Table of Contents
ISRN Signal Processing
Volume 2013, Article ID 605035, 18 pages
http://dx.doi.org/10.1155/2013/605035
Research Article

About a Partial Differential Equation-Based Interpolator for Signal Envelope Computing: Existence Results and Applications

1Département Informatique et Télécommunications, Ecole Polytechnique de Thiès (EPT), Thiès BP A10, Senegal
2Laboratoire Images, Signaux et Systèmes Intelligents (LISSI-E.A.3956), Université Paris-Est Créteil Val-de-Marne, Créteil, France
3Laboratoire d’Analyse Numérique et d’Informatique (LANI), Université Gaston Berger (UGB), Saint-Louis BP 234, Senegal
4Département de Mathématique et Informatique, Faculté des Sciences et Technique, Université Cheikh Anta Diop de Dakar, Dakar BP 5005, Senegal

Received 6 November 2012; Accepted 3 December 2012

Academic Editors: H. Hu and S. Li

Copyright © 2013 Oumar Niang et al. 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. N. E. Huang, Z. Shen, S. R. Long et al., “The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proceedings of the Royal Society A, vol. 545, no. 1971, pp. 903–995, 1998. View at Google Scholar
  2. E. Deléchelle, J. Lemoine, and O. Niang, “Empirical mode decomposition: an analytical approach for sifting process,” IEEE Signal Processing Letters, vol. 12, no. 11, pp. 764–767, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. O. Niang, Empirical mode decomposition: contribution à la modélisation mathématique et application en traitement du signal et l'image [Ph.D. thesis], University Paris-Est Créteil, Paris, France, 2007.
  4. O. Niang, E. Deléchelle, and J. Lemoine, “A spectral approach for sifting process in empirical mode decomposition,” IEEE Transactions on Signal Processing, vol. 58, no. 11, pp. 5612–5623, 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. O. Niang, A. Thioune, E. Delechelle, and J. Lemoine, “Spectral intrinsic decomposition method for adaptive signal representation,” ISRN Signal Processing, vol. 2012, Article ID 457152, 10 pages, 2012. View at Publisher · View at Google Scholar
  6. O. Niang, A. Thioune, M. C. El Gueirea, E. Deléchelle, and J. Lemoine, “Partial differential equation-based approach for empirical mode decomposition: application on image analysis,” IEEE Transactions on Image Processing, vol. 21, no. 9, pp. 3991–4001, 2012. View at Publisher · View at Google Scholar
  7. F. Catte, P. L. Lions, J. M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM Journal on Numerical Analysis, vol. 29, no. 1, pp. 182–193, 1992. View at Google Scholar · View at Scopus
  8. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990. View at Publisher · View at Google Scholar · View at Scopus
  9. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22 of University Lecture Series, AMS, Providence, RI, USA, 2002.
  10. S. Osher, A. Sole, and L. Vese, “Image decomposition and restoration using total variation minimization and the H1 norm,” Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, vol. 1, no. 3, pp. 349–3370, 2003. View at Google Scholar
  11. M. Lysaker, A. Lundervold, and X. C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time,” IEEE Transactions on Image Processing, vol. 12, no. 12, pp. 1579–1589, 2003. View at Publisher · View at Google Scholar · View at Scopus
  12. J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Transactions on Image Processing, vol. 7, no. 3, pp. 398–410, 1998. View at Google Scholar · View at Scopus
  13. J. Weickert, “A review of nonlinear diffusion filtering,” in Scale-Space Theory for Computer Vision, B. H. Romeny, Ed., vol. 1252 of Lecture Notes in Computer Science, pp. 3–28, Springer, New York, NY, USA, 1997. View at Google Scholar
  14. D. W. Peaceman and H. H. Rachford, “The numerical solution of parabolic and elliptic differential equations,” Journal of the Society For Industrial and Applied Mathematics, vol. 3, no. 1, pp. 28–41, 1955. View at Google Scholar
  15. D. Barash and R. Kimmel, An Accurate Operator Splitting Scheme for Nonlinear Diffusion Filter, HP Company, 2000.
  16. J. Tumblin and G. Turk, “LCIS: a boundary hierarchy for detail-preserving contrast reduction,” in Proceedings of the SIGGRAPH 1999 Annual Conference on Computer Graphics, Los Angeles, Calif, USA, 1999.
  17. G. W. Wei, “Generalized Perona-Malik equation for image restoration,” IEEE Signal Processing Letters, vol. 6, no. 7, pp. 165–167, 1999. View at Publisher · View at Google Scholar · View at Scopus
  18. Y. L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Transactions on Image Processing, vol. 9, no. 10, pp. 1723–1730, 2000. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Tumblin, Private Communication, 2003.
  20. H. Brezis, Analyse fonctionnelle. Théorie et Applications, Masson, Paris, France, 1983.
  21. J. L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Springer, Berlin, Germany, 1961.
  22. J. D. Murray, Mathematical Biology, Springer, Berlin, Germany, 1993.
  23. P. Flandrin, G. Rilling, and P. Gonçalvés, “Empirical mode decomposition as a filter bank,” IEEE Signal Processing Letters, vol. 11, no. 2, pp. 112–114, 2004. View at Google Scholar
  24. R. Sperb, Maximum Principle and Their Applications, vol. 157 of Mathematics in Science and Engineering Series, Academic Press, New York, NY, USA, 1981.
  25. H. Brezis, Operateurs Maximaux monotones et semigropes de contraction dans les espaces de Hilbert, North-Holland, Amsterdam, The Netherlands, 1972.
  26. G. Aubert and P. Kornprobst, Mathematical Problems in Images Processing, Partial Differential Equations and the Calculus of Variations, vol. 147 of Applied Mathematical Sciences Series, Springer, Berlin, Germany, 2002.
  27. C. Vogel and M. Oman, “Iteration methods for total variation denoising,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 227–238, 1996. View at Publisher · View at Google Scholar
  28. G. Engeln-Muellges and F. Uhlig, Numerical Algorithms with C, chapter 4, Springer, Berlin, Germany, 1996.
  29. T. P. Witelski and M. Bowen, “ADI schemes for higher-order nonlinear diffusion equations,” Applied Numerical Mathematics, vol. 45, no. 2-3, pp. 331–351, 2003. View at Google Scholar