Explicit Nonflat Time Evolution for PDE-Based Image Restoration

  • Seongjai Kim
  • Song-Hwa Kwon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4338)


This article is concerned with new strategies with which explicit time-stepping procedures of PDE-based restoration models converge with a similar efficiency to implicit algorithms. Conventional explicit algorithms often require hundreds of iterations to converge. In order to overcome the difficulty and to further improve image quality, the article introduces new spatially variable constraint term and timestep size, as a method of nonflat time evolution (MONTE). It has been verified that the explicit time-stepping scheme incorporating MONTE converges in only 4-15 iterations for all restoration examples we have tested. It has proved more effective than the additive operator splitting (AOS) method in both computation time and image quality (measured in PSNR), for most cases. Since the explicit MONTE procedure is efficient in computer memory, requiring only twice the image size, it can be applied particularly for huge data sets with a great efficiency in computer memory as well.


Image Restoration Computer Memory Constraint Parameter Spatial Scheme Explicit Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. on Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  2. 2.
    Alvarez, L., Lions, P., Morel, M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal. 29, 845–866 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Catte, F., Lions, P., Morel, M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Kim, S.: PDE-based image restoration: A hybrid model and color image denoising. IEEE Trans. Image Processing 15, 1163–1170 (2006)CrossRefGoogle Scholar
  5. 5.
    Kim, S., Lim, H.: A non-convex diffusion model for simultaneous image denoising and edge enhancement. Electronic Journal of Differential Equations (accepted, 2006)Google Scholar
  6. 6.
    Marquina, A., Osher, S.: Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal. SIAM J. Sci. Comput. 22, 387–405 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. on Pattern Anal. Mach. Intell. 14, 826–833 (1992)CrossRefGoogle Scholar
  8. 8.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MATHCrossRefGoogle Scholar
  9. 9.
    You, Y.L., Xu, W., Tannenbaum, A., Kaveh, M.: Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Process. 5, 1539–1553 (1996)CrossRefGoogle Scholar
  10. 10.
    Chan, T., Shen, J.: Image Processing and Analysis. SIAM, Philadelphia (2005)MATHCrossRefGoogle Scholar
  11. 11.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)MATHGoogle Scholar
  12. 12.
    Sapiro, G.: Geometric partial differential equations and image analysis. Cambridge University Press, Cambridge (2001)MATHCrossRefGoogle Scholar
  13. 13.
    Weickert, J., ter Haar Romeny, B., Viergever, M.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. on Image Processing 7, 398–410 (1998)CrossRefGoogle Scholar
  14. 14.
    Douglas Jr., J., Gunn, J.: A general formulation of alternating direction methods Part I. Parabolic and hyperbolic problems. Numer. Math. 6, 428–453 (1964)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Douglas Jr., J., Kim, S.: Improved accuracy for locally one-dimensional methods for parabolic equations. Mathematical Models and Methods in Applied Sciences 11, 1563–1579 (2001)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cha, Y., Kim, S.: Edge-forming methods for image zooming. J. Mathematical Imaging and Vision (in press, 2006)Google Scholar
  17. 17.
    Cha, Y., Kim, S.: Edge-forming methods for color image zooming. IEEE Trans. Image Process. 15, 2315–2323 (2006)CrossRefGoogle Scholar
  18. 18.
    Kim, S., Kwon, S.H.: Efficiency and reliability in nonlinear diffusion filtering (in preparation)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Seongjai Kim
    • 1
  • Song-Hwa Kwon
    • 2
  1. 1.Department of Mathematics & StatisticsMississippi State UniversityMississippi StateUSA
  2. 2.IMAUniversity of MinnesotaMinneapolisUSA

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