camera. This case is of particular importance in applications, where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects.

keywords

  • shape
  • invariants
  • densities
  • error analysis
  • recognition
  • distribution of invariants

Download conference paper PDF

References

  1. K. åström. Invariancy Methods for Points, Curves and Surfaces in Computer Vision. PhD thesis, Lund University, Lund Institute of Technology, Department of Mathematics, 1996.

    Google Scholar 

  2. R. Berthilsson. A statistical theory of shape. Technical report, Department of Mathematics, Lund Institute of Technology, 1997. Licentiate Thesis, www.maths.lth.se/matematiklth/personal/rikard/index.html.

    Google Scholar 

  3. F. L. Bookstein. Size and shape spaces for landmark data in two dimensions. Statistical Science, 1(2):181–242, 1986.

    MATH  Google Scholar 

  4. W Grimson, D Huttenlocher, and D Jacobs. A study of affine matching with bounded sensor error. In G. Sandini, editor, Proc. 2nd European Conf. on Computer Vision, volume 588, pages 291–306. Springer-Verlag, 1992.

    Google Scholar 

  5. Anders Heyden. Geometry and Algebra of Multiple Projective Transforms. PhD thesis, Lund University, Lund Institute of Technology, Department of Mathematics, 1995.

    Google Scholar 

  6. David G. Kendall. A survey of the statistical theory of shape. Statistical Science, 4(2):87–120, 1989.

    MATH  MathSciNet  Google Scholar 

  7. S. J. Maybank. Probabilistic analysis of the application of the cross ratio to model based vision: Misclassification. International Journal of Computer Vision, 14(3), 1995.

    Google Scholar 

  8. V. Mardia, K. and I. L. Dryden. Shape distribution for landmark data. Adv. Appl. Prob., pages 742–755, 1989.

    Google Scholar 

  9. I. Rigoutsos and R. Hummel. A bayesian approach to model matching with geometric hashing. Computer Vision and Image Understanding, 62(1):11–26, July 1995.

    CrossRef  Google Scholar 

  10. G. Sparr. Depth-computations from polyhedral images. In G. Sandini, editor, Proc. 2nd European Conf. on Computer Vision, pages 378–386. Springer-Verlag, 1992. Also in Image and Vision Computing, Vol 10., 1992, pages 683–688.

    Google Scholar 

  11. G. Sparr. Simultaneous reconstruction of scene structure and camera locations from uncalibrated image sequences. In 13th International Conference on Pattern Recognition, Vienna, Austria, volume 1, pages 328–333, 1996.

    Google Scholar 

  12. Carl-Gustav Werner. Selective geometric hashing, by means of determinants of transformations. In SCIA '93, Proceedings of the 8th Scandinavian Conference on Image Analysis, volume 2, pages 715–718. NOBIM, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept of Mathematics, Lund University, Box 118, S-221 00, Lund, Sweden

    Rikard Berthilsson & Anders Heyden

Authors
  1. Rikard Berthilsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anders Heyden
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

    Rights and permissions

    Reprints and Permissions

    Copyright information

    © 1998 Springer-Verlag Berlin Heidelberg

    About this paper

    Cite this paper

    Berthilsson, R., Heyden, A. (1998). Recognition of planar point configurations using the density of affine shape. In: Burkhardt, H., Neumann, B. (eds) Computer Vision — ECCV'98. ECCV 1998. Lecture Notes in Computer Science, vol 1406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055660

    Download citation

    • DOI: https://doi.org/10.1007/BFb0055660

    • Published:

    • Publisher Name: Springer, Berlin, Heidelberg

    • Print ISBN: 978-3-540-64569-6

    • Online ISBN: 978-3-540-69354-3

    • eBook Packages: Springer Book Archive