Interpolating Orientation Fields: An Axiomatic Approach

  • Anatole Chessel
  • Frederic Cao
  • Ronan Fablet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3954)


We develop an axiomatic approach of vector field interpolation, which is useful as a feature extraction preprocessing step. Two operators will be singled out: the curvature operator, appearing in the total variation minimisation for image restoration and inpainting/disocclusion, and the Absolutely Minimizing Lipschitz Extension (AMLE), already known as a robust and coherent scalar image interpolation technique if we relax slightly the axioms. Numerical results, using a multiresolution scheme, show that they produce fields in accordance with the human perception of edges.


Curvature Operator Level Line Scalar Case Extension Operator Illusory Contour 
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.


  1. 1.
    Alvarez, L., Guichard, F., Lions, P.L., Morel, J.: Axioms and fondamental equations of image processing. Arch. Rational Mechanics and Anal. 16, 200–257 (1993)zbMATHGoogle Scholar
  2. 2.
    Caselles, V., Morel, J., Sbert, C.: An axiomatic approach to image interpolation. IEEE Trans. Image Processing 7, 376–386 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kaniza, G.: La grammaire du voir. Diderot (1996)Google Scholar
  4. 4.
    Wertheimer, M.: Untersuchungen zur Lehre der Gestalt II. Psychologische Forschung 4, 301–350 (1923)CrossRefGoogle Scholar
  5. 5.
    Parent, P., Zucker, W.: Trace inference, curvature consistency, and curve detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (1989)Google Scholar
  6. 6.
    Sha’ashua, A., Ullman, S.: Structual saliency: The detection of globally salient structures using a locally connected network. In: Second Int. Conf. Comp. Vision, Tarpon Springs, FL, pp. 321–327 (1988)Google Scholar
  7. 7.
    Medioni, G., Lee, M.: Grouping.,-,-> into regions, curves, and junctions. Computer Vision and Image Understanding 76, 54–69 (1999)CrossRefGoogle Scholar
  8. 8.
    Medioni, G., Lee, M., Tang, C.: A computational framework for segmentation and grouping. Elsevier Science, Amsterdam (2000)zbMATHGoogle Scholar
  9. 9.
    Zweck, J., Williams, L.: Euclidian group invariant computation of stochastic completion fields using shiftable-twistable basis function. J. Math. Imaging and Vision 21, 135–154 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mumford, D.: Elastica and computer vision. In: Bajaj, C. (ed.) Algebraic Geometry and Its Applications, Springer, Heidelberg (1994)Google Scholar
  11. 11.
    Ballester, C., Caselles, V., Verdera, J.: Disocclusion by joint interpolation of vector fields and gray levels. Multiscale Modelling and Simulation 2, 80–123 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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
  13. 13.
    Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H 1 norm. Multiscale Modeling and Simulation 1, 349–370 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chessel, A., Cao, F., Fablet, R.: Orientation interpolation: an axiomatic approach. Technical report, IRISA (in preparation)Google Scholar
  15. 15.
    Granas, A., Dugundji, J.: Fixed point theory. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  16. 16.
    Masnou, S., Morel, J.: Level-line based disocclusion. IEEE ICIP (October 1998)Google Scholar
  17. 17.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation bsaed noise removal algorithms. Phisica D 60, 259–268 (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Chan, T., Shen, J.: Local inpainting model and TV inpainting. SIAM J. Appl. Math. 62, 1019–1043 (2001)MathSciNetGoogle Scholar
  19. 19.
    Tang, B., Sapiro, G., Caselles, V.: Diffusion of general data on non-flat manifolds. Int. J. Computer Vision 36, 149–161 (2000)CrossRefGoogle Scholar
  20. 20.
  21. 21.
    Perona, P.: Orientation diffusion. IEEE Trans. Image Processing 7, 457–467 (1998)CrossRefGoogle Scholar
  22. 22.
    Cecil, T., Osher, S., Vese, L.: Numerical methods for minimization problems constrained to S 1 and S 2. J. of Computational Physics 198, 567–579 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cabral, B., Leedom, L.: Imaging vector field using line integral convolution. In: Computer Graphics Proceedings, pp. 263–270 (1993)Google Scholar
  24. 24.
    Deriche, R.: Using canny’s criteria to derive a recursively implemented optimal edge detector. Int. J. Computer Vision 1, 167–187 (1987)CrossRefGoogle Scholar
  25. 25.
    Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. 41, 439–505 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jensen, R.: Uniqueness of lipschitz extentions: minimizing the sup norm of gradient. Arch. Rational Mechanics and Anal. 123, 51–74 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Aronsson, G.: Extention of function satisfying lipschtiz conditions. Ark. Mat. 6, 551–561 (1997)CrossRefGoogle Scholar
  28. 28.
    Sethian, J.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  29. 29.
    Tschumperlé, D.: LIC-based regularization of multi-valued images. In: ICIP 2005, Genoa, Italy (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Anatole Chessel
    • 1
  • Frederic Cao
    • 2
  • Ronan Fablet
    • 1
  1. 1.IFREMER/LASAATechnopole Brest-IroisePlouzaneFrance
  2. 2.Campus de BeaulieuIRISA/VISTARennesFrance

Personalised recommendations