Advertisement

International Journal of Computer Vision

, Volume 65, Issue 1–2, pp 29–42 | Cite as

Detecting Codimension—Two Objects in an Image with Ginzburg-Landau Models

  • Gilles Aubert
  • Jean-François Aujol
  • Laure Blanc-Féraud
Article

Abstract

In this paper, we propose a new mathematical model for detecting in an image singularities of codimension greater than or equal to two. This means we want to detect isolated points in a 2-D image or points and curves in a 3-D image. We drew one's inspiration from Ginzburg-Landau (G-L) models which have proved their efficiency for modeling many phenomena in physics. We introduce the model, state its mathematical properties and give some experimental results demonstrating its capability in image processing.

Keywords

Ginzburg-Landau model points detection segmentation PDE biological images SAR images 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alberti, G., Baldo, S., and Orlandi, G. 2003. Variational convergence for functionals of Ginzburg-Landau type. Preprint.Google Scholar
  2. Ambrosio, L. and Soner, H.M. 1996. Level set approach to mean curvature flow in arbitrary dimension. Journal of Differential Geometry, 43.Google Scholar
  3. Aubert, G. and Blanc-Feraud, L. 1999. Some remarks on the equivalence between 2D and 3D classical snakes and geodesic active Contours. IJCV, 34(1):19–28.Google Scholar
  4. Aubert, G., Blanc-Féraud, L., and March, R. 2004. Γ-convergence of discrete functionals with non-convex perturbation for image classification, to appear in the SIAM. Journal on Numerical Analysis.Google Scholar
  5. Aubert, G. and Kornprobst, P. 2002. Mathematical problems in image processing, vol. 147 of Applied Mathematical Sciences, Springer-Verlag.Google Scholar
  6. Bethuel, F., Brezis, H., and Helein, F. 1994. Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and Their Applications. Birkhauser.Google Scholar
  7. Caselles, V., Catte, F., Coll, T., and Dibos, F. 1993. A geometric model for active contours. Numerische Mathematik 66:1–31.Google Scholar
  8. Caselles, V., Kimmel, R., and Sapiro, G. 1997. Geodesic active contours. IJCV 22(1):61–79.Google Scholar
  9. Chen, X.Y., Jimbo, S., and Morita, Y. 1998. Stabilization of vortices in the Ginzburg-Landau equation with variable diffusion coefficients. SIAM Journal Math. Anal.Google Scholar
  10. Gilboa, G., Zeevi, Y.Y., and Sochen, N. 2001. Complex diffusion processes for image filtering. In Scale-Space '01, vol. 2106 of Lecture Notes in Computer Science.Google Scholar
  11. Gilboa, G., Sochen, N.A., and Zeevi, Y.Y. 2004. Image enhancement and denoising by complex diffusion processes. IEEE Trans. Pattern Anal. Mach. Intell. 26(8):1020–1036.Google Scholar
  12. Ginzburg, V. and Landau, L. 1950. On the theory of superconductivity. Zheksper. Teo. Fiz, 20.Google Scholar
  13. Grossauer, H. and Scherzer, O. 2003. Using the complex Ginzburg-Landau equation for digital inpainting in 2D and 3D. In Scale-Space '03, vol. 1682 of Lecture Notes in Computer Science.Google Scholar
  14. C. Harris and M. Stephens 1988. A combined corner and edge detector. In Proceedings of the Fourth alvey vision conference. Manchester, pp. 147–151.Google Scholar
  15. Henderson L. 1998. Principle and Applications of Imaging Radar, vol. 2 of 3rd edition. J. Wiley and Sons.Google Scholar
  16. Lorigo, L., Faugeras, O., Grimson, W., Keriven, R., and Westin, C.F. 1999. Co-dimension-two geodesic active contours for the segmentation of tubular structures. In Int. Conf. Information Processing in Medical Imaging. Visegrad, Hungary.Google Scholar
  17. Malladi, R., Sethian, J.A., and Vemuri, B.C. 1994. Evolutionary fronts for topology-independent shape modeling and recovery. In ECCV 1994, vol. 800 of Lecture Notes in Computer Science, pp. 3–13.Google Scholar
  18. Osher, S. and Fedkiw, R.P. 2001. Level set methods: An overview and some recent results. Journal of Computational Physics 169:463–502.Google Scholar
  19. Ruuth, S., Merriman, B., Xin, J., and Osher, S. 1998. Diffusion-generated motion by mean curvatures for filaments, Technical Report 98-47, UCLA Computational and Applied Mathematics.Google Scholar
  20. Sussman, M., Smereka, P., and Osher, S. 1994. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics 114:146–159.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Gilles Aubert
    • 1
  • Jean-François Aujol
    • 1
    • 2
  • Laure Blanc-Féraud
    • 2
  1. 1.Laboratoire J.A. DieudonnéUMR CNRS 6621, Université de Nice Sophia-Antipolis, ParcNice Cedex 2France
  2. 2.ARIANAprojet commun CNRS/INRIA/UNSASophia Antipolis CedexFrance

Personalised recommendations