Using deformable surfaces to segment 3-D images and infer differential structures

  • Isaac Cohen
  • Laurent D. Cohen
  • Nicholas Ayache
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 588)


In this paper, we generalize the deformable model [4, 7] to a 3-D model, which evolves in 3-D images, under the action of internal forces (describing some elasticity properties of the surface), and external forces attracting the surface toward some detected edgels. Our formalism leads to the minimization of an energy which is expressed as a functional. We use a variational approach and a finite element method to actually express the surface in a discrete basis of continuous functions. This leads to a reduced computational complexity and a better numerical stability.

The power of the present approach to segment 3-D images is demonstrated by a set of experimental results on various complex medical 3-D images.

Another contribution of this approach is the possibility to infer easily the differential structure of the segmented surface. As we end-up with an analytical description of the surface, this allows to compute for instance its first and second fundamental forms. From this, one can extract a curvature primal sketch of the surface, including some intrinsic features which can be used as landmarks for 3-D image interpretation.


Finite Element Method Fundamental Form Conjugate Gradient Method Principal Curvature Deformable Model 
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.
    Isaac Cohen, Laurent D. Cohen, and Nicholas Ayache. Using deformable surfaces to segment 3-D images and infer differential structures. Computer Vision, Graphics, and Image Processing: Image Understanding, 1992. In press.Google Scholar
  2. 2.
    Laurent D. Cohen and Isaac Cohen. Finite element methods for active contour models and balloons from 2-D to 3-D. Technical Report 9124, CEREMADE, U.R.A. CNRS 749, Université Paris DC — Dauphine, November 1991. Cahiers de Mathematiques de la Decision.Google Scholar
  3. 3.
    A. Guéziec and N. Ayache. Smoothing and matching of 3D-space curves. In Proceedings of the Second European Conference on Computer Vision 199S, Santa Margherita Ligure, Italy, May 1992.Google Scholar
  4. 4.
    Michael Kass, Andrew Witkin, and Demetri Terzopoulos. Snakes: Active contour models. In Proceedings of the First International Conference on Computer Vision, pages 259–268, London, June 1987.Google Scholar
  5. 5.
    F. Leitner and P. Cinquin. Dynamic segmentation: Detecting complex topology 3D-object. In Proceedings of International Conference of the IEEE Engineering in Medicine and Biology Society, pages 295–296, Orlando, Florida, November 1991.Google Scholar
  6. 6.
    O. Monga, N. Ayache, and P. Sander. From voxel to curvature. In Proc. Computer Vision and Pattern Recognition, pages 644–649. IEEE Computer Society Conference, June 1991. Lahaina, Maui, Hawaii.Google Scholar
  7. 7.
    Demetri Terzopoulos, Andrew Witkin, and Michael Kass. Constraints on deformable models: recovering 3-D shape and nonrigid motion. AI Journal, 36:91–123, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Isaac Cohen
    • 1
  • Laurent D. Cohen
    • 2
  • Nicholas Ayache
    • 1
  1. 1.INRIALe Chesnay CedexFrance
  2. 2.CEREMADE, U.R.A. CNRS 749, Université Paris IX-DauphineParis CedexFrance

Personalised recommendations