Advertisement

Using the Vector Distance Functions to Evolve Manifolds of Arbitrary Codimension

  • José Gomes
  • Olivier Faugeras
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)

Abstract

We present a novel method for representing and evolving objects of arbitrary dimension. The method, called the Vector Distance Function (VDF) method, uses the vector that connects any point in space to its closest point on the object. It can deal with smooth manifolds with and without boundaries and with shapes of different dimensions. It can be used to evolve such objects according to a variety of motions, including mean curvature. If discontinuous velocity fields are allowed the dimension of the objects can change. The evolution method that we propose guarantees that we stay in the class of VDF’s and therefore that the intrinsic properties of the underlying shapes such as their dimension, curvatures can be read off easily from the VDF and its spatial derivatives at each time instant. The main disadvantage of the method is its redundancy: the size of the representation is always that of the ambient space even though the object we are representing may be of a much lower dimension. This disadvantage is also one of its strengths since it buys us flexibility.

Keywords

Normal Space IEEE Computer Society Smooth Manifold Ambient Space Active Contour 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Ambrosio and C. Mantegazza. Curvature and distance function from a manifold. J. Geom. Anal., 1996. To appear.Google Scholar
  2. 2.
    Luigi Ambrosio and Halil M. Soner. Level set approach to mean curvature flow in arbitrary codimension. J. of Diff. Geom., 43:693–737, 1996.MathSciNetzbMATHGoogle Scholar
  3. 3.
    V.I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag New York Inc., 1983.zbMATHCrossRefGoogle Scholar
  4. 4.
    M. Bertalmio, G. Sapiro, and G. Randall. Region Tacking on Surfaces Deforming via Level-Sets Methods. In Mads Nielsen, P. Johansen, O.F. Olsen, and J. Weickert, editors, Scale-Space Theories in Computer Vision, volume 1682 of Lecture Notes in Computer Science, pages 58–69. Springer, September 1999.CrossRefGoogle Scholar
  5. 5.
    Paul Burchard, Li-Tien Cheng, Barry Merriman, and Stanley Osher. Motion of curves in three spatial dimensions using a level set approach. Technical report, Department of Mathematics, University of California Los Angeles, September 2000.Google Scholar
  6. 6.
    V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. In Proceedings of the 5th International Conference on Computer Vision, pages 694–699, Boston, MA, June 1995. IEEE Computer Society, IEEE Computer Society Press.Google Scholar
  7. 7.
    Y.G. Chen, Y. Giga, and S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geometry, 33:749–786, 1991.MathSciNetzbMATHGoogle Scholar
  8. 8.
    M.P. DoCarmo. Riemannian Geometry. Birkhäuser, 1992.Google Scholar
  9. 9.
    L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, 1998.Google Scholar
  10. 10.
    M. Gage and R.S. Hamilton. The heat equation shrinking convex plane curves. J. of Differential Geometry, 23:69–96, 1986.MathSciNetzbMATHGoogle Scholar
  11. 11.
    J. Gomes and O. Faugeras. Shape representation as the intersection of n-k hypersurfaces. Technical Report 4011, INRIA, 2000.Google Scholar
  12. 12.
    M. Grayson. The heat equation shrinks embedded plane curves to round points. J. of Differential Geometry, 26:285–314, 1987.MathSciNetzbMATHGoogle Scholar
  13. 14.
    M. Kass, A. Witkin, and D. Terzopoulos. SNAKES: Active contour models. The International Journal of Computer Vision, 1:321–332, January 1988.CrossRefGoogle Scholar
  14. 15.
    S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi. Gradient flows and geometric active contour models. In Proceedings of the 5th International Conference on Computer Vision, Boston, MA, June 1995. IEEE Computer Society, IEEE Computer Society Press.Google Scholar
  15. 16.
    S. Osher and J. Sethian. Fronts propagating with curvature dependent speed: algorithms based on the Hamilton-Jacobi formulation. Journal of Computational Physics, 79:12–49, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 17.
    S.J. Ruuth, B. Merriman, J. Xin, and S. Osher. Difusion-Generated Motion by Mean Curvature for filaments. Technical Report 98-47, UCLA Computational and Applied Mathematics Reports, November 1998.Google Scholar
  17. 18.
    S.J. Ruuth, B. Merriman, and S. Osher. A fixed grid method for capturing the motion of self-intersecting interfaces and related pdes. Technical Report 99-22, UCLA Computational and Applied Mathematics Reports, July 1999.Google Scholar
  18. 19.
    Michael Spivak. A Comprehensive Introduction to Differential Geometry, volume I-V. Publish or Perish, Berkeley, CA, 1979. Second edition.Google Scholar
  19. 20.
    J. Steinhoff, M. Fan, and L. Wang. A new eulerian method for the computation of propagating short acoustic and electromagnetic pulses. Journal of Computational Physics, 157:683–706, 2000.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • José Gomes
    • 1
  • Olivier Faugeras
    • 2
  1. 1.I.B.M Watson Research CenterNew YorkUSA
  2. 2.I.N.R.I.A Sophia Antipolis, France and M.I.TBostonUSA

Personalised recommendations