Journal of Mathematical Imaging and Vision

, Volume 24, Issue 2, pp 259–276 | Cite as

Segmentation of a Vector Field: Dominant Parameter and Shape Optimization

  • Tristan Roy
  • Éric DebreuveEmail author
  • Michel Barlaud
  • Gilles Aubert


Vector field segmentation methods usually belong to either of three classes: methods which segment regions homogeneous in direction and/or norm, methods which detect discontinuities in the vector field, and region growing or classification methods. The first two classes of method do not allow segmentation of complex vector fields and control of the type of fields to be segmented, respectively. The third class does not directly allow a smooth representation of the segmentation boundaries. In the particular case where the vector field actually represents an optical flow, a fourth class of methods acts as a detector of main motion. The proposed method combines a vector field model and a theoretically founded minimization approach. Compared to existing methods following the same philosophy, it relies on an intuitive, geometric way to define the model while preserving a general point of view adapted to the segmentation of potentially complex vector fields with the condition that they can be described by a finite number of parameters. The energy to be minimized is deduced from the choice of a specific class of field lines, e.g. straight lines or circles, described by the general form of their parametric equations. In that sense, the proposed method is a principled approach for segmenting parametric vector fields. The minimization problem was rewritten into a shape optimization and implemented by spline-based active contours. The algorithm was applied to the segmentation of precomputed optical flow fields given by an external, independent algorithm.


segmentation vector field dominant parameter shape optimization optical flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Adalsteinsson and J.A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comp. Phys., Vol. 148, pp. 2–22, 1999.MathSciNetGoogle Scholar
  2. 2.
    G. Aubert, M. Barlaud, O. Faugeras, and S. Jehan-Besson, “Image segmentation using active contours: Calculus of variations or shape gradients?” SIAM J. Appl. Math., Vol. 63, pp. 2128–2154, 2003.CrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Aubert, R. Deriche, and P. Kornprobst, “Computing optical flow via variational techniques,” SIAM J. Appl. Math., Vol. 60, pp. 156–182, 1999.CrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Barles, “Remarks on a flame propagation model,” Technical Report 464, Team Sinus, INRIA, Sophia Antipolis, France, 1985.Google Scholar
  5. 5.
    L. Blance-Féraud, M. Barlaud, and T. Gaidon, “Motion estimation involving discontinuities in a multiresolution scheme,” Optical Engineering, Vol. 32, pp. 1475–1482, 1993.Google Scholar
  6. 6.
    P. Brigger, J. Hoeg, and M. Unser, “B-spline snakes: A flexible tool for parametric contour detection,” IEEE Trans. Imag. Proc., Vol. 9, pp. 1484–1496, 2000.CrossRefMathSciNetGoogle Scholar
  7. 7.
    V. Caselles, F. Catté, T. Coll, and F. Dibos, “A geometric model for active contours,” Nümerische Mathematik, Vol. 66, pp. 1–31, 1993.Google Scholar
  8. 8.
    V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comp. Vision, Vol. 22, pp. 61–79, 1997.Google Scholar
  9. 9.
    J.G. Choi and S.-D. Kim, “Multi-stage segmentation of optical flow field,” Signal Processing, Vol. 54, pp. 109–118, 1996.MathSciNetGoogle Scholar
  10. 10.
    D.L. Chopp, “Computing minimal surfaces via level set curvature flow,” J. Comp. Phys., Vol. 106, pp. 77–91, 1993.zbMATHMathSciNetGoogle Scholar
  11. 11.
    J. Condell, B. Scotney, and P. Morrow, “Adaptative grid refinement procedures for efficient optical flow computation,” Int. J. Comp. Vision, Vol. 61, pp. 31–54, 2005.Google Scholar
  12. 12.
    D. Cremers and C. Schnörr, “Motion competition: Variational integration of motion segmentation and shape regularization,” in Pattern Recognition, Zürich, Switzerland, 2002, pp. 472–480.Google Scholar
  13. 13.
    D. Cremers and C. Soatto, “Motion competition: A variational approach to piecewise parametric motion segmentation,” Int. J. Comp. Vision, Vol. 62, pp. 249–265, 2005.Google Scholar
  14. 14.
    M.C. Delfour and J.-P. Zolesio, “Shapes and geometries: Analysis, differential calculus and optimization,” in Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001.Google Scholar
  15. 15.
    A. Giachetti and V. Torre, “Refinement of optical flow estimation and detection of motion edges,” in European Conference on Computer Vision, Cambridge, UK, 1996, pp. 151–160.Google Scholar
  16. 16.
    J. Gomes and O.D. Faugeras, “Reconciling distance functions and level sets,” Journal of Visual Communication and Image Representation, Vol. 11, pp. 209–223, 2000.CrossRefGoogle Scholar
  17. 17.
    M. Hintermüller and W. Ring, “A second order shape optimization approach for image segmentation,” SIAM J. Appl. Math., Vol. 64, pp. 442–467, 2003.CrossRefMathSciNetGoogle Scholar
  18. 18.
    B.K.P. Horn and B.G. Schunck, “Detemining optical flow,” Artificial Intelligence, Vol. 17, pp. 185–203, 1981.CrossRefGoogle Scholar
  19. 19.
    M. Jacob, T. Blu, and M. Unser, “A unifying approach and interface for spline-based snakes,” in SPIE International Symposium on Medical Imaging: Image Processing, San Diego, CA, USA, 2001, pp. 340–347.Google Scholar
  20. 20.
    S. Jehan-Besson, M. Barlaud, and G. Aubert, “Dream2s: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation,” Int. J. Comp. Vision, Vol. 53, pp. 45–70, 2003.Google Scholar
  21. 21.
    M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” Int. J. Comp. Vision, Vol. 1, pp. 321–332, 1988.Google Scholar
  22. 22.
    B. Lucas and T. Kanade, “An iterative image registration technique with an application to stereo vision,” in International Joint Conference on Artificial Intelligence, Vancouver, Canada, 1981, pp. 674–679.Google Scholar
  23. 23.
    D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Comm. Pure Appl. Math., Vol. 42, pp. 577–685, 1989.MathSciNetGoogle Scholar
  24. 24.
    S. Osher and J.A. Sethian, “Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comp. Phys., Vol. 79, pp. 12–49, 1988.MathSciNetGoogle Scholar
  25. 25.
    F. Precioso, M. Barlaud, T. Blu, and M. Unser, “Smoothing B-spline active contour for fast and robust image and video segmentation,” in International Conference on Image Processing, Barcelona, Spain, 2003, pp. 137–140.Google Scholar
  26. 26.
    R. Ronfard, “Region-based strategies for active contour models,” Int. J. Comp. Vis., Vol. 13, pp. 229–251, 1994.Google Scholar
  27. 27.
    T. Roy, M. Barlaud, E. Debreuve, and G. Aubert, “Vector field segmentation using active contours: Regions of vectors with the same direction,” in Workshop on Variational, Geometric and Level Set Methods in Computer Vision, Nice, France, 2003.Google Scholar
  28. 28.
    O. Sanchez and F. Dibos, “Displacement following of hidden objects in a video sequence,” Int. J. Comp. Vision, Vol. 57, pp. 91–105, 2004.Google Scholar
  29. 29.
    G. Sapiro, “Vector (self) snakes: A geometric framework for color, texture and multiscale image segmentation,” in International Conference on Image Processing, Lausanne, Switzerland, 1996, pp. 817–820.Google Scholar
  30. 30.
    C. Schnörr, “Computation of discontinuous optical flow by domain decomposition and shape optimization,” Int. J. Comp. Vision, Vol. 8, pp. 153–165, 1992.Google Scholar
  31. 31.
    M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing: Part I — Theory,” IEEE Trans. Signal Proc., Vol, 41, pp. 821–833, 1993.Google Scholar
  32. 32.
    J.Y.A. Wang and E.H. Adelson, “Spatio-temporal segmentation of video data,” in SPIE on Image and Video Processing II, Vol. 2182, San Jose (CA), USA, 1994, pp. 120–131.Google Scholar
  33. 33.
    J. Weickert and C. Schnörr, “Variational optic flow computation with a spatio-temporal smoothness constraint,” Journal of Mathematical Imaging and Vision, Vol. 14, pp. 245–255, 2001.CrossRefGoogle Scholar
  34. 34.
    S.F. Wu and J. Kittler, “A gradient-based method for general motion estimation and segmentation,” Journal of Visual Communication and Image Representation, Vol. 4, pp. 25–38, 1993.CrossRefGoogle Scholar
  35. 35.
    S. Zhu and K.-K. Ma, “A new diamond search algorithm for fast block matching motion estimation,” IEEE Imag. Proc., Vol. 9, pp. 287–290, 2000.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Tristan Roy
    • 1
  • Éric Debreuve
    • 1
    Email author
  • Michel Barlaud
    • 1
  • Gilles Aubert
    • 2
  1. 1.Laboratoire I3S, UMR CNRS 6070Les Algorithmes, BâtSophia Antipolis CedexFrance
  2. 2.Laboratoire Dieudonné, UMR CNRS 6621Université de Nice-Sophia AntipolisNice Cedex 2France

Personalised recommendations