Imposing hard constraints on soft snakes

  • P. Fua
  • C. Brechbühler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1065)


An approach is presented for imposing generic hard constraints on deformable models at a low computational cost, while preserving the good convergence properties of snake-like models. We believe this capability to be essential not only for the accurate modeling of individual objects that obey known geometric and semantic constraints but also for the consistent modeling of sets of objects.

Many of the approaches to this problem that have appeared in the vision literature rely on adding penalty terms to the objective functions. They rapidly become untractable when the number of constraints increases. Applied mathematicians have developed powerful constrained optimization algorithms that, in theory, can address this problem. However, these algorithms typically do not take advantage of the specific properties of snakes. We have therefore designed a new algorithm that is tailored to accommodate the particular brand of deformable models used in the Image Understanding community.

We demonstrate the validity of our approach first in two dimensions using synthetic images and then in three dimensions using real aerial images to simultaneously model terrain, roads, and ridgelines under consistency constraints.


Hard Constraint Deformable Model Newton Step Gradient Projection Method Semantic Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A.A. Amini, S. Tehrani, and T.E. Weymouth. Using Dynamic Programming for Minimizing the Energy of Active Contours in the Presence of Hard Constraints. In International Conference on Computer Vision, pages 95–99, 1988.Google Scholar
  2. 2.
    C. Brechbühler, G. Gerig, and O. Kübler. Parametrization of Closed Surfaces for 3-D Shape Description. Computer Vision, Graphics, and Image Processing: Image Understanding, 61(2):154–170, March 1995.Google Scholar
  3. 3.
    L. Cohen. Auxiliary Variables for Deformable Models. In International Conference on Computer Vision, pages 975–980, Cambridge, MA, June 1995.Google Scholar
  4. 4.
    R. Fletcher. Practical Methods of Optimization. John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore, 2nd edition, 1987. A Wiley-Interscience Publication.Google Scholar
  5. 5.
    P. Fua. Parametric Models are Versatile: The Case of Model Based Optimization. In ISPRS WG III/2 Joint Workshop, Stockholm, Sweden, September 1995.Google Scholar
  6. 6.
    P. Fua and Y. G. Leclerc. Model Driven Edge Detection. Machine Vision and Applications, 3:45–56, 1990.Google Scholar
  7. 7.
    P. Fua and Y. G. Leclerc. Object-Centered Surface Reconstruction: Combining Multi-Image Stereo and Shading. International Journal of Computer Vision, 16:35–56, September 1995.Google Scholar
  8. 8.
    P.E. Gill, W. Murray, and M.H. Wright. Practical Optimization. Academic Press, London a.o., 1981.Google Scholar
  9. 9.
    M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active Contour Models. International Journal of Computer Vision, 1(4):321–331, 1988.Google Scholar
  10. 10.
    D. Metaxas and D. Terzopoulos. Shape and Norigid Motion Estimation through Physics-Based Synthesis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(6):580–591, 1991.Google Scholar
  11. 11.
    Rosen. Gradient projection method for nonlinear programming. SIAM Journal of Applied Mathematics, 8:181–217, 1961.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • P. Fua
    • 1
  • C. Brechbühler
    • 2
  1. 1.SRI InternationalMenlo Park
  2. 2.ETH-ZürichZürichSwitzerland

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