Qualitative and Quantitative Behaviour of Geometrical PDEs in Image Processing

  • Arjan Kuijper
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4843)


We analyse a series of approaches to evolve images. It is motivated by combining Gaussian blurring, the Mean Curvature Motion (used for denoising and edge-preserving), and maximal blurring (used for inpainting). We investigate the generalised method using the combination of second order derivatives in terms of gauge coordinates.

For the qualitative behaviour, we derive a solution of the PDE series and mention its properties briefly. Relations with general diffusion equations are discussed. Quantitative results are obtained by a novel implementation whose stability and convergence is analysed.

The practical results are visualised on a real-life image, showing the expected qualitative behaviour. When a constraint is added that penalises the distance of the results to the input image, one can vary the desired amount of blurring and denoising.


Order Derivative Qualitative Behaviour Large Time Step Curvature Motion Geometrical Evolution 
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.
    Koenderink, J.J.: The structure of images. Biological Cybernetics 50, 363–370 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    Alvarez, L., Lions, P., Morel, J.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 29, 845–866 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Caselles, V., Morel, J.M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Transactions on Image Processing 7, 376–386 (1996)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bertalmio, M., Vese, L., Sapiro, G., Osher, S.: Simultaneous structure and texture image inpainting. IEEE Transactions on Image Processing 12, 882–889 (2003)CrossRefGoogle Scholar
  6. 6.
    Haar Romeny, B.M.t.: Front-end vision and multi-scale image analysis. Kluwer Academic Publishers, Dordrecht, The Netherlands (2003)Google Scholar
  7. 7.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  8. 8.
    Kornprobst, P., Deriche, R., Aubert, G.: Image coupling, restoration and enhancement via PDE’s. In: Proc. Int. Conf. on Image Processing, vol. 4, pp. 458–461 (1997)Google Scholar
  9. 9.
    Griffin, L.: Mean, median and mode filtering of images. Proceedings of the Royal Society Series A 456, 2995–3004 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Yezzi, A.: Modified curvature motion for image smoothing and enhancement. IEEE Transactions on Image Processing 7, 345–352 (1998)CrossRefGoogle Scholar
  11. 11.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edn. Dover, New York (1972)zbMATHGoogle Scholar
  12. 12.
    Weickert, J.A.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  13. 13.
    Aronsson, G.: On the partial differential equation \(u^2_x u_{xx} + 2u_x u_y u_{xy} + u^2_ y u_{yy} = 0\). Arkiv für Matematik 7, 395–425 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kuijper, A.: p-laplacian driven image processing. In: ICIP 2007 (2007)Google Scholar
  15. 15.
    Niessen, W.J., ter Haar Romeny, B.M., Florack, L.M.J., Viergever, M.A.: A general framework for geometry-driven evolution equations. International Journal of Computer Vision 21, 187–205 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Arjan Kuijper
    • 1
  1. 1.Radon Institute for Computational and Applied Mathematics, LinzAustria

Personalised recommendations