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Journal of Mathematical Imaging and Vision

, Volume 43, Issue 2, pp 135–142 | Cite as

Application of Lattice Boltzmann Method to Image Filtering

  • Wenhuan Zhang
  • Baochang Shi
Article

Abstract

In this paper, lattice Boltzmann D2Q5 (two dimensions and five discrete velocity directions) and D2Q9 (two dimensions and nine discrete velocity directions) models are used to solve Perona-Malik equation, which is widely used in image filtering. A set of images added three types of noise are processed using these models. Then the processed images are compared in aspects of peak signal to noise ratio (PSNR) and visual effect. The comparison show that two models have almost the same filtering effect. Simultaneously, it is validated that D2Q5 model is more efficient. Other findings are: (1) D2Q5 and D2Q9 models are more effective in dealing with some images than others; (2) salt and pepper noise is relatively difficult to remove compared with gaussian noise and speckle noise; (3) lattice Boltzmann method shows good stability in the image filtering.

Keywords

Lattice Boltzmann method Perona-Malik equation Image filtering Nonlinear diffusion equation 

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References

  1. 1.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990) CrossRefGoogle Scholar
  2. 2.
    Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 845–866 (1992) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    You, Y.L., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9, 1723–1730 (2000) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chen, S., Dawson, S.P., Doolen, G.D., Janecky, D.R., Lawniczak, A.: Lattice methods and their applications to reacting systems. Comput. Chem. Eng. 19(6–7), 617–646 (1995) CrossRefGoogle Scholar
  6. 6.
    Zhang, C.Y., Tan, H.L., Liu, M.R., Kong, L.J.: A lattice Boltzmann model and simulation of KdV-Burgers equation. Commun. Theor. Phys. 42(2), 281–284 (2004) MATHGoogle Scholar
  7. 7.
    Ma, C.: A new lattice Boltzmann model for KdV-Burgers equation. Chin. Phys. Lett. 29(9), 2313–2315 (2005) Google Scholar
  8. 8.
    Chai, Z.H., Shi, B.C., Zheng, L.: A unified lattice Boltzmann model for some nonlinear partial differential equations. Chaos Solitons Fractals 36, 874–882 (2008) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Yan, G.: A lattice Boltzmann equation for waves. J. Comput. Phys. 161, 61–69 (2000) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hirabayashi, M., Chen, Y., Ohashi, H.: The lattice BGK model for the Poisson equation. JSME Int. J. Ser. B 44(1), 45–52 (2001) CrossRefGoogle Scholar
  11. 11.
    Chai, Z.H., Shi, B.C.: A novel lattice Boltzmann model for the Poisson equation. Appl. Math. Model. 32, 2050–2058 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Guo, Z.L., Shi, B.C., Wang, N.C.: Fully Lagrangian and lattice Boltzmann methods for the advection-diffusion equation. J. Sci. Comput. 14(3), 291–300 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Deng, B., Shi, B.C., Wang, G.C.: A new lattice Bhatnagar-Gross-Krook model for convection-diffusion equation with a source term. Chin. Phys. Lett. 22(2), 267–270 (2005) CrossRefGoogle Scholar
  14. 14.
    Shi, B.C., Guo, Z.L.: Lattice Boltzmann model for nonlinear convection-diffusion equations. Phys. Rev. E 79, 016701 (2009) CrossRefGoogle Scholar
  15. 15.
    Jawerth, B., Lin, P., Sinzinger, E.: Lattice Boltzmann models for anisotropic diffusion of images. J. Math. Imaging Vis. 11, 231–237 (1999) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Zhao, Y.: Lattice Boltzmann based PDE solver on the GPU. Vis. Comput. 24(5), 323–333 (2008) CrossRefGoogle Scholar
  17. 17.
    Zhao, Y.: GPU-Accelerated surface denoising and morphing with lattice Boltzmann scheme. In: IEEE International Conference on Shape Modeling and Applications (SMI), pp. 19–28. (2008) CrossRefGoogle Scholar
  18. 18.
    Chen, Y., Yan, Z.Z., Qian, Y.H.: An anisotropic diffusion model for medical image smoothing by using the lattice Boltzmann method. APCMBE 2008. IFMBE Proc. 19, 255–259 (2008) CrossRefGoogle Scholar
  19. 19.
    Chen, Y., Yan, Z.Z., Qian, Y.H.: The lattice Boltzmann method based image denoising. Acta Electron. Sinica (in Chinese) 37(3), 574–580 (2009) Google Scholar
  20. 20.
    Chang, Q.S., Yang, T.: A lattice Boltzmann method for image denoising. IEEE Trans. Image Process. 18(12), 2797–2802 (2009) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qian, Y.H.: Fractional propagation and the elimination of staggered invariants in lattice-BGK models. Int. J. Mod. Phys. C 8, 753–761 (1997) CrossRefGoogle Scholar
  22. 22.
    Guo, Z.L., Zheng, C.G., Shi, B.C.: Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Phys. 11(4), 366–374 (2002) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Canny, J.: computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986) CrossRefGoogle Scholar
  24. 24.
    Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, New York (1977) MATHGoogle Scholar
  25. 25.
    Wolf-Gladrow, D.: A lattice Boltzmann equation for diffusion. J. Stat. Phys. 79(5/6), 1023–1032 (1995) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.State Key Laboratory of Coal CombustionHuazhong University of Science and TechnologyWuhanP.R. China
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanP.R. China

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