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

International Journal of Computer Vision volume 65, pages 29–42 (2005)Cite this 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.

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References

  • Alberti, G., Baldo, S., and Orlandi, G. 2003. Variational convergence for functionals of Ginzburg-Landau type. Preprint.

  • Ambrosio, L. and Soner, H.M. 1996. Level set approach to mean curvature flow in arbitrary dimension. Journal of Differential Geometry, 43.

  • 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.

  • 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.

  • Aubert, G. and Kornprobst, P. 2002. Mathematical problems in image processing, vol. 147 of Applied Mathematical Sciences, Springer-Verlag.

  • Bethuel, F., Brezis, H., and Helein, F. 1994. Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and Their Applications. Birkhauser.

  • Caselles, V., Catte, F., Coll, T., and Dibos, F. 1993. A geometric model for active contours. Numerische Mathematik 66:1–31.

    Google Scholar 

  • Caselles, V., Kimmel, R., and Sapiro, G. 1997. Geodesic active contours. IJCV 22(1):61–79.

    Google Scholar 

  • 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.

  • 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.

  • 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 

  • Ginzburg, V. and Landau, L. 1950. On the theory of superconductivity. Zheksper. Teo. Fiz, 20.

  • 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.

  • C. Harris and M. Stephens 1988. A combined corner and edge detector. In Proceedings of the Fourth alvey vision conference. Manchester, pp. 147–151.

  • Henderson L. 1998. Principle and Applications of Imaging Radar, vol. 2 of 3rd edition. J. Wiley and Sons.

  • 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.

  • 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.

  • 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 

  • 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.

  • 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 

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Authors and Affiliations

  1. Laboratoire J.A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc, Valrose, 06108, Nice Cedex 2, France

    Gilles Aubert & Jean-François Aujol

  2. ARIANA, projet commun CNRS/INRIA/UNSA, INRIA Sophia Antipolis 2004, route des Lucioles, BP93, 06902, Sophia Antipolis Cedex, France

    Jean-François Aujol & Laure Blanc-Féraud

Authors
  1. Gilles Aubert
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  2. Jean-François Aujol
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  3. Laure Blanc-Féraud
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Corresponding author

Correspondence to Jean-François Aujol.

Additional information

First online version published in October, 2005

Author is now with CMLA (CNRS UMR 8536), ENS Cachan, France.

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Aubert, G., Aujol, JF. & Blanc-Féraud, L. Detecting Codimension—Two Objects in an Image with Ginzburg-Landau Models. Int J Comput Vision 65, 29–42 (2005). https://doi.org/10.1007/s11263-005-3847-y

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  • DOI: https://doi.org/10.1007/s11263-005-3847-y

Keywords

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