Computing the Local Continuity Order of Optical Flow Using Fractional Variational Method

  • K. Kashu
  • Y. Kameda
  • A. Imiya
  • T. Sakai
  • Y. Mochizuki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5681)


We introduce variational optical flow computation involving priors with fractional order differentiations. Fractional order differentiations are typical tools in signal processing and image analysis. The zero-crossing of a fractional order Laplacian yields better performance for edge detection than the zero-crossing of the usual Laplacian. The order of the differentiation of the prior controls the continuity class of the solution. Therefore, using the square norm of the fractional order differentiation of optical flow field as the prior, we develop a method to estimate the local continuity order of the optical flow field at each point. The method detects the optimal continuity order of optical flow and corresponding optical flow vector at each point. Numerical results show that the Horn-Schunck type prior involving the n + ε order differentiation for 0 < ε< 1 and an integer n is suitable for accurate optical flow computation.


Optical Flow Fractional Order Fractional Derivative Total Variational Regularisation Continuity Order 
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.
    Papenberg, N., Bruhn, A., Brox, T., Didas, S., Weickert, J.: Highly accurate optic flow computation with theoretically justified warping. IJCV 67, 141–158 (2006)CrossRefGoogle Scholar
  2. 2.
    Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. J. Visual Communication and Image Representation 18, 240–252 (2007)CrossRefGoogle Scholar
  3. 3.
    Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. of Computational Physics 220, 813–823 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 55, 293–318 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis, J.A., Smith, D.A., McNamara, D.E., Cottrell, D.M., Campos, J.: Fractional derivatives-analysis and experimental implementation. Applied Optics 32, 5943–5948 (2001)CrossRefGoogle Scholar
  6. 6.
    Tseng, C.-C., Pei, S.-C., Hsia, S.-C.: Computation of fractional derivatives using Fourier transform and digital FIR differentiator. Signal Processing 80, 151–159 (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Zhang, J., Wei, Z.-H.: Fractional variational model and algorithm for image denoising. In: Proceedings of 4th International Conference on Natural Computation, vol. 5, pp. 524–528 (2008)Google Scholar
  8. 8.
    Mathieu, B., Melchior, P., Oustaloup, A., Ceyral, Cn.: Fractional differentiation for edge detection. Signal Processing 83, 2421–2432 (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Sabatier, J., Agrawel, O.P., Tenreiro Machado, I.A.: Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering. Springer, Netherlands (2007)CrossRefGoogle Scholar
  10. 10.
    Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory And Applications of Differentiation And Integration to Arbitrary Order (Dover Books on Mathematics). Dover (2004)Google Scholar
  11. 11.
    Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, London (1999)zbMATHGoogle Scholar
  12. 12.
    Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artificial Intelligence 17, 185–204 (1981)CrossRefGoogle Scholar
  13. 13.
    Beauchemin, S.S., Barron, J.L.: The computation of optical flow. ACM Computer Surveys 26, 433–467 (1995)CrossRefGoogle Scholar
  14. 14.
    Nagel, H.-H., Enkelmann, W.: An investigation of smoothness constraint for the estimation of displacement vector fields from image sequences. IEEE Trans. on PAMI 8, 565–593 (1986)CrossRefGoogle Scholar
  15. 15.
    Nagel, H.-H.: On the estimation of optical flow:Relations between different approaches and some new results. Artificial Intelligence 33, 299–324 (1987)CrossRefGoogle Scholar
  16. 16.
    Momani, S., Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons and Fractals 31, 1248–1255 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Murio, D.A.: Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional IHCP. Inverse Problems in Science and Engineering 17, 229–243 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gorenfloa, R., Abdel-Rehimb, E.A.: Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. J. of Computational and Applied Mathematics 205, 871–881 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Debbi. L., Explicit solutions of some fractional partial differential equations via stable subordinators. J. of Applied Mathematics and Stochastic Analysis, Article ID 93502, 1–18 (2006)Google Scholar
  20. 20.
    Debbi, L.: On some properties of a high order fractional differential operator which is not in general selfadjoint. Applied Mathematical Sciences 1, 1325–1339 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Fractional diffusion in inhomogeneous media. J. Phys. A: Math. Gen. 38, L679–L684 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Duits, R., Felsberg, M., Florack, L.M.J., Platel, B.: α scale spaces on a bounded domain. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 502–518. Springer, Heidelberg (2003)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • K. Kashu
    • 1
  • Y. Kameda
    • 1
  • A. Imiya
    • 2
  • T. Sakai
    • 2
  • Y. Mochizuki
    • 1
  1. 1.School of Advanced Integration ScienceChiba UniversityChibaJapan
  2. 2.Institute of Media and Information TechnologyChiba UniversityChibaJapan

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