Two-Frame Motion Estimation Based on Polynomial Expansion

  • Gunnar Farnebäck
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2749)


This paper presents a novel two-frame motion estimation algorithm. The first step is to approximate each neighborhood of both frames by quadratic polynomials, which can be done efficiently using the polynomial expansion transform. From observing how an exact polynomial transforms under translation a method to estimate displacement fields from the polynomial expansion coefficients is derived and after a series of refinements leads to a robust algorithm. Evaluation on the Yosemite sequence shows good results.


Computer Vision Motion Model Quadratic Polynomial Polynomial Expansion Orientation Tensor 
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  1. 1.
    Farnebäck, G.: Fast and Accurate Motion Estimation using Orientation Tensors and Parametric Motion Models. In: Proceedings of 15th International Conference on Pattern Recognition. Volume 1., Barcelona, Spain, IAPR (2000) 135–139Google Scholar
  2. 2.
    Farnebäck, G.: Very High Accuracy Velocity Estimation using Orientation Tensors, Parametric Motion, and Simultaneous Segmentation of the Motion Field. In: Proceedings of the Eighth IEEE International Conference on Computer Vision. Volume I., Vancouver, Canada (2001) 171–177CrossRefGoogle Scholar
  3. 3.
    URL: Scholar
  4. 4.
    Knutsson, H., Westin, C.F.: Normalized and Differential Convolution: Methods for Interpolation and Filtering of Incomplete and Uncertain Data. In: Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York City, USA, IEEE (1993) 515–523CrossRefGoogle Scholar
  5. 5.
    Westin, C.F.: A Tensor Framework for Multidimensional Signal Processing. PhD thesis, Linköping University, Sweden, SE-581 83 Linköping, Sweden (1994) Dissertation No 348, ISBN 91-7871-421-4.Google Scholar
  6. 6.
    Farnebäck, G.: Polynomial Expansion for Orientation and Motion Estimation. PhD thesis, Linköping University, Sweden, SE-581 83 Linköping, Sweden (2002) Dissertation No 790, ISBN 91-7373-475-6.Google Scholar
  7. 7.
    Heeger, D.J.: Model for the extraction of image flow. J. Opt. Soc. Am. A 4 (1987) 1455–1471CrossRefGoogle Scholar
  8. 8.
    Barron, J.L., Fleet, D.J., Beauchemin, S.S.: Performance of optical flow techniques. Int. J. of Computer Vision 12 (1994) 43–77CrossRefGoogle Scholar
  9. 9.
    Lucas, B., Kanade, T.: An Iterative Image Registration Technique with Applications to Stereo Vision. In: Proc. Darpa IU Workshop. (1981) 121–130Google Scholar
  10. 10.
    Uras, S., Girosi, F., Verri, A., Torre, V.: A computational approach to motion perception. Biological Cybernetics (1988) 79–97Google Scholar
  11. 11.
    Fleet, D.J., Jepson, A.D.: Computation of Component Image Velocity from Local Phase Information. Int. Journal of Computer Vision 5 (1990) 77–104CrossRefGoogle Scholar
  12. 12.
    Black, M.J., Anandan, P.: The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields. Computer Vision and Image Understanding 63 (1996) 75–104CrossRefGoogle Scholar
  13. 13.
    Szeliski, R., Coughlan, J.: Hierarchical spline-based image registration. In: Proc. IEEE Conference on Computer Vision Pattern Recognition, Seattle, Washington (1994) 194–201Google Scholar
  14. 14.
    Black, M.J., Jepson, A.: Estimating optical flow in segmented images using variable-order parametric models with local deformations. IEEE Transactions on Pattern Analysis and Machine Intelligence 18 (1996) 972–986CrossRefGoogle Scholar
  15. 15.
    Ju, S.X., Black, M.J., Jepson, A.D.: Skin and bones: Multi-layer, locally affine, optical flow and regularization with transparency. In: Proceedings CVPR’96, IEEE (1996) 307–314Google Scholar
  16. 16.
    Karlholm, J.: Local Signal Models for Image Sequence Analysis. PhD thesis, Linköping University, Sweden, SE-581 83 Linköping, Sweden (1998) Dissertation No 536, ISBN 91-7219-220-8.Google Scholar
  17. 17.
    Lai, S.H., Vemuri, B.C.: Reliable and efficient computation of optical flow. International Journal of Computer Vision 29 (1998) 87–105CrossRefGoogle Scholar
  18. 18.
    Bab-Hadiashar, A., Suter, D.: Robust optic flow computation. International Journal of Computer Vision 29 (1998) 59–77CrossRefGoogle Scholar
  19. 19.
    Mémin, E., Pérez, P.: Hierarchical estimation and segmentation of dense motion fields. International Journal of Computer Vision 46 (2002) 129–155zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Gunnar Farnebäck
    • 1
  1. 1.Computer Vision LaboratoryLinköping UniversityLinköpingSweden

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