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Equalized Net Diffusion (END) for the Preservation of Fine Structures in PDE-based Image Restoration

  • Cha, Youngjoon (Department of Applied Mathematics, Sejong University) ;
  • Kim, Seongjai (Department of Mathematics and Statistics, Mississippi State University)
  • 투고 : 2013.11.21
  • 심사 : 2013.12.09
  • 발행 : 2013.12.31

초록

The article is concerned with a mathematical modeling which can improve performances of PDE-based restoration models. Most PDE-based restoration models tend to lose fine structures due to certain degrees of nonphysical dissipation. Sources of such an undesirable dissipation are analyzed for total variation-based restoration models. Based on the analysis, the so-called equalized net diffusion (END) modeling is suggested in order for PDE-based restoration models to significantly reduce nonphysical dissipation. It has been numerically verified that the END-incorporated models can preserve and recover fine structures satisfactorily, outperforming the basic models for both quality and efficiency. Various numerical examples are shown to demonstrate effectiveness of the END modeling.

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참고문헌

  1. L. Alvarez, P. Lions, and M. Morel, "Image selective smoothing and edge detection by nonlinear diffusion. II," SIAM J. Numerical Anal., vol. 29, no. 3, pp. 845-866, June 1992. https://doi.org/10.1137/0729052
  2. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Series: Applied Math. Sci., vol. 147, Springer-Verlag, 2002.
  3. P. V. Blomgren and T. F. Chan, "Color TV: Total variation methods for restoration of vector valued images," IEEE Trans. Image Process., vol. 7, no. 3, pp. 304-309, Mar. 1998. https://doi.org/10.1109/83.661180
  4. F. Catte, P. Lions, M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM J. Numerical Anal., vol. 29, no. 3, pp. 182-193, June 1992. https://doi.org/10.1137/0729012
  5. Y. Cha and S. Kim, "Diffusion and constraint terms in PDE-based image restoration, zooming, deconvolution, and segmentation," in preparation.
  6. Y. Cha and S. Kim, "Edge-forming methods for color image zooming," IEEE Trans. Image Process., vol. 15, no. 8, pp. 2315-2323, Aug. 2006. https://doi.org/10.1109/TIP.2006.875182
  7. Y. Cha and S. Kim, "Edge-forming methods for image zooming," J. Math. Imaging Vision, vol. 25, no. 3, pp. 353-364, Oct. 2006. https://doi.org/10.1007/s10851-006-7250-2
  8. Y. Cha and S. Kim, "The method of nonflat time evolution (MONTE) in PDE-based image restoration," J. Korea Inform. Commun. Soc. (KICS), vol. 37A, no. 11, pp. 961-971, Nov. 2012. https://doi.org/10.7840/kics.2012.37A.11.961
  9. T. Chan, S. Osher, and J. Shen, "The digital TV filter and nonlinear denoising," IEEE Trans. Image Process., vol. 10, no. 2, pp. 231-241, Feb. 2001. https://doi.org/10.1109/83.902288
  10. T. F. Chan and J. Shen, "Variational restoration of non-flat image features: Models and algorithms," SIAM J. Appl. Math., vol. 61, no. 4, pp. 1338-1361, Jan. 2001. https://doi.org/10.1137/S003613999935799X
  11. J. Douglas, Jr. and J. Gunn, "A general formulation of alternating direction methods Part I. Parabolic and hyperbolic problems," Numerical Math., vol. 6, no. 1, pp. 428-453, Dec. 1964. https://doi.org/10.1007/BF01386093
  12. J. Douglas, Jr. and S. Kim, "Improved accuracy for locally one-dimensional methods for parabolic equations," Math. Models Methods Appl. Sci., vol. 11, no. 9, pp. 1563-1579, Dec. 2001. https://doi.org/10.1142/S0218202501001471
  13. R. Gonzalez and R. Woods, Digital Image Processing, 2nd Ed., Prentice-Hall, 2002.
  14. S. Kim, "Equalized net diffusion (END) in image denoising," in Proc. 10th WSEAS Int. Conf. Appl. Math., pp. 349-354, Istanbul, Turkey, May 2006.
  15. S. Kim, "PDE-based image restoration: A hybrid model and color image denoising," IEEE Trans. Image Process., vol. 15, no. 5, pp. 1163-1170, May 2006. https://doi.org/10.1109/TIP.2005.864184
  16. S. Kim and H. Lim, "A non-convex diffusion model for simultaneous image denoising and edge enhancement," Electron. J. Differential Equations (EJDE), Conference Special Issue, vol. 15, pp. 175-192, Feb. 2007.
  17. R. Kimmel and N. Sochen, "Orientation diffusion or how to comb a porcupine?" J. Visual Commun. Image Representation, vol. 13, no. 1-2, pp. 238-248, Mar. 2002. https://doi.org/10.1006/jvci.2001.0501
  18. A. Marquina and S. Osher, "Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal," SIAM J. Sci. Comput., vol. 22, no. 2, pp. 387-405, Aug. 2000. https://doi.org/10.1137/S1064827599351751
  19. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Series: University Lecture Series, vol. 22, American Mathematical Society, 2001.
  20. S. Mitra and G. Sicuranza, Nonlinear Image Processing, Academic Press, 2001.
  21. J.-M. Morel and S. Solimini, Variational Methods in Image Segmentation, Series: Progress in Nonlinear Differential Equations and Their Applications, vol. 14, Birkhauser, 1995.
  22. M. Nitzberg and T. Shiota, "Nonlinear image filtering with edge and corner enhancement," IEEE Trans. Pattern Anal. Mach. Intell., vol. 14, no. 8, pp. 826-833, Aug. 1992. https://doi.org/10.1109/34.149593
  23. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, "Using geometry and iterated refinement for inverse problems (1): Total variation based image restoration," CORC Technical Report 2004-03, Mar. 2004.
  24. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, 2003.
  25. P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Mach. Intell., vol. 12, no. 7, pp. 629-639, July 1990. https://doi.org/10.1109/34.56205
  26. L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D: Nonlinear Phenomena, vol. 60, no. 1-4, pp. 259-268, Nov. 1992. https://doi.org/10.1016/0167-2789(92)90242-F
  27. G. Sapiro, Geometric partial differential equations and image analysis, Cambridge University Press, 2001.
  28. N. Sochen, R. Kimmel, and R. Malladi, "A general framework for low level vision," IEEE Trans. Image Process., vol. 7, no. 3, pp. 310-318, Mar. 1998. https://doi.org/10.1109/83.661181
  29. R. Weinstock, Calculus of Variations, Dover Pubilcations, 1974.
  30. Y.-L. You, W. Xu, A. Tannenbaum, and M. Kaveh, "Behavioral analysis of anisotropic diffusion in image processing," IEEE Trans. Image Process., vol. 5, no. 11, pp. 1539-1553, Nov. 1996. https://doi.org/10.1109/83.541424