Skip to main content
SpringerLink
Log in

The Local Projective Shape of Smooth Surfaces and Their Outlines

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

Abstract

This article examines projectively-invariant local geometric properties of smooth curves and surfaces. Oriented projective differential geometry is proposed as a general framework for establishing such invariants and characterizing the local projective shape of surfaces and their outlines. It is applied to two problems: (1) the projective generalization of Koenderink’s famous characterization of convexities, concavities, and inflections of the apparent contours of solids bounded by smooth surfaces, and (2) the image-based construction of rim meshes, which provide a combinatorial description of the arrangement induced on the surface of an object by the contour generators associated with multiple cameras observing it.

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

  • Arbogast, E. and Mohr, R. 1991. 3D structure inference from image sequences. Journal of Pattern Recognition and Artificial Intelligence, 5(5).

  • Baumgart, B. 1974. Geometric modeling for computer vision. Technical Report AIM-249, Stanford University. Ph.D. Thesis. Department of Computer Science.

  • Belhumeur, P., Kriegman, D., and Yuille, A. 1997. The bas-relief ambiguity. In Proc. IEEE Conf. Comp. Vision Patt. Recog., San Juan, Puerto Rico, pp. 1060–1066.

  • Berger, M. 1987. Geometry. Springer-Verlag.

  • Blaschke, W. 1967. Differential Geometrie. Chelsea: New York, 2 vols.

  • Bol, G. 1950. Projektive Differentialgeometrie. Vandenhoeck & Ruprecht: Göttingen.

    Google Scholar 

  • Boyer, E. 1996. Object models from contour sequences. In Proceedings of Fourth European Conference on Computer Vision, Cambridge, (England), Lecture Notes in Computer Science, vol. 1065, pp. 109–118.

    Google Scholar 

  • Brady, J., Ponce, J., Yuille, A., and Asada, H. 1985. Describing surfaces. Computer Vision, Graphics and Image Processing, 32(1):1–28.

    Google Scholar 

  • Calabi, E., Olver, P., Shakiban, C., Tannenbaum, A., and Haker, S. 1998. Differential and numerically invariant signature curves applied to object recognition. Int. J. of Comp. Vision, 26(2):107–135.

    Google Scholar 

  • Cartan, E. 1937. La théorie des groupes finis et continus et la. géométrie différentielle traitée par la méthode du repère mobile. Jacques Gabay, 1992. Original edition, Gauthier-Villars, 1937.

  • Chum, O., Werner, T., and Pajdla, T. 2003. Joint orientation of epipoles. In Proc. British Machine Vision Conference, pp. 73–82.

  • Cipolla, R., Åström, K., and Giblin, P. 1995. Motion from the frontier of curved surfaces. In Proc. Int. Conf. Comp. Vision, pp. 269–275.

  • Cipolla, R. and Blake, A. 1992. Surface shape from the deformation of the apparent contour. Int. J. of Comp. Vision, 9(2):83–112.

    Google Scholar 

  • Cross, G. and Zisserman, A. 2000. Surface reconstruction from multiple views using apparent contours and surface texture. In NATO Advanced Research Workshop on Confluence of Computer Vision and Computer Graphics, pp. 25–47.

  • do Carmo, M. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall: Englewood Cliffs, NJ.

    Google Scholar 

  • Durand, F., Drettakis, G., and Puech, C. 2002. The 3D visibility complex. ACM Transactions on Graphics, 21(2):176–206.

    Google Scholar 

  • Faugeras, O. 1994. Cartan’s moving frame method and its application to the geometry and evolution of curves in the euclidean, affine and projective planes. In Applications of Invariance in Computer Vision, J. Mundy, A. Zisserman, and D. Forsyth (Eds.), vol. 825 of Lecture Notes in Computer Science. Springer-Verlag, pp. 11-46.

  • Faugeras, O., Luong, Q.-T., and Papadopoulo, T. 2001. The Geometry of Multiple Images. MIT Press.

  • Faugeras, O. and Papadopoulo, T. 1993. A theory of the motion fields of curves. Int. J. of Comp. Vision, 10(2):125–156.

    Google Scholar 

  • Gansner, E. and North, S. 1999. An open graph visualization system and its applications to software engineering. Software—Practice and Experience, 30(11):1203–1233.

    Google Scholar 

  • Hartley, R. 1998. Cheirality. Int. J. of Comp. Vision, 26(1):41–61.

