Motion Estimation from Image Sequences: A Fractional Order Total Variation Model

  • Pushpendra KumarEmail author
  • Balasubramanian Raman
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 460)


In this paper, a fractional order total variation model is introduced in the estimation of motion field. In particular, the proposed model generalizes the integer order total variation models. The motion estimation is carried out in terms optical flow. The presented model is made using a quadratic and total variation terms. This mathematical formulation makes the model robust against outliers and preserves discontinuities. However, it is difficult to solve the presented model due to the non-differentiability nature of total variation term. For this purpose, the Grünwald-Letnikov derivative is used as a discretization scheme to discretize the fractional order derivative. The resulting formulation is solved by using a more efficient algorithm. Experimental results on various datasets verify the validity of the proposed model.


Fractional derivative Image sequence Optical flow Total variation regularization 



The author, Pushpendra Kumar gratefully acknowledges the financial support provided by Council of Scientific and Industrial Research(CSIR), New Delhi, India to carry out this work.


  1. 1.
    Alvarez, L., Weickert, J., Sánchez, J.: Reliable estimation of dense optical flow fields with large displacements. International Journal of Computer Vision 39(1), 41–56 (2000)Google Scholar
  2. 2.
    Baker, S., Scharstein, D., Lewis, J.P., Roth, S., Black, M.J., Szeliski, R.: A database and evaluation methodology for optical flow. International Journal of Computer Vision 92, 1–31 (2011)Google Scholar
  3. 3.
    Barron, J.L., Fleet, D.J., Beauchemin, S.: Performance of optical flow techniques. International Journal of Computer Vision 12, 43–77 (1994)Google Scholar
  4. 4.
    Black, M.J., Anandan, P.: The robust estimation of multiple motions: Parametric and piecewise smooth flow. Computer Vision and Image Understanding 63(1), 75–104 (1996)Google Scholar
  5. 5.
    Brox, T., Bruhn, A., Papenberg, N., Weickert, J.: High accuracy optical flow estimation based on a theory for warping. Computer Vision - ECCV 4, 25–36 (2004)Google Scholar
  6. 6.
    Chambolle, A.: An algorithm for total variation minimization and applications. Journal of Mathematical imaging and vision 20(1–2), 89–97 (2004)Google Scholar
  7. 7.
    Chen, D., Chen, Y., Xue, D.: Fractional-order total variation image restoration based on primal-dual algorithm. Abstract and Applied Analysis 2013 (2013)Google Scholar
  8. 8.
    Chen, D., Sheng, H., Chen, Y., Xue, D.: Fractional-order variational optical flow model for motion estimation. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 371(1990), 20120148 (2013)Google Scholar
  9. 9.
    Drulea, M., Nedevschi, S.: Total variation regularization of local-global optical flow. In: 14th International Conference on Intelligent Transportation Systems (ITSC). pp. 318–323 (2011)Google Scholar
  10. 10.
    Horn, B., Schunck, B.: Determining optical flow. Artificial Intelligence 17, 185–203 (1981)Google Scholar
  11. 11.
    Lucas, B., Kanade, T.: An iterative image registration technique with an application to stereo vision. In: Seventh International Joint Conference on Artificial Intelligence, Vancouver, Canada. vol. 81, pp. 674–679 (1981)Google Scholar
  12. 12.
    Miller, K.S.: Derivatives of noninteger order. Mathematics magazine pp. 183–192 (1995)Google Scholar
  13. 13.
    Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley New York (1993)Google Scholar
  14. 14.
    Motai, Y., Jha, S.K., Kruse, D.: Human tracking from a mobile agent: optical flow and kalman filter arbitration. Signal Processing: Image Communication 27(1), 83–95 (2012)Google Scholar
  15. 15.
    Niese, R., Al-Hamadi, A., Farag, A., Neumann, H., Michaelis, B.: Facial expression recognition based on geometric and optical flow features in colour image sequences. IET computer vision 6(2), 79–89 (2012)Google Scholar
  16. 16.
    Pu, Y.F., Zhou, J.L., Yuan, X.: Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement. IEEE Transactions on Image Processing 19(2), 491–511 (2010)Google Scholar
  17. 17.
    Riemann, B.: Versuch einer allgemeinen auffassung der integration und differentiation. Gesammelte Werke 62 (1876)Google Scholar
  18. 18.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60(1), 259–268 (1992)Google Scholar
  19. 19.
    Schneider, R.J., Perrin, D.P., Vasilyev, N.V., Marx, G.R., Pedro, J., Howe, R.D.: Mitral annulus segmentation from four-dimensional ultrasound using a valve state predictor and constrained optical flow. Medical image analysis 16(2), 497–504 (2012)Google Scholar
  20. 20.
    Weickert, J.: On discontinuity-preserving optic flow. In: Proceeding of Computer Vision and Mobile Robotics Workshop (1998)Google Scholar
  21. 21.
    Werlberger, M., Trobin, W., Pock, T., Wedel, A., Cremers, D., Bischof, H.: Anisotropic huber-l1 optical flow. In: BMVC. vol. 1, p. 3 (2009)Google Scholar
  22. 22.
    Zach, C., Pock, T., Bischof, H.: A duality based approach for realtime tv-l 1 optical flow. In: Pattern Recognition, pp. 214–223. Springer (2007)Google Scholar

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© Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Department of Computer Science & EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia

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