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Image Denoising Using Generalized Anisotropic Diffusion

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Abstract

Motivated by some recent works in fractional-order anisotropic diffusions, we introduce a class of generalized anisotropic diffusion equations for image denoising. We first define a new type of derivative (called G-derivative), which contains the fractional derivative as a special case, using the Fourier transform, then the generalized anisotropic diffusion equations are Euler–Lagrange equations of a cost functional which is an increasing function of the absolute value of the G-derivative of the image intensity function. All the G-derivative operators constitute a ring, and the semigroup property of the G-derivative consists with the semigroup property of the fractional derivative, so the resulting generalized anisotropic diffusions can be seen as generalizations of the fractional-order anisotropic diffusions. We also discuss the generalized Sobolev space described by the G-derivative and some variants of generalized anisotropic diffusions. The discretization of the G-derivative is computed in the frequency domain, and the stability analysis of the difference scheme is given. We list some generalized anisotropic diffusions and apply them to image denoising. Numerical results show that new models have great potentials in image denoising.

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References

  1. Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Tran. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  2. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Lysaker, M., Osher, S., Tai, X.C.: Noise removal using smoothed normals and surface fitting. IEEE Trans. Image Process. 13(10), 1345–1357 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Carmona, R.A., Zhong, S.: Adaptive smoothing respecting feature directions. IEEEE Trans. Image Process. 7(3), 353–358 (1998)

    Article  Google Scholar 

  7. Lysaker, M., Tai, X.C.: Iterative image restoration combined total variation minimization and a second-order functional. Int. J. Comput. Vis. 66(1), 5–18 (2006)

    Article  MATH  Google Scholar 

  8. Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhang, J., Wei, Z., Xiao, L.: Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43(1), 39–49 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Janev, M., Pilipovic, S., Atanaokovic, T., Obradovic, P., Ralevic, N.: Fully fractional anisotropic diffusion for image denoising. Math. Comput. Model. 54(1–2), 729–741 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ren, Z., He, C., Zhang, Q.: Fractional order total variation regularization for image super resolution. Signal Process. 93(9), 2408–2421 (2013)

    Article  Google Scholar 

  12. Zhang, J., Chen, K.: A total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution. SIAM J. Imaging Sci. 8(4), 2487–2518 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, J., Wei, Z.: A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising. Appl. Math. Model. 35(5), 2516–2528 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Y., Pu, Y.F., Hu, J.R., Zhou, J.L.: A class of fractional-order variational image inpainting models. Appl. Math. Inf. Sci. 6(2), 299–306 (2012)

    MathSciNet  Google Scholar 

  15. Zhang, Y., Cheng, H.D., Tian, J., Huang, J., Tang, X.: Fractional subpixel diffusion and fuzzy logic approach for ultrasound speckle reduction. Pattern Recognit. 43(8), 2962–2970 (2010)

    Article  MATH  Google Scholar 

  16. Lee, S.H., Seo, J.K.: Noise removal with Gauss curvature-driven diffusion. IEEE Trans. Image Process. 14(7), 904–909 (2005)

    Article  MathSciNet  Google Scholar 

  17. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ren, A.Z., He, C., Li, M.: Fractional order bidirectional diffusion for image up-sampling. J. Electro Imaging 21(3), 3006–3018 (2012)

    Google Scholar 

  20. Zhang, Y.T., Cheng, H.D., Chen, Y.Q., Huang, J.: A novel noise removal method based on fractional anisotropic diffusion and subpixel approach. New Math. Nat. Comput. 7(1), 173–185 (2011)

    Article  MATH  Google Scholar 

  21. Dong, F., Chen, Y.: A fractional-order derivative based variational framework for image denoising. Inverse Probl. Imaging 10(1), 27–50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, D., Sun, S., Zhang, C., Chen, Y.Q., Xue, D.: Fractional-order TV-L2 model for image denoising. Cent. Eur. J. Phy. 11(10), 1414–1422 (2013)

    Google Scholar 

  23. Ross, B.: A brief history and exposition of fundamental theory of fractional calculus (vol. 457, pp. 40–130 ). In: Lecture Notes in Math. Springer, New York (1975)

  24. Adams, R. A.: Sobolev space. In Pure and Applied Mathematics, Ser. Monograph and Textbooks (vol. 65). Academic, New York (1975)

  25. Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal. 29(3), 845–866 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Osher, S., Sole, A., Fatemi, E.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\) norm. Multiscale Model. Simul. 1(3), 349–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Osher, S., Rudin, L.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27(4), 919–940 (1990)

    Article  MATH  Google Scholar 

  28. Chen, Q., Montesinos, P., Sun, Q.S., Heng, P.A., Xia, D.S.: Adaptive total variation denoising based on difference curvature. Image Vis. Comput. 28(3), 298–306 (2010)

    Article  Google Scholar 

  29. Chan, R.H., Lanza, A., Morigi, S., Sgallari, F.: An adaptive strategy for the restoration of textured images using fractional order regulation. Numer. Math. Theory Methods Appl. 6(1), 276–296 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Xidian University, Xian, China

    Jian Bai & Xiang-Chu Feng

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  1. Jian Bai
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  2. Xiang-Chu Feng
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Correspondence to Jian Bai.

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Bai, J., Feng, XC. Image Denoising Using Generalized Anisotropic Diffusion. J Math Imaging Vis 60, 994–1007 (2018). https://doi.org/10.1007/s10851-018-0790-4

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  • DOI: https://doi.org/10.1007/s10851-018-0790-4

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