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A Variational Approach to Joint Denoising, Edge Detection and Motion Estimation

  • Alexandru Telea
  • Tobias Preusser
  • Christoph Garbe
  • Marc Droske
  • Martin Rumpf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4174)

Abstract

The estimation of optical flow fields from image sequences is incorporated in a Mumford–Shah approach for image denoising and edge detection. Possibly noisy image sequences are considered as input and a piecewise smooth image intensity, a piecewise smooth motion field, and a joint discontinuity set are obtained as minimizers of the functional. The method simultaneously detects image edges and motion field discontinuities in a rigorous and robust way. It comes along with a natural multi–scale approximation that is closely related to the phase field approximation for edge detection by Ambrosio and Tortorelli. We present an implementation for 2D image sequences with finite elements in space and time. It leads to three linear systems of equations, which have to be iteratively in the minimization procedure. Numerical results underline the robustness of the presented approach and different applications are shown.

Keywords

Optical Flow Edge Detection Motion Estimation Image Denoising Variational Framework 
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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References

  1. 1.
    Horn, B.K.P., Schunk, B.: Determining optical flow. Artificial Intelligence 17, 185–204 (1981)CrossRefGoogle Scholar
  2. 2.
    Nagel, H.H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of dispalcement vector fields from image sequences. IEEE Trans. on PAMI 8(5), 565–593 (1986)Google Scholar
  3. 3.
    Weickert, J., Schnörr, C.: A theoretical framework for convex regularizers in pde-based computation of image motion. Int. J. of Comp. Vision 45(3), 245–264 (2001)MATHCrossRefGoogle Scholar
  4. 4.
    Bruhn, A., Weickert, J., Feddern, C., Kohlberger, T., Schnörr, C.: Real-Time Optical Flow Computation with Variational Methods. In: Petkov, N., Westenberg, M.A. (eds.) CAIP 2003. LNCS, vol. 2756, pp. 222–229. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Wang, J.Y.A., Adelson, E.H.: Representating moving images with layers. IEEE Trans. on Im. Proc. 3(5), 625–638 (1994)CrossRefGoogle Scholar
  6. 6.
    Schnörr, C.: Segmentation of visual motion by minimizing convex non-quadratic functionals. In: 12th ICPR (1994)Google Scholar
  7. 7.
    Odobez, J.M., Bouthemy, P.: Robust multiresolution estimation of parametric motion models. J. of Vis. Comm. and Image Rep. 6(4), 348–365 (1995)CrossRefGoogle Scholar
  8. 8.
    Odobez, J.M., Bouthemy, P.: Direct incremental model-based image motion segmentation for video analysis. Sig. Proc. 66, 143–155 (1998)MATHCrossRefGoogle Scholar
  9. 9.
    Caselles, V., Coll, B.: Snakes in movement. SIAM J. Num. An. 33, 2445–2456 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Memin, E., Perez, P.: A multigrid approach for hierarchical motion estimation. In: ICCV, pp. 933–938 (1998)Google Scholar
  11. 11.
    Paragios, N., Deriche, R.: Geodesic active contours and level sets for the detection and tracking of moving objects. IEEE Trans. on PAMI 22(3), 266–280 (2000)Google Scholar
  12. 12.
    Droske, M., Ring, W.: A Mumford-Shah level-set approach for geometric image registration. SIAM Appl. Math. (to appear, 2005)Google Scholar
  13. 13.
    Mumford-shah based registration. Computing and Visualization in Science (submitted, 2005)Google Scholar
  14. 14.
    Kapur, T., Yezzi, L., Zöllei, L.: A variational framework for joint segmentation and registration. IEEE CVPR, 44–51 (2001)Google Scholar
  15. 15.
    Unal, G., Slabaugch, G., Yezzi, A., Tyan, J.: Joint segmentation and non-rigid registration without shape priors (2004)Google Scholar
  16. 16.
    Vemuri, B., Ye, J., Chen, Y., Leonard, C.: Image registration via level-set motion: Applications to atlas-based segmentation. Med. Im. Analysis 7, 1–20 (2003)CrossRefGoogle Scholar
  17. 17.
    Davatzikos, C.A., Bryan, R.N., Prince, J.L.: Image registration based on boundary mapping. IEEE Trans. Med. Imaging 15(1), 112–115 (1996)CrossRefGoogle Scholar
  18. 18.
    Cremers, D., Soatto, S.: Motion competition: A variational framework for piecewise parametric motion segmentation. Int. J. of Comp. Vision 62(3), 249–265 (2005)CrossRefGoogle Scholar
  19. 19.
    Cremers, D., Kohlberger, T., Schnörr, C.: Nonlinear Shape Statistics in Mumford-Shah Based Segmentation. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2351, pp. 93–108. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Brox, T., Bruhn, A., Papenberg, N., Weickert, J.: High Accuracy Optical Flow Estimation Based on a Theory for Warping. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 25–36. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  22. 22.
    Amiaz, T., Kiryati, N.: Dense discontinuous optical flow via contour-based segmentation. In: Proc. ICIP 2005, vol. III, pp. 1264–1267 (2005)Google Scholar
  23. 23.
    Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. Computer Vision 50, 271–293 (2002)MATHCrossRefGoogle Scholar
  24. 24.
    Nir, T., Kimmel, R., Bruckstein, A.: Variational approach for joint optic-flow computation and video restoration. Technical report, Dep. of C. S. - Israel Inst. of Tech., Haifa, Israel (2005)Google Scholar
  25. 25.
    Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B 6(7), 105–123 (1992)MATHMathSciNetGoogle Scholar
  26. 26.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  27. 27.
    Bourdin, B.: Image segmentation with a Finite Element method. ESIAM: Math. Modelling and Num. Analysis 33(2), 229–244 (1999)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Bourdin, B., Chambolle, A.: Implementation of an adaptive Finite-Element approximation of the Mumford-Shah functional. Numer. Math. 85(4), 609–646 (2000)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Group, C.V.R.: Optical flow datasets. Univ. of Otago, New Zealand (2005), http://www.cs.otago.ac.nz/research/vision

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexandru Telea
    • 1
  • Tobias Preusser
    • 2
  • Christoph Garbe
    • 3
  • Marc Droske
    • 4
  • Martin Rumpf
    • 5
  1. 1.Eindhoven University of Technology 
  2. 2.CeVisUniversity of Bremen 
  3. 3.IWR, University of Heidelberg 
  4. 4.UCLALosAngeles
  5. 5.INSUniversity of Bonn 

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