Advertisement

An Efficient and Stable Two-Pixel Scheme for 2D Forward-and-Backward Diffusion

  • Martin WelkEmail author
  • Joachim Weickert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

Image enhancement with forward-and-backward (FAB) diffusion is numerically very challenging due to its negative diffusivities. As a remedy, we first extend the explicit nonstandard scheme by Welk et al. (2009) from the 1D scenario to the practically relevant two-dimensional setting. We prove that under a fairly severe time step restriction, this 2D scheme preserves a maximum–minimum principle. Moreover, we find an interesting Lyapunov sequence which guarantees convergence to a flat steady state. Since a global application of the time step size restriction leads to very slow algorithms and is more restrictive than necessary for most pixels, we introduce a much more efficient scheme with locally adapted time step sizes. It applies diffusive two-pixel interactions in a randomised order and adapts the time step size to the specific pixel pair. These space-variant time steps are synchronised at sync times. Our experiments show that our novel two-pixel scheme allows to compute FAB diffusion with guaranteed \(L^\infty \)-stability at a speed that can be three orders of magnitude larger than its explicit counterpart with a global time step size.

References

  1. 1.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  2. 2.
    Burgeth, B., Weickert, J., Tari, S.: Minimally stochastic schemes for singular diffusion equations. In: Tai, X.C., Lie, K.A., Chan, T.F., Osher, S. (eds.) Image Processing Based on Partial Differential Equations. (MATHVISUAL), pp. 325–339. Springer, Berlin (2007). doi: 10.1007/978-3-540-33267-1_18 CrossRefGoogle Scholar
  3. 3.
    Gilboa, G., Sochen, N., Zeevi, Y.Y.: Image sharpening by flows based on triple well potentials. J. Math. Imaging Vis. 20, 121–131 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Trans. Image Process. 11, 689–703 (2002)CrossRefGoogle Scholar
  5. 5.
    Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recogn. 7, 53–58 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mrázek, P., Weickert, J., Steidl, G.: Diffusion-inspired shrinkage functions and stability results for wavelet denoising. Int. J. Comput. Vis. 64, 171–186 (2005)CrossRefGoogle Scholar
  7. 7.
    Osher, S., Rudin, L.: Shocks and other nonlinear filtering applied to image processing. In: Tescher, A.G. (ed.) Applications of Digital Image Processing XIV. Proceedings of SPIE, vol. 1567, pp. 414–431. SPIE Press, Bellingham (1991)Google Scholar
  8. 8.
    Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27, 919–940 (1990)CrossRefzbMATHGoogle Scholar
  9. 9.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  10. 10.
    Pollak, I., Willsky, A.S., Krim, H.: Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Trans. Image Process. 9, 256–266 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Smolka, B.: Combined forward and backward anisotropic diffusion filtering of color images. In: Van Gool, L. (ed.) DAGM 2002. LNCS, vol. 2449, pp. 314–322. Springer, Heidelberg (2002). doi: 10.1007/3-540-45783-6_38 CrossRefGoogle Scholar
  12. 12.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  13. 13.
    Weickert, J., Benhamouda, B.: A semidiscrete nonlinear scale-space theory and its relation to the Perona–Malik paradox. In: Solina, F., Kropatsch, W.G., Klette, R., Bajcsy, R. (eds.) Advances in Computer Vision. (ACS), pp. 1–10. Springer, Vienna (1997). doi: 10.1007/978-3-7091-6867-7_1 CrossRefGoogle Scholar
  14. 14.
    Welk, M., Gilboa, G., Weickert, J.: Theoretical foundations for discrete forward-and-backward diffusion filtering. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 527–538. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02256-2_44 CrossRefGoogle Scholar
  15. 15.
    Welk, M., Weickert, J., Galić, I.: Theoretical foundations for spatially discrete 1-D shock filtering. Image Vis. Comput. 25, 455–463 (2007)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Biomedical Image AnalysisPrivate University for Health Sciences, Medical Informatics and TechnologyHall/TyrolAustria
  2. 2.Mathematical Image Analysis GroupSaarland UniversitySaarbrückenGermany

Personalised recommendations