Curvature-Preserving Regularization of Multi-valued Images Using PDE’s

  • David Tschumperlé
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3952)


We are interested in diffusion PDE’s for smoothing multi-valued images in an anisotropic manner. By pointing out the pros and cons of existing tensor-driven regularization methods, we introduce a new constrained diffusion PDE that regularizes image data while taking curvatures of image structures into account. Our method has a direct link with a continuous formulation of the Line Integral Convolutions, allowing us to design a very fast and stable algorithm for its implementation. Besides, our smoothing scheme numerically performs with a sub-pixel accuracy and is then able to preserves very thin image structures contrary to classical PDE discretizations based on finite difference approximations. We illustrate our method with different applications on color images.


Integral Curve Anisotropic Diffusion Integral Curf Smoothing Process Line Integral Convolution 
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.
    Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations of image processing. Arch. for Rational Mechanics and Analysis 123(3), 199–257 (1993)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: PDE’s and the Calculus of Variations, January 2002. Applied Math. Sciences, vol. 147. Springer, Heidelberg (2002)MATHGoogle Scholar
  3. 3.
    Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational Problems and PDE’s on Implicit Surfaces. Comput. and Visual. in Science 174(2), 759–780 (2001)MATHGoogle Scholar
  4. 4.
    Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: ACM SIGGRAPH, Int. Conf. on Computer Graphics and Interactive Techniques, pp. 417–424 (2000)Google Scholar
  5. 5.
    Black, M.J., Sapiro, G., Marimont, D.H., Heeger, D.: Robust anisotropic diffusion. IEEE Transaction on Image Processing 7(3), 421–432 (1998)CrossRefGoogle Scholar
  6. 6.
    Cabral, B., Leedom, L.C.: Imaging vector fields using line integral convolution. In: SIGGRAPH 1993, in Computer Graphics, vol. 27, pp. 263–272 (1993)Google Scholar
  7. 7.
    Carmona, R., Zhong, S.: Adaptive Smoothing Respecting Feature Directions. IEEE Transactions on Image Processing 7(3), 353–358 (1998)CrossRefGoogle Scholar
  8. 8.
    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Nümerische Mathematik 76(2), 167–188 (1997)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chan, T., Shen, J.: Variational restoration of non-flat image features: Models and algorithms. SIAM Journal of Applied Mathematics 61(4), 1338–1361 (2000)MATHGoogle Scholar
  10. 10.
    Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. on Image Proc. 6(2) (1997)Google Scholar
  11. 11.
    Chefd’hotel, C., Tschumperlé, D., Deriche, R., Faugeras, O.: Regularizing Flows for Constrained Matrix-Valued Images. Journ. of Math. Imaging and Vision 20(2) (January 2004)Google Scholar
  12. 12.
    Di Zenzo, S.: A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing 33, 116–125 (1986)CrossRefMATHGoogle Scholar
  13. 13.
    Kimmel, R., Malladi, R., Sochen, N.: Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. IJCV 39(2) (September 2000)Google Scholar
  14. 14.
    Kimmel, R., Sochen, N.: Orientation diffusion or how to comb a porcupine. Journal of Visual Communication and Image Representation 13, 238–248 (2002)CrossRefGoogle Scholar
  15. 15.
    Koenderink, J.J.: The structure of images. Biological Cybernetics 50, 363–370 (1984)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kornprobst, P., Deriche, R., Aubert, G.: Nonlinear operators in image restoration. In: IEEE Conference on Computer Vision and Pattern Recognition, June 1997, pp. 325–331 (1997)Google Scholar
  17. 17.
    Krissian, K.: Multiscale Analysis: Application to Medical Imaging and 3D Vessel Detection. Ph.D. Thesis, INRIA-Sophia Antipolis/France (2000)Google Scholar
  18. 18.
    Lindeberg, T.: Scale-Space Theory in Computer Vision. Kluwer Academic Publishers, Dordrecht (1994)CrossRefMATHGoogle Scholar
  19. 19.
    Nielsen, M., Florack, L., Deriche, R.: Regularization, scale-space and edge detection filters. Journal of Mathematical Imaging and Vision 7(4), 291–308 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Perona, P.: Orientation diffusions. IEEE Trans. on Image Proc. 7(3), 457–467 (1998)CrossRefGoogle Scholar
  21. 21.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. on Pattern Anal. and Machine Intell. 12(7), 629–639 (1990)CrossRefGoogle Scholar
  22. 22.
    Preusser, T., Rumpf, M.: Anisotropic nonlinear diffusion in flow visualization. In: IEEE Visualization Conference (1999)Google Scholar
  23. 23.
    Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Runge-Kutta Method. In: Numerical Recipes, pp. 704–716. Cambridge University Press, Cambridge (1992)Google Scholar
  24. 24.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  26. 26.
    Sapiro, G., Ringach, D.L.: Anisotropic diffusion of multi-valued images with applications to color filtering. IEEE Transactions on Image Processing 5(11), 1582–1585 (1996)CrossRefGoogle Scholar
  27. 27.
    Sochen, N., Kimmel, R., Bruckstein, A.M.: Diffusions and confusions in signal and image processing. Journal of Mathematical Imaging and Vision 14(3), 195–209 (2001)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Stalling, D., Hege, H.C.: Fast and Resolution Independent Line Integral Convolution. In: SIGGRAPH, 22nd Ann. Conf. on Computer Graphics, pp. 249–256 (1995)Google Scholar
  29. 29.
    Tang, B., Sapiro, G., Caselles, V.: Direction diffusion. In: IEEE International Conference on Computer Vision, pp. 12–45 (1999)Google Scholar
  30. 30.
    Tang, B., Sapiro, G., Caselles, V.: Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case. IJCV 36(2), 149–161 (2000)CrossRefGoogle Scholar
  31. 31.
    Tschumperlé, D.: PDE’s Based Regularization of Multi-valued Images and Applications. PhD Thesis, Université de Nice-Sophia Antipolis/France (December 2002)Google Scholar
  32. 32.
    Tschumperlé, D., Deriche, R.: Vector-Valued Image Regularization with PDE’s: A Common Framework for Different Applications. IEEE Trans. on PAMI 27(4) (April 2005)Google Scholar
  33. 33.
    Tschumperlé, D.: The CImg Library. The C++ Template Image Processing Library,
  34. 34.
    Vemuri, B., Chen, Y., Rao, M., McGraw, T., Mareci, T., Wang, Z.: Fiber tract mapping from diffusion tensor MRI. In: IEEE Workshop on Variat. and Level Set Methods (July 2001)Google Scholar
  35. 35.
    Weickert, J.: Anisotropic Diffusion Filters for Image Processing Based Quality Control. In: 7th European Conference on Mathematics in Industry, pp. 355–362 (1994)Google Scholar
  36. 36.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart (1998)MATHGoogle Scholar
  37. 37.
    Weickert, J.: Coherence-Enhancing Diffusion of Colour Images. Image and Vision Computing 17, 199–210 (1999)CrossRefGoogle Scholar
  38. 38.
    Weickert, J., Brox, T.: Diffusion and Regularization of Vector and Matrix-valued Images. In: Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, vol. 313, pp. 251–268 (2002)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David Tschumperlé
    • 1
  1. 1.GREYC Image (CNRS UMR 6072)CaenFrance

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