Advertisement

On a generalized 5 × 5 stencil scheme for nonlinear diffusion filtering

  • J. Mahipal
  • S. K. Sharma
  • S. Sundar
Article
  • 202 Downloads

Abstract

Optimized kernels for approximating first order spatial derivatives in the regularized Perona–Malik diffusion filter reveal an excellent degree of rotational invariance but produce small checkerboard artifacts in the processed image. In this work, our focus is to reduce these checkerboard artifacts, without a compromise in rotational invariance. To achieve the same, we propose a new 5 × 5 stencil scheme based on generalized finite difference idea. The free parameter in the proposed scheme can be set to recover various well-known schemes. Based on the numerical experiments, we suggest a value of the parameter to have less checkerboard artifacts. We also prove the consistency and the stability of the proposed scheme. A pre-smoothing with 3 × 3 Gaussian mask at every iteration step further improves the quality of the processed image. Simulation results show that the hybrid scheme obtained with such a pre-smoothing is more efficient in computational cost.

Keywords

Nonlinear diffusion Rotational invariance Numerical dissipativity Finite difference method Checkerboard artifacts 

Mathematics Subject Classification

65N06 35K55 94A08 

Notes

Acknowledgments

Second author thanks the Council of Scientific and Industrial Research, India (Grant number: 09/084(0584)/2011-EMR-I) for financial support. We are thankful to the High Performance Computing Center, IIT Madras, India for providing their computing facility to carry out these experiments. We are also thankful to Dr. Arun Kumar and Ms. Raisa Dsouza for final proof reading of this draft.

References

  1. 1.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chan, T.F., Shen, J.J.: Image processing and analysis: variational, PDE, wavelet, and stochastic methods. Siam, Philadelphia (2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Patt. Anal. Mach. Intell. 12(7), 629–639 (1990)CrossRefGoogle Scholar
  4. 4.
    Kichenassamy, S.: The Perona–Malik method as an edge pruning algorithm. J. Math. Imag. Vis. 30(2), 209–219 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kichenassamy, S.: The Perona–Malik paradox. SIAM J. Appl. Math. 57(5), 1328–1342 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Weickert, J., Benhamouda, B.: A semidiscrete nonlinear scale-space theory and its relation to the Perona–Malik paradox. In: Advances in Computer Vision, Advances in Computing Science, pp. 1–10. Springer, Vienna (1997). doi: 10.1007/978-3-7091-6867-7_1
  7. 7.
    You, Y.L., Xu, W., Tannenbaum, A., Kaveh, M.: Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Process. 5(11), 1539–1553 (1996)CrossRefGoogle Scholar
  8. 8.
    Catté, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Weickert, J.: Anisotropic diffusion in image processing, vol. 1. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  10. 10.
    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)CrossRefzbMATHGoogle Scholar
  11. 11.
    You, Y.L., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hajiaboli, M.R.: An anisotropic fourth-order diffusion filter for image noise removal. Int. J. Comput. Vis. 92(2), 177–191 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Aubert, G., Kornprobst, P.: Mathematical problems in image processing: partial differential equations and the calculus of variations, vol. 147. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  14. 14.
    Barash, D., Israeli, M., Kimmel, R.: An accurate operator splitting scheme for nonlinear diffusion filtering. In: Scale-Space and Morphology in Computer Vision, 3rd International Conference. pp. 281–289. Springer, Berlin, Heidelberg (2001)Google Scholar
  15. 15.
    Chen, D., MacLachlan, S., Kilmer, M.: Iterative parameter-choice and multigrid methods for anisotropic diffusion denoising. SIAM J. Sci. Comput. 33(5), 2972–2994 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mahipal, J.: A scheme with discrete maximum principle and optimized rotation invariance for coherence enhancing diffusion. Comput. Math. Appl. 68(8), 859–871 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Weickert, J., Romeny, BMtH, Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)CrossRefGoogle Scholar
  18. 18.
    Weickert, J., Scharr, H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J. Vis. Commun. Image Represent. 13(1), 103–118 (2002)CrossRefGoogle Scholar
  19. 19.
    Welk, M., Steidl, G., Weickert, J.: Locally analytic schemes: a link between diffusion filtering and wavelet shrinkage. Appl. Comput. Harmon. Anal. 24(2), 195–224 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wenyuan, W.: On the design of optimal derivative filters for coherence-enhancing diffusion filtering. In: International Conference on Computer Graphics, Imaging and Visualization, 2004. CGIV 2004. Proceedings. pp. 35–40. IEEE (2004)Google Scholar
  21. 21.
    Simoncelli, E.P.: Design of multi-dimensional derivative filters. ICIP 1, 790–794 (1994)Google Scholar
  22. 22.
    Kroon, D.J., Slump, C.H., Maal, T.J.: Optimized anisotropic rotational invariant diffusion scheme on cone-beam CT. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2010, pp. 221–228. Springer, Heidelberg (2010)Google Scholar
  23. 23.
    Mendrik, A.M., Vonken, E.J., Rutten, A., Viergever, M., Van Ginneken, B., et al.: Noise reduction in computed tomography scans using 3-D anisotropic hybrid diffusion with continuous switch. IEEE Trans. Med. Imag. 28(10), 1585–1594 (2009)CrossRefGoogle Scholar
  24. 24.
    Didas, S., Weickert, J., Burgeth, B.: Properties of higher order nonlinear diffusion filtering. J. Math. Imag. Vis. 35(3), 208–226 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tiwari, S., Kuhnert, J.: Finite pointset method based on the projection method for simulations of the incompressible Navier–Stokes equations. Meshfree methods for partial differential equations, pp. 373–387. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  26. 26.
    Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(1–4), 3–47 (1996)CrossRefzbMATHGoogle Scholar
  27. 27.
    Liu, G.R.: Meshfree methods: moving beyond the finite element method. CRC, Boca Raton (2003)Google Scholar
  28. 28.
    Bulirsch, R., Stoer, J.: Introduction to numerical analysis. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  29. 29.
    Tsiotsios, C., Petrou, M.: On the choice of the parameters for anisotropic diffusion in image processing. Pattern Recognit. 46(5), 1369–1381 (2013)CrossRefGoogle Scholar
  30. 30.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar

Copyright information

© Indian Institute of Technology Madras 2016

Authors and Affiliations

  1. 1.School of Natural SciencesMahindra École CentraleHyderabadIndia
  2. 2.Department of MathematicsIIT MadrasChennaiIndia

Personalised recommendations