Journal of Mathematical Imaging and Vision

, Volume 28, Issue 3, pp 259–278 | Cite as

Fast Image Inpainting Based on Coherence Transport

  • Folkmar Bornemann
  • Tom März


High-quality image inpainting methods based on nonlinear higher-order partial differential equations have been developed in the last few years. These methods are iterative by nature, with a time variable serving as iteration parameter. For reasons of stability a large number of iterations can be needed which results in a computational complexity that is often too large for interactive image manipulation.

Based on a detailed analysis of stationary first order transport equations the current paper develops a fast noniterative method for image inpainting. It traverses the inpainting domain by the fast marching method just once while transporting, along the way, image values in a coherence direction robustly estimated by means of the structure tensor. Depending on a measure of coherence strength the method switches continuously between diffusion and directional transport. It satisfies a comparison principle. Experiments with the inpainting of gray tone and color images show that the novel algorithm meets the high level of quality of the methods of Bertalmio et al. (SIG-GRAPH ’00: Proc. 27th Conf. on Computer Graphics and Interactive Techniques, New Orleans, ACM Press/Addison-Wesley, New York, pp. 417–424, 2000), Masnou (IEEE Trans. Image Process. 11(2):68–76, 2002), and Tschumperlé (Int. J. Comput. Vis. 68(1):65–82, 2006), while being faster by at least an order of magnitude.


Image inpainting Disocclusion Hyperbolic equation Eikonal equation Skeleton Coherence direction Structure tensor Fast marching 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Springer, New York (2002) zbMATHGoogle Scholar
  2. 2.
    Bertalmio, M., Bertozzi, A.L., Sapiro, G.: Navier-Stokes, fluid dynamics, and image and video inpainting. In: CVPR’01: Proc. IEEE Int. Conf. on Computer Vision and Pattern Recognition, Kauai, vol. I, pp. 355–362. IEEE Press, New York (2001) Google Scholar
  3. 3.
    Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: SIGGRAPH ’00: Proc. 27th Conf. on Computer Graphics and Interactive Techniques, New Orleans, pp. 417–424. ACM Press/Addison-Wesley, New York (2000) CrossRefGoogle Scholar
  4. 4.
    Caselles, V., Masnou, S., Morel, J.-M., Sbert, C.: Image interpolation. In: Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, 1997–1998. Exp. No. XII, p. 15 (1998), downloadable at the url
  5. 5.
    Chan, T.F., Kang, S.H., Shen, J.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan, T.F., Shen, J.: Inpainting based on nonlinear transport and diffusion. In: Inverse problems, image analysis, and medical imaging, New Orleans, LA, 2001. Contemporary Mathematics, vol. 313, pp. 53–65. AMS, Providence (2002) Google Scholar
  7. 7.
    Chan, T.F., Shen, J.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chan, T.F., Shen, J.: Image Processing and Analysis. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005) zbMATHGoogle Scholar
  9. 9.
    Chan, T.F., Shen, J.: Variational image inpainting. Commun. Pure Appl. Math. 58(5), 579–619 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Esedoglu, S., Shen, J.: Digital inpainting based on the Mumford-Shah-Euler image model. Eur. J. Appl. Math. 13(4), 353–370 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fuchs, F.G.: Eulers Elastica- und krümmungsbasiertes Inpainting. Master’s thesis, Technische Universität München (2006) Google Scholar
  13. 13.
    Gonzalez, R.C., Woods, R.E., Eddins, S.L.: Digital Image Processing Using Matlab. Pearson Prentice Hall, Upper Saddle River (2004) Google Scholar
  14. 14.
    Guichard, F., Morel, J.-M.: Image Analysis and PDEs. Unpublished book. Manuscript version 15/07/2000, 345 pp. (2000), downloadable at the url:
  15. 15.
    Kimmel, R.: Numerical Geometry of Images. Springer, New York (2004) zbMATHGoogle Scholar
  16. 16.
    Masnou, S.: Disocclusion: a variational approach using level lines. IEEE Trans. Image Process. 11(2), 68–76 (2002) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: ICIP’98: Proc. IEEE Int. Conf. on Image Processing, Chicago, pp. 259–263. IEEE Press, New York (1998) Google Scholar
  18. 18.
    Oliveira, M.M., Bowen, B., McKenna, R., Chang, Y.-S.: Fast digital image inpainting. In: VIIP ’01: Proc. Int. Conf. on Visualization, Imaging, and Image Processing, Marbella, Spain, pp. 261–266 (2001) Google Scholar
  19. 19.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001) zbMATHGoogle Scholar
  20. 20.
    Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. U.S.A. 93(4), 1591–1595 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  22. 22.
    Soille, P.: Spatial distributions from contour lines: an efficient methodology based on distance transforms. J. Vis. Commun. Image Rep. 2(2), 138–150 (1991) CrossRefGoogle Scholar
  23. 23.
    Soille, P.: Morphological Image Analysis. 2nd edn. Springer, Berlin (2003) zbMATHGoogle Scholar
  24. 24.
    Telea, A.: An image inpainting technique based on the fast marching method. J. Graph. Tools 9(1), 23–34 (2004) Google Scholar
  25. 25.
    Tschumperlé, D.: Fast anisotropic smoothing of multi-valued images using curvature-preserving PDE’s. Int. J. Comput. Vis. 68(1), 65–82 (2006) CrossRefGoogle Scholar
  26. 26.
    Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Control 40(9), 1528–1538 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998) zbMATHGoogle Scholar
  28. 28.
    Weickert, J.: Coherence-enhancing diffusion of color images. Image Vis. Comput. 17(3–4), 201–212 (1999) CrossRefGoogle Scholar
  29. 29.
    Weickert, J.: Coherence-enhancing shock filters. In: Michaelis, B., Krell, G. (eds.) Pattern Recognition. Lecture Notes in Computer Science, vol. 2781. Springer, New York (2003) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarchingGermany

Personalised recommendations