Skip to main content
SpringerLink
Log in

Quasi-Invariant Parameterisations and Matching of Curves in Images

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

Abstract

In this paper, we investigate quasi-invariance on a smooth manifold, and show that there exist quasi-invariant parameterisations which are not exactly invariant but approximately invariant under group transformations and do not require high order derivatives. The affine quasi-invariant parameterisation is investigated in more detail and exploited for defining general affine semi-local invariants from second order derivatives only. The new invariants are implemented and used for matching curve segments under general affine motions and extracting symmetry axes of objects with 3D bilateral symmetry.

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

Access this article

Log in via an institution

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

References

  • Abu-Mostafa, Y. and Psaltis, D. 1984. Recognitive aspects of moment invariants. IEEE Trans. Pattern Analysis and Machine Intelligence, 6(6):698–706.

    Google Scholar 

  • Barrett, E. and Payton, P. 1991. General methods for determining projective invariants in imagery. Computer Vision, Graphics and Image Processing, 53(1):46–65.

    Google Scholar 

  • Binford, T. and Levitt, T. 1993. Quasi-invariant: Theory and exploitation. In Proc. DARPA Image Understanding Workshop, pages 819–829.

  • Blake, A. 1995. A symmetry theory of planar grasp. International Journal of Robotics Research, 14(5):425–444.

    Google Scholar 

  • Brady, J. and Asada, H. 1984. Smoothed local symmetries and their implementation. International Journal of Robotics Research, 3(3):36–61.

    Google Scholar 

  • Bruckstein, A., Holt, R., Netravali, A., and Richardson, T. 1993. Invariant signatures for planar shape recognition under partial occlusion. ComputerVision, Graphics and Image Processing, 58(1):49–65.

    Google Scholar 

  • Canny, J. 1986. A computational approach to edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence, 8(6):679–698.

    Google Scholar 

  • Cham, T. and Cipolla, R. 1996. Automated B-spline curve representation with MDL-based active contours. In Proc. British Machine Vision Conference, volume 2, pages 363–372, Edinburgh.

    Google Scholar 

  • Cipolla, R. and Blake, A. 1992. Surface orientation and time to contact from image divergence and deformation. In Sandini, G., editor, Proc. 2nd European Conference on Computer Vision, pages 187–202, Santa Margherita, Italy. Springer–Verlag.

    Google Scholar 

  • Cyganski, D., Orr, J., Cott, T., and Dodson, R. 1987. An affine transformation invariant curvature function. In Proc. 1st International Conference on Computer Vision, pages 496–500, London.

  • Friedberg, S. 1986. Finding axes of skewed symmetry. Computer Vision, Graphics and Image Processing, 34:138–155.

    Google Scholar 

  • Gelfand, I. and Formin, S. 1963. Calculus of Variations. Prentice-Hall, New Jersey.

    Google Scholar 

  • Giblin, P. and Brassett, S. 1985. Local symmetry of plane curves. American Mathematical Monthly, 92:689–707.

    Google Scholar 

  • Gross, A. and Boult, T. 1994. Analyzing skewed symmetries. International Journal of Computer Vision, 13(1):91–111.

    Google Scholar 

  • Guggenheimer, H. 1977. Differential Geometry. Dover.

  • Hu, M. 1962. Visual pattern recognition by moment invariants. IRE Transaction on Information Theory, IT-8:179–187.

    Google Scholar 

  • Jacobson, N. 1962. Lie algebras. New York.

  • Kanade, T. 1981. Recovery of the three-dimensional shape of an object from a single view. Artificial Intelligence, 17:409–460.

    Google Scholar 

  • Kanade, T. and Kender, J. 1983. Mapping image properties into shape constraints: Skewed symmetry, affine-transformable patterns, and the shape-from-texture paradigm. In Beck, J. et al, editors, Human and Machine Vision, pages 237–257. Academic Press, NY.

    Google Scholar 

  • Kanatani, K. 1990. Group-Theoretical Methods in Image Understanding. Springer-Verlag.

  • Koenderink, J. and van Doorn, A. 1976. Geometry of binocular vision and a model for stereopsis. Biological Cybernetics, 21:29–35.

    Google Scholar 

  • Lie, S. 1927. Gesammelte Abhandlungen, volume 6. Teubner, Leipzig.

  • Mohan, R. and Nevatia, R. 1992. Perceptual organization for scene segmentation and description. IEEE Trans. Pattern Analysis and Machine Intelligence, 146(6):616–635.

    Google Scholar 

  • Moons, T., Pauwels, E., Van Gool, L. and Oosterlinck, A. 1995. Foundations of semi-differential invariants. International Journal of Computer Vision, 14(1):25–47.

    Google Scholar 

  • Mundy, J. and Zisserman, A. 1992. Geometric Invariance in Computer Vision. MIT Press, Cambridge, USA.

    Google Scholar 

  • Olver, P. 1986. Applications of Lie Groups to Differential Equations. Springer-Verlag.

  • Olver, P. 1995. Equivalence, Invariants, and Symmetry. Cambridge University Press.

  • Olver, P., Sapiro, G. and Tannenbaum, A. 1994. Differential invariant signatures and flows in computer vision. In ter Haar Romeny, B., editor, Geometry-Driven Diffusion in Computer Vision, pages 255–306. Kluwer Academic Publishers.

  • Pauwels, E., Moons, T., Van Gool, L., Kempenaers, P. and Oosterlinck, A. 1995. Recognition of planar shapes under affine distortion. International Journal of Computer Vision, 14(1):49–65.

    Google Scholar 

  • Reisfeld, D., Wolfson, H. and Yeshurun, Y. 1995. Contex-free attention operators: The generalized symmetry transform. International Journal of Computer Vision, 14(2):119–130.

    Google Scholar 

  • Reiss, T. 1993. Recognizing planar objects using invariant image features. (LNCS 676), Springer-Verlag.

  • Rothwell, C., Zisserman, A., Forsyth, D. and Mundy, J. 1995. Planar object recognition using projective shape representation. International Journal of Computer Vision, 16(1):57–99.

    Google Scholar 

  • Sato, J. and Cipolla, R. 1996a. Affine integral invariants and matching of curves. In Proc. International Conference on Pattern Recognition, volume 1, pages 915–919, Vienna, Austria.

    Google Scholar 

  • Sato, J. and Cipolla, R. 1996b. Affine integral invariants for extracting symmetry axes. In Proc. British Machine Vision Conference, volume 1, pages 63–72, Edinburgh.

    Google Scholar 

  • Sattinger, D. and Weaver, O. 1986. Lie groups and algebras with applications to physics, geometry and mechanics. Springer-Verlag, New York.

    Google Scholar 

  • Taubin, G. and Cooper, D. 1992. Object recognition based on moment (or algebraic) invariants. In Mundy, J. and Zisserman, A., editors, Geometric Invariance in Computer Vision, pages 375–397. MIT Press.

  • Van Gool, L., Moons, T., Ungureanu, D. and Oosterlinck, A. 1995a. The characterization and detection of skewed symmetry. Computer Vision and Image Understanding, 61(1):138–150.

    Google Scholar 

  • Van Gool, L., Moons, T., Ungureanu, D. and Pauwels, E. 1995b. Symmetry from shape and shape from symmetry. International Journal of Robotics Research, 14(5):407–424.

    Google Scholar 

  • Van Gool, L., Moons, T., Pauwels, E. and Oosterlinck, A. 1992. Semi-differential invariants. In Mundy, J. and Zisserman, A., editors, Geometric Invariance in Computer Vision, pages 157–192. MIT Press.

  • Weiss, I. 1988. Projective invariants of shapes. In Proc. Image Understanding workshop, volume 2, pages 1125–1134.

    Google Scholar 

  • Weiss, I. 1993. Geometric invariants and object recognition. International Journal of Computer Vision, 10(3):207–231.

    Google Scholar 

  • Zerroug, M. and Nevatia, R. 1993. Quasi-invariant properties and 3-D shape recovery of non-straight, non-constant generalized cylinders. In Proc. DARPA Image Understanding Workshop, pages 725–735.

  • Zerroug, M. and Nevatia, R. 1996. Three-dimensional descriptions based on the analysis of the invariant and quasi-invariant properties of come curved-axis generalized cylinders. IEEE Trans. Pattern Analysis and Machine Intelligence, 18(3):237–253.

    Google Scholar 

  • Zisserman, A., Forsyth, D., Mundy, J. and Rothwell, C. 1992.Recognizing general curved objects efficiently. In Mundy, J. and Zisserman, A., editors, Geometric Invariance in Computer Vision, pages 228– 251. MIT Press.

Download references

Author information

Authors and Affiliations

  1. Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK. E-mail

    Jun Sato & Roberto Cipolla

Authors
  1. Jun Sato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Cipolla
    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

Sato, J., Cipolla, R. Quasi-Invariant Parameterisations and Matching of Curves in Images. International Journal of Computer Vision 28, 117–136 (1998). https://doi.org/10.1023/A:1008011016516

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008011016516

Search

Navigation