Combining Curvature Motion and Edge-Preserving Denoising

  • Stephan Didas
  • Joachim Weickert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4485)


In this paper we investigate a family of partial differential equations (PDEs) for image processing which can be regarded as isotropic nonlinear diffusion with an additional factor on the right-hand side. The one-dimensional analogues to this filter class have been motivated as scaling limits of one-dimensional adaptive averaging schemes. In 2-D, mean curvature motion is one of the most prominent examples of this family of PDEs. Other representatives of the filter class combine properties of curvature motion with the enhanced edge preservation of Perona-Malik diffusion. It becomes appearent that these PDEs require a careful discretisation. Numerical experiments display the differences between Perona-Malik diffusion, classical mean curvature motion and the proposed extensions. We consider, for example, enhanced edge sharpness, the question of morphological invariance, and the behaviour with respect to noise.


Curvature Motion Morphological Invariance Small Time Step Size Shrinkage Time Convex Plane Curf 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  2. 2.
    Kimmel, R.: Numerical Geometry of Images: Theory, Algorithms, and Applications. Springer, New York (2003)Google Scholar
  3. 3.
    Cao, F.: Geometric Curve Evolution and Image Processing. Lecture Notes in Mathematics, vol. 1805. Springer, Berlin (2003)MATHGoogle Scholar
  4. 4.
    Gabor, D.: Information theory in electron microscopy. Laboratory Investigation 14(6), 801–807 (1965)Google Scholar
  5. 5.
    Lindenbaum, M., Fischer, M., Bruckstein, A.: On Gabor’s contribution to image enhancement. Pattern Recognition 27(1), 1–8 (1994)CrossRefGoogle Scholar
  6. 6.
    Gage, M.: Curve shortening makes convex curves circular. Inventiones Mathematicae 76, 357–364 (1984)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. Journal of Differential Geometry 23, 69–96 (1986)MathSciNetMATHGoogle Scholar
  8. 8.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. Journal of Differential Geometry 20, 237–266 (1984)MathSciNetMATHGoogle Scholar
  9. 9.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics 79, 12–49 (1988)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Kimia, B.B., Tannenbaum, A., Zucker, S.W.: Toward a computational theory of shape: an overview. In: Faugeras, O. (ed.) ECCV 1990. LNCS, vol. 427, pp. 402–407. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  11. 11.
    Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion ii. SIAM Journal on Numerical Analysis 29(3), 845–866 (1992)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Sapiro, G.: Vector (self) snakes: a geometric framework for color, texture and multiscale image segmentation. In: Proc. 1996 IEEE International Conference on Image Processing, vol. 1, Lausanne, Switzerland, Sep. 1996, pp. 817–820. IEEE Computer Society Press, Los Alamitos (1996)CrossRefGoogle Scholar
  13. 13.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)CrossRefGoogle Scholar
  14. 14.
    Catté, F., et al.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 29(1), 182–193 (1992)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Alvarez, L., et al.: Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis 123, 199–257 (1993)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Carmona, R.A., Zhong, S.: Adaptive smoothing respecting feature directions. IEEE Transactions on Image Processing 7, 353–358 (1998)CrossRefGoogle Scholar
  17. 17.
    Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDE’s: A common framework for different applications. IEEE Transactions on Image Processing 27(4), 1–12 (2005)Google Scholar
  18. 18.
    Didas, S., Weickert, J.: From adaptive averaging to accelerated nonlinear diffusion filtering. In: Franke, K., et al. (eds.) DAGM 2006. LNCS, vol. 4174, pp. 101–110. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Weickert, J.: Applications of nonlinear diffusion in image processing and computer vision. Acta Mathematica Universitatis Comenianae LXX(1), 33–50 (2001)MathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Stephan Didas
    • 1
  • Joachim Weickert
    • 1
  1. 1.Mathematical Image Analysis Group, Department of Mathematics and Computer Science, Saarland University, Building E1 1, 66041 Saarbrücken 

Personalised recommendations