    Google Scholar 

  • Hartley, R. and Zisserman A. 2000. Multiple View Geometry in Computer Vision. Cambridge University Press.

  • Helmholtz, H. 1909. Physiological Optics. Dover. 1962 edition of the English translation of the 1909 German original, first published by the Optical Society of America in 1924.

  • Koenderink, J. 1984. What does the occluding contour tell us about solid shape? Perception, 13:321–330.

    Google Scholar 

  • Koenderink, J. 1990. Solid Shape. MIT Press: Cambridge, MA.

    Google Scholar 

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

    Google Scholar 

  • Koenderink, J. and Van Doorn, A. 1979. The internal representation of solid shape with respect to vision. Biological Cybernetics, 32:211–216.

    Google Scholar 

  • Lane, E. 1932. Projective Differential Geometry of Curves and Surfaces. The University of Chicago Press.

  • Laurentini, A. 1994. The visual hull concept for silhouette-based image understanding. IEEE Trans. Patt. Anal. Mach. Intell., 16(2):150–162.

    Google Scholar 

  • Laveau, S. and Faugeras, O. 1996. Oriented projective geometry for computer vision. In Proc. European Conf. Comp. Vision, pp. 147–156.

  • Lazebnik, S. 2002. Projective visual hulls. Master’s thesis, University of Illinois at Urbana-Champaign. Also Beckman CVR Technical Report CVR-TR-2002-01, available at http://www-cvr.ai.uiuc.edu/ponce_ grp.

  • Lazebnik, S., Boyer, E., and Ponce, J. 2001. On computing exact visual hulls of solids bounded by smooth surfaces. In Proc. IEEE Conf. Comp. Vision Patt. Recog., pp. 156–161.

  • Lazebnik, S. and Ponce, J. 2003. The local projective shape of smooth surfaces and their outlines. In Proc. Int. Conf. Comp. Vision. To appear.

  • Luong, Q.-T. and Faugeras, O. 1996. The fundamental matrix: Theory, algorithms, and stability analysis. Int. J. of Comp. Vision, 17(1):43–76.

    Google Scholar 

  • Moons, T., Pauwels, E., VanGool, L., and Oosterlinck, A. 1995. Foundations of semi-differential invariants. Int. J. of Comp. Vision, 14(1):25–47.

    Google Scholar 

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

    Google Scholar 

  • Mundy, J., Zisserman, A., and Forsyth, D. 1994. Applications of Invariance in Computer Vision, vol. 825 of Lecture Notes in Computer Science, Springer-Verlag.

  • Porrill, J. and Pollard, S. 1991. Curve matching and stereo calibration. Image and Vision Computing, 9(1):45–50.

    Google Scholar 

  • Rieger, J. 1986. Three-dimensional motion from fixed points of a deforming profile curve. Optics Letters, 11:123–125.

    Google Scholar 

  • Salden, A., Romeny, B.T., and Vierverger, M. 1993. Affine and projective differential geometric invariants of space curves. In Geometric Methods in Computer Vision II, B. Vemuri (Ed.), SPIE, pp. 60–74.

  • Stolfi, J. 1991. Oriented Projective Geometry: A Framework for Geometric Computations. Academic Press.

  • Vaillant, R. and Faugeras, O. 1992. Using extremal boundaries for 3D object modeling. IEEE Trans. Patt. Anal. Mach. Intell., 14(2):157–173.

    Google Scholar 

  • Weiss, I. 1988. Projective invariants of shapes. In Proc. IEEE Conf. Comp. Vision Patt. Recog., Ann Arbor, MI, pp. 291–297.

  • Werner, T. and Pajdla, T. 2001a. Cheirality in epipolar geometry. In ICCV, pp. 548–553.

  • Werner, T. and Pajdla, T 2001b. Oriented matching constraints. In Proc. British Machine Vision Conference, pp. 441–450.

  • Werner, T., Pajdla, T., and Hlavac, V. 1998. Oriented projective reconstruction. In Proc. meeting of the Austrian Association for Pattern Recognition, pp. 245–254.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Beckman Institute, University of Illinois, Urbana, IL, 61801, USA

    Svetlana Lazebnik & Jean Ponce

Authors
  1. Svetlana Lazebnik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean Ponce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Svetlana Lazebnik.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lazebnik, S., Ponce, J. The Local Projective Shape of Smooth Surfaces and Their Outlines. Int J Comput Vision 63, 65–83 (2005). https://doi.org/10.1007/s11263-005-4947-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-005-4947-4

Keywords

Search

Navigation