International Journal of Computer Vision

, Volume 73, Issue 3, pp 345–366 | Cite as

Sobolev Active Contours

  • Ganesh Sundaramoorthi
  • Anthony Yezzi
  • Andrea C. Mennucci


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; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are 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.


active contours gradient flows Sobolev norm global flows shape optimization 


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  1. Adalsteinsson, D. and Sethian, J. 1995. A fast level set method for propagating interfaces. J. Comp. Phys., 118:269–277.CrossRefMathSciNetzbMATHGoogle Scholar
  2. Blake, A. and Isard, M. 1998, Active Contours. Springer Verlag.Google Scholar
  3. Burger, M. 2003. A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Boundaries, 5:301–329.MathSciNetzbMATHGoogle Scholar
  4. Burger, M. and Osher, S. 2005. A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math., 16.Google Scholar
  5. Caselles, V., Catte, F., Coll, T., and Dibos, F. 1993. A geometric model for edge detection. Num. Mathematik, 66:1–31.CrossRefMathSciNetzbMATHGoogle Scholar
  6. Caselles, V., Kimmel, R., and Sapiro, G. 1995. Geodesic active contours. In: Proc. of the IEEE Int. Conf. on Computer Vision, Cambridge, MA, USA, pp. 694–699.Google Scholar
  7. Chan, T. and Vese, L. 2001. Active contours without edges. IEEE Transactions on Image Processing, 10(2):266–277.CrossRefzbMATHGoogle Scholar
  8. Charpiat, G., Faugeras, O.D., and Keriven, R. 2005a. Approximations of shape metrics and application to shape warping and empirical shape statistics. Foundations Comput. Math., 5(1):1–58.CrossRefMathSciNetzbMATHGoogle Scholar
  9. Charpiat, G., Keriven, R., Pons, J., and Faugeras, O. 2005b. Designing spatially coherent minimizing flows for variational problems base d on Active Contours. In: ICCV.Google Scholar
  10. Chen, Y., Tagare, H., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K., Briggs, R., and Geiser, E. 2002. Using prior shapes in geometric active contours in a variational framework. Int. J. Comput. Vision, 50(3):315–328.CrossRefzbMATHGoogle Scholar
  11. Chopp, D.L. and Sethian, J.A. 1999. Motion by intrinsic laplacian of curvature. Interfaces Free Boundaries, 1:107–123.MathSciNetzbMATHGoogle Scholar
  12. Cohen, L.D. 1991. On active contour models and ballons. Comput. Vision, Graphics, and Image Processing: Image Processing, 53(2).Google Scholar
  13. Cremers, D. and Soatto, S. 2003. A pseuso distance for shape priors in level set segmentation. In: IEEE Int. Workshop on Variational, Geometric and Level Set Methods, pp. 169–176.Google Scholar
  14. do Carmo, M. 1992. Riemannian Geometry. Birkhäuser Boston.zbMATHGoogle Scholar
  15. Droske, M. and Rumpf, M. 2004. A level set formulation for the willmore flow. Interfaces and Boundaries, 6(3):361–378.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hintermüller, M. and Ring, W. 2004. An inexact Newton-CG-type active contour approach for the minimization of the mumford-shah functional. J. Math. Imaging Vision, 20(1):19–42.CrossRefMathSciNetGoogle Scholar
  17. Kass, M., Witkin, A., and Terzopoulos, D. 1987. Snakes: Active contour models. Int. J. Comput. Vision, 1:321–331.CrossRefGoogle Scholar
  18. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., and Yezzi, A. 1995. Gradient flows and geometric active contour models. In: Proc. of the IEEE Int. Conf. on Comput. Vision, pp. 810–815.Google Scholar
  19. Lang, S. 1999. Fundamentals of Differential Geometry. Springer-Verlag.Google Scholar
  20. Leventon, M., Grimson, E., and Faugeras, O. 2000. Statistical Shape influence in geodesic active contours. In: IEEE Conf. on Comp. Vision and Patt. Recog., vol. 1, pp. 316–323.Google Scholar
  21. Malladi, R., Sethian, J., and Vemuri, B. 1995. Shape modeling with front propagation: a level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17:158–175.CrossRefGoogle Scholar
  22. Mansouri, A.-R., Mukherjee, D.P., and Acton, S.T. 2004. Constraining active contour evolution via Lie Groups of transformation. IEEE Transactions on Image Processing, 13(6):853–863.CrossRefMathSciNetGoogle Scholar
  23. Mennucci, A.C.G., Yezzi, A., and Sundaramoorthi, G. 2006. Sobolev–type metrics in the space of curves. Preprint, arXiv:math.DG/0605017.Google Scholar
  24. Michor, P. and Mumford, D. 2003. Riemannian geometries on the space of plane curves. ESI Preprint 1425, arXiv:math.DG/0312384.Google Scholar
  25. Mio, W. and Srivastava, A. 2004. Elastic-string models for representation and analysis of planar shapes. In: CVPR, vol. 2, pp. 10–15.Google Scholar
  26. Mumford, D. and Shah, J. 1985. Boundary detection by minimizing functionals. In: Proc. IEEE Conf. Computer Vision Pattern Recognition.Google Scholar
  27. Mumford, D. and Shah