Two Enhanced Fourth Order Diffusion Models for Image Denoising

1. Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge-detection by non-linear diffusion II. SIAM J. Numer. Anal. 29(3), 845–866 (1992)

   Article  MathSciNet  MATH  Google Scholar 

2. Amann, H.: Time-delayed Perona-Malik problems. Acta Math. Univ. Comen. LXXVI, 15–38 (2007)

   MathSciNet  Google Scholar 

3. Bai, J., Feng, X.C.: Fractional order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)

   Article  MathSciNet  Google Scholar 

4. Belahmidi, A.: Equations aux dérivées partielles appliquées à la restoration et à l’agrandissement des images. Ph.D. Thesis. Université Paris-Dauphine, Paris (2003)

5. Belahmidi, A., Chambolle, A.: Time-delay regularization of anisotropic diffusion and image processing. Modél. Math. Anal. Numér. 39(2), 231–251 (2005)

   Article  MathSciNet  MATH  Google Scholar 

6. Bertozzi, A., Greer, J.: Low-curvature image simplifiers: global regularity of smooth solutions and Laplacian limiting schemes. Commun. Pure Appl. Math. LVII, 0764–0790 (2004)

   MathSciNet  Google Scholar 

7. Buades, A., Coll, B., Morel, J.: A review of image denoising algorithms with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

   Article  MathSciNet  MATH  Google Scholar 

8. Catté, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge-detection by non-linear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

   Article  MathSciNet  MATH  Google Scholar 

9. Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000) (electronic). doi:10.1137/S1064827598344169

   Article  MathSciNet  MATH  Google Scholar 

10. Chan, T.F., Esedoglu, S., Park, F.E.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. UCLA CAM report 05-28 (April 2005)

11. Chen, Y., Bose, P.: On the incorporation of time-delay regularization into curvature-based diffusion. J. Math. Imaging Vis. 14(2), 149–164 (2001)

    Article  MathSciNet  Google Scholar 

12. Cottet, G.H., Ayyadi, M.E.: A Volterra type model for image processing. IEEE Trans. Image Process. 7, 292–303 (1998)

    Article  Google Scholar 

13. Didas, S., Burgeth, B., Imiya, A., Weickert, J.: Regularity and scale-space properties of fractional high order linear filtering. In: Scale Space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3459. Springer, Berlin (2005)

    Chapter  Google Scholar 

14. Didas, S., Weickert, J., Burgeth, B.: Stability and local feature enhancement of higher order nonlinear diffusion filtering. Pattern Recognit. 3663, 451–458 (2005)

    Article  Google Scholar 

15. Didas, S., Weickert, J., Burgeth, B.: Properties of higher order nonlinear diffusion filtering. J. Math. Imaging Vis. 35(3), 208–226 (2009). doi:10.1007/s10851-009-0166-x

    Article  MathSciNet  Google Scholar 

16. Guidotti, P.: A new nonlocal nonlinear diffusion of image processing. J. Differ. Equ. 246(12), 4731–4742 (2009)

    Article  MathSciNet  MATH  Google Scholar 

17. Guidotti, P.: A new well-posed nonlinear nonlocal diffusion. Nonlinear Anal. 72, 4625–4637 (2010)

    Article  MathSciNet  MATH  Google Scholar 

18. Guidotti, P., Lambers, J.: Two new nonlinear nonlocal diffusions for noise reduction. J. Math. Imaging Vis. 33(1), 25–37 (2009)

    Article  MathSciNet  Google Scholar 

19. Guidotti, P., Longo, K.: Well-posedness for a class of fourth order diffusions for image processing. Nonlinear Differ. Equ. Appl. (to appear)

20. Hajiaboli, M.: A self-governing hybrid model for noise removal. In: Advances in Image and Video Technology. Lecture Notes in Computer Science, vol. 5414. Springer, Berlin (2008)

    Google Scholar 

21. Hajiaboli, M.: An anisotropic fourth-order partial differential equation for noise removal. In: Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 5567. Springer, Berlin (2009)

    Chapter  Google Scholar 

22. Kichenassamy, S.: The Perona-Malik paradox. SIAM J. Appl. Math. 57(5), 1328–1342 (1997)

    Article  MathSciNet  MATH  Google Scholar 

23. Lambers, J.V.: Enhancement of Krylov subspace spectral methods by block Lanczos iteration. Electron. Trans. Numer. Anal. 31, 86–109 (2008)

    MathSciNet  MATH  Google Scholar 

24. Li, F., Shen, C., Fan, J., Shen, C.: Image restoration combining a total variational filter and a fourth-order filter. J. Vis. Commun. Image Represent. 18(4), 322–330 (2007)

    Article  Google Scholar 

25. Lysaker, M., Lundervold, A., Tai, X.: Noise removal using fourth order differential equations with applications to medical magnetic resonance images in space-time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  Google Scholar 

26. Mathieu, B., Melchior, P., Outstaloup, A., Ceyral, C.: Fractional differentiation for edge detection. Signal Process. 83, 2421–2432 (2003)

    Article  MATH  Google Scholar 

27. Nitzberg, M., Shiota, T.: Nonlinear image smoothing with edge and corner enhancement. Tech. Report 90-2. Harvard University, Cambridge, MA (1990)

28. Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14, 826–833 (1992)

    Article  Google Scholar 

29. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 161–192 (1990)

    Article  Google Scholar 

30. Tumblin, J., Turk, G.: LCIS: A boundary hierarchy for detail-preserving contrast reduction. In: Proceedings of the SIGGRAPH 1999 Annual Conference on Computer Graphics, August 8–13, 1999, Los Angeles, CA, USA, Siggraph Annual Conference Series, pp. 83–90. ACM Siggraph, Addison-Wesley, Longman, Harlow (1999)

    Google Scholar 

31. Wei, G.: Generalized Perona-Malik equation for image restoration. IEEE Signal Process. Lett. 6(7), 165–167 (1999)

    Article  Google Scholar 

32. Weickert, J.: Anisotropic Diffusion in Image Processing. ECMI Series. Teubner Verlag, Stuttgart (1998)

    MATH  Google Scholar 

33. Witkin, A.P.: Scale-space filtering. In: Proc. IJCAI, Karlsruhe, pp. 1021–1019 (1983)

    Google Scholar 

34. You, Y., Kaveh, M.: Fourth order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)

    Article  MathSciNet  MATH  Google Scholar 

35. You, Y., Xu, W., Tannenbaum, A., Kaveh, M.: Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Process. 5(11), 1539–1553 (1996)

    Article  Google Scholar 

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