Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2010, Article ID 761783, 19 pages
http://dx.doi.org/10.1155/2010/761783
Research Article

An Overview of the Topological Gradient Approach in Image Processing: Advantages and Inconveniences

1Dammam University, Faculty of Sciences, PO Box 838, 31113 Dammam, Saudi Arabia
2ENIT-LAMSIN, Université de Tunis El Manar, BP 37, 1002 Tunis-Bélvédère, Tunisia

Received 27 July 2010; Accepted 21 November 2010

Academic Editor: Ke Chen

Copyright © 2010 Lamia Jaafar Belaid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Amstutz, I. Horchani, and M. Masmoudi, “Crack detection by the topological gradient method,” Control and Cybernetics, vol. 34, no. 1, pp. 81–101, 2005. View at Google Scholar · View at Zentralblatt MATH
  2. D. Auroux and M. Masmoudi, “A one-shot inpainting algorithm based on the topological asymptotic analysis,” Computational & Applied Mathematics, vol. 25, no. 2-3, pp. 251–267, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. D. Auroux, L. Jaafar Belaid, and M. Masmoudi, “Image restoration and classification by topological asymptotic expansion,” in Variational Formulations in Mechanics: Theory and Applications, E. Taroco, E. A. de Souza Neto, and A. A. Novotny, Eds., pp. 23–42, Catalonia, Spain, CIMNE, 2006. View at Google Scholar
  4. D. Auroux, L. Jaafar Belaid, and M. Masmoudi, “A topological asymptotic analysis for the regularized grey-level image classification problem,” Mathematical Modelling and Numerical Analysis, vol. 41, no. 3, pp. 607–625, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. L. Jaafar Belaid, M. Jaoua, M. Masmoudi, and L. Siala, “Image restoration and edge detection by topological asymptotic expansion,” Comptes Rendus Mathématique, vol. 342, no. 5, pp. 313–318, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. L. Jaafar Belaid, M. Jaoua, M. Masmoudi, and L. Siala, “Application of the topological gradient to image restoration and edge detection,” Engineering Analysis with Boundary Elements, vol. 32, no. 11, pp. 891–899, 2008. View at Publisher · View at Google Scholar
  7. M. Masmoudi, “The topological asymptotic,” in Computational Methods for Control Applications, R. Glowinski, H. Karawada, and J. Periaux, Eds., vol. 16 of GAKUTO International Series. Mathematical Sciences and Applications, pp. 53–72, GAKUTO, Tokyo, Japan, 2001. View at Google Scholar
  8. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, vol. 147 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 2001.
  9. S. Di Zenzo, “A note on the gradient of a multi-image,” Computer Vision, Graphics, and Image Processing, vol. 33, no. 1, pp. 116–125, 1986. View at Google Scholar
  10. A. Brook, R. Kimmel, and N. A. Sochen, “Variational restoration and edge detection for color images,” Journal of Mathematical Imaging and Vision, vol. 18, no. 3, pp. 247–268, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. R. Kimmel, R. Malladi, and N. Sochen, “Image processing via the Beltrami operator,” in Proceedings of the 3rd Asian Conference on Computer Vision, pp. 574–581, Hong Kong, 1998.
  12. N. A. Sochen, G. Gilboa, and Y. Y. Zeevi, “Color image enhancement by a forward-and-backward adaptive Beltrami °ow,” in Proceedings of the International Workshop on Algebraic Frames for the Perception-Action Cycle (AFPAC '00), G. Sommer and Y. Y. Zeevi, Eds., vol. 1888 of Lecture Notes in Computer Science, pp. 319–328, Springer, 2000.
  13. D. Auroux, L. Jaafar Belaid, and B. Rjaibi, “Application of the topological gradient method to color image restoration,” SIAM Journal on Imaging Sciences, vol. 3, no. 2, pp. 153–175, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, “A variational model for image classification and restoration,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 5, pp. 460–472, 2000. View at Google Scholar
  15. T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 503–516, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. Serra, Image Analysis and Mathematical Morphology I, Academic Press, London, UK, 1984.
  17. J. Serra, Image Analysis and Mathematical Morphology II: Theoretical Advances, Academic Press, London, UK, 1984.
  18. L. Vincent and P. Soille, “Watersheds in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 6, pp. 583–598, 1991. View at Publisher · View at Google Scholar
  19. L. Jaafar Belaid and W. Mourou, “Image segmentation: a watershed transformation algorithm,” Image Analysis & Stereology, vol. 28, no. 2, pp. 93–102, 2009. View at Google Scholar · View at Zentralblatt MATH
  20. M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: an active contour models,” International Journal of Computer Vision, vol. 1, pp. 133–144, 1987. View at Google Scholar
  21. V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” International Journal of Computer Vision, vol. 22, no. 1, pp. 61–79, 1997. View at Google Scholar