A Rotation-Invariant Regularization Term for Optical Flow Related Problems

  • Roberto P. PalomaresEmail author
  • Gloria Haro
  • Coloma Ballester
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9007)


This paper proposes a new regularization term for optical flow related problems. The proposed regularizer properly handles rotation movements and it also produces good smoothness conditions on the flow field while preserving discontinuities. We also present a dual formulation of the new term that turns the minimization problem into a saddle-point problem that can be solved using a primal-dual algorithm. The performance of the new regularizer has been compared against the Total Variation (TV) in three different problems: optical flow estimation, optical flow inpainting, and optical flow completion from sparse samples. In the three situations the new regularizer improves the results obtained with the TV as a smoothing term.



We acknowledge partial support by MICINN project, reference MTM2012-30772, by GRC reference 2009 SGR 773 funded by the Generalitat de Catalunya, and by the ERC Advanced Grant INPAINTING (Grant agreement no.: 319899). The second author acknowledges partial support to the Ramón y Cajal program of the MINECO.


  1. 1.
    Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artif. Intell. 17(1–3), 185–203 (1981)CrossRefGoogle Scholar
  2. 2.
    Black, M.J., Anandan, P.: The robust estimation of multiple motions: parametric and piecewise-smooth flow fields. Comput. Vis. Image Underst. 63(1), 75–104 (1996)CrossRefGoogle Scholar
  3. 3.
    Zimmer, H., Bruhn, A., Weickert, J.: Optic flow in harmony. Int. J. Comput. Vis. 93(3), 368–388 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Nagel, H.H., Enkelmann, W.: An Investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. Pattern Anal. Mach. Intell. 8(5), 565–593 (1986)CrossRefGoogle Scholar
  5. 5.
    Wedel, A., Cremers D., Pock, T., Bischof, H.: Structure-and motion-adaptative regularization for high accuracy optic flow. In: IEEE 12th International Conference on Computer Vision, pp. 1663–1668 (2009)Google Scholar
  6. 6.
    Werlberger, M., Trobin, W., Pock, T., Wedel, A., Cremers, D., Bischof, H.: Anisotropic Huber-L1 optical flow. In: Proceedings of the British Machine Vision Conference (2009)Google Scholar
  7. 7.
    Weickert, J., Schnörr, C.: A theoretical framework for convex regularizers in PDE-based computation of image motion. Int. J. Comput. Vis. 45(3), 245–264 (2001)CrossRefzbMATHGoogle Scholar
  8. 8.
    Trobin, W., Pock, T., Cremers, D., Bischof, H.: An unbiased second-order prior for high-accuracy motion estimation. In: Rigoll, G. (ed.) DAGM 2008. LNCS, vol. 5096, pp. 396–405. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  9. 9.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. J. Phys. D Appl. Phys. 60, 259–268 (1992)zbMATHGoogle Scholar
  10. 10.
    Zach, C., Pock, T., Bischof, H.: A duality based approach for realtime TV-L1 optical flow. In: Hamprecht, F.A., Schnörr, C., Jähne, B. (eds.) DAGM 2007. LNCS, vol. 4713, pp. 214–223. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  11. 11.
    Wedel, A., Pock, T., Braun, J., Franke, U., Cremers, D.: Duality TV-L1 flow with fundamental matrix prior. In: 23rd International Conference Image and Vision Computing, pp. 1–6, New Zealand (2008)Google Scholar
  12. 12.
    Rosman, G., Shem-Tov, S., Bitton, D., Nir, T., Adiv, G., Kimmel, R., Feuer, A., Bruckstein, A.M.: Over-parameterized optical flow using a stereoscopic constraint. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 761–772. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  13. 13.
    Kondermann, C., Kondermann, D., Garbe, C.S.: Postprocessing of optical flows via surface measures and motion inpainting. In: Rigoll, G. (ed.) DAGM 2008. LNCS, vol. 5096, pp. 355–364. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  14. 14.
    Matsushita, Y., Ofek, E., Ge, W., Tang, X., Shum, H.-Y.: Full-frame video stabilization with motion inpainting. IEEE Trans. Pattern Anal. Mach. Intell. 28(7), 1150–1163 (2006)CrossRefGoogle Scholar
  15. 15.
    Shiratori, T, Matsushita, Y.: Video completion by motion field transfer. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 411–418 (2006)Google Scholar
  16. 16.
    Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics: Texts in Applied Mathematics. Springer, New York (1990) CrossRefzbMATHGoogle Scholar
  17. 17.
    Strekalovskiy, E., Chambolle, A., Cremers, D.: Convex relaxation of vectorial problems with coupled regularization. SIAM J. Imaging Sci. 7, 294–336 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Berkels, B., Rätz, A., Rumpf, M., Voigt, A.: Extracting Grain boundaries and macroscopic deformations from images on atomic scale. J. Sci. Comput. 35(1), 1–23 (2007)CrossRefGoogle Scholar
  19. 19.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sánchez Pérez, J., Meinhardt-Llopis, E., Facciolo, G.: TV-L1 Optical Flow Estimation. Image Processing On Line (IPOL), pp. 137–150 (2013)Google Scholar
  22. 22.
    Baker, S., Scharstein, D., Lewis, J.P., Roth, S., Black, M.J., Szeliski, R.: A database and evaluation methodology for optical flow. Int. J. Comput. Vis. 92, 1–31 (2010)CrossRefGoogle Scholar
  23. 23.
    Butler, D.J., Wulff, J., Stanley, G.B., Black, M.J.: A naturalistic open source movie for optical flow evaluation. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part VI. LNCS, vol. 7577, pp. 611–625. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  24. 24.
    Bredies, K.: Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. In: Bruhn, A., Pock, T., Tai, X.-C. (eds.) Efficient Algorithms for Global Optimization Methods in Computer Vision. LNCS, vol. 8293, pp. 44–77. Springer, Heidelberg (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Roberto P. Palomares
    • 1
    Email author
  • Gloria Haro
    • 1
  • Coloma Ballester
    • 1
  1. 1.DTICPompeu Fabra UniversityBarcelonaSpain

Personalised recommendations