Sobolev Active Contours

  • Ganesh Sundaramoorthi
  • Anthony Yezzi
  • Andrea Mennucci
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3752)


All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions. This is particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and region-based active contour models as Sobolev active contours and show the substantial improvements gained in segmentation and tracking applications.


Active Contour Active Contour Model Initial Contour Tracking Application Geodesic Active Contour 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision 1, 321–331 (1987)CrossRefGoogle Scholar
  2. 2.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proceedings of the IEEE Int. Conf. on Computer Vision, Cambridge, MA, USA, pp. 694–699 (1995)Google Scholar
  3. 3.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: Proceedings of the IEEE Int. Conf. on Computer Vision, pp. 810–815 (1995)Google Scholar
  4. 4.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Yezzi, A., Tsai, A., Willsky, A.: A statistical approach to snakes for bimodal and trimodal imagery. In: Int. Conf. on Computer Vision, pp. 898–903 (1999)Google Scholar
  6. 6.
    Chan, T., Vese, L.: Active contours without edges. IEEE Transactions on Image Processing 10, 266–277 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Michor, P., Mumford, D.: Riemannian geometries on the space of plane curves. ESI Preprint 1425 (2003) arXiv:math.DG/0312384Google Scholar
  8. 8.
    Yezzi, A., Mennucci, A.: Metrics in the space of curves. Preprint (2005) arXiv:math.DG /0412454Google Scholar
  9. 9.
    Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: algorithms based on the Hamilton-Jacobi equations. J. Comp. Physics 79, 12–49 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Younes, L.: Computable elasic distances between shapes. SIAM J. Appl. Math. 58, 565–586 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kriegl, A., Michor, P.: The Convenient Setting of Global Analysis. In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society (1997)Google Scholar
  12. 12.
    Soatto, S., Yezzi, A.J.: DEFORMOTION: Deforming motion, shape average and the joint registration and segmentation of images. In: ECCV, vol. (3), pp. 32–57 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ganesh Sundaramoorthi
    • 1
  • Anthony Yezzi
    • 1
  • Andrea Mennucci
    • 2
  1. 1.School of Electrical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsScuola Normale SuperiorePisaItaly

Personalised recommendations