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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
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.
Caselles, V., Kimmel, R., and Sapiro, G. 1997. Geodesic active contours. IJCV 22(1):61–79.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
First online version published in October, 2005
Author is now with CMLA (CNRS UMR 8536), ENS Cachan, France.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-005-3847-y