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Discrete Orthogonal Decomposition and Variational Fluid Flow Estimation

  • Jing Yuan
  • Paul Ruhnau
  • Etinne Mémin
  • Christoph Schnörr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3459)

Abstract

The decomposition of motion vector fields into components of orthogonal subspaces is an important representation for both the analysis and the variational estimation of complex motions. Common finite differencing or finite element methods, however, do not preserve the basic identities of vector analysis. Therefore, we introduce in this paper the mimetic finite difference method for the estimation of fluid flows from image sequences. Using this discrete setting, we represent the motion components directly in terms of potential functions which are useful for motion pattern analysis. Additionally, we analyze well-posedness which has been lacking in previous work. Experimental results, including hard physical constraints like vanishing divergence of the flow, validate the theory.

Keywords

Orthogonal Subspace Motion Vector Field Subspace Correction Real Image Sequence Scale Space Method 
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.
    Amodei, L., Benbourhim, M.N.: A vector spline approximation. J. Approx. Theory 67(1), 51–79 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Suter, D.: Motion estimation and vector splines. In: Proceedings of the Conference on Computer Vision and Pattern Recognition, June 1994, pp. 939–942. IEEE Computer Society Press, Los Alamitos (1994)CrossRefGoogle Scholar
  3. 3.
    Gupta, S., Prince, J.: Stochastic models for div-curl optical flow methods. Signal Proc. Letters 3(2), 32–34 (1996)CrossRefGoogle Scholar
  4. 4.
    Corpetti, T., Mémin, É., Pérez, P.: Dense motion analysis in fluid imagery. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 676–691. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Corpetti, T., Mémin, E., Pérez, P.: Dense estimation of fluid flows. IEEE Trans. Pattern Anal. Machine Intell. 24(3), 365–380 (2002)CrossRefGoogle Scholar
  6. 6.
    Corpetti, T., Mémin, E., Pérez, P.: Extraction of singular points from dense motion fields: an analytic approach. J. of Math. Imag. Vision 19(3), 175–198 (2003)zbMATHCrossRefGoogle Scholar
  7. 7.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer, Heidelberg (1986)zbMATHGoogle Scholar
  8. 8.
    Kohlberger, T., Mémin, E., Schnörr, C.: Variational dense motion estimation using the helmholtz decomposition. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 432–448. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Hyman, J.M., Shashkov, M.J.: Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl. 33(4), 81–104 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hyman, J.M., Shashkov, M.J.: Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids. Appl. Numer. Math. 25(4), 413–442 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hyman, J.M., Shashkov, M.J.: The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal. 36(3), 788–818 (1999) (electronic)CrossRefMathSciNetGoogle 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(1), 75–104 (1996)CrossRefGoogle Scholar
  13. 13.
    Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
  14. 14.
    Xu, J.: Iterative methods by space decomposition and subspace correction: A unifying approach. SIAM Review 34, 581–613 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tai, X.-C., Xu, J.: Global and uniform convergence of subspace correction methods for some convex optimization problems. Math. Comp. 71(237), 105–124 (2002) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1995) (1999)zbMATHGoogle Scholar
  17. 17.
    Barron, J.L., Fleet, D.J., Beauchemin, S.S.: Performance of optical flow techniques. International Journal of Computer Vision 12(1), 43–77 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jing Yuan
    • 1
  • Paul Ruhnau
    • 1
  • Etinne Mémin
    • 2
  • Christoph Schnörr
    • 1
  1. 1.Department of Mathematics and Computer Science, Computer Vision, Graphics, and Pattern Recognition GroupUniversity of MannheimMannheimGermany
  2. 2.IRISA RennesRennes CedexFrance

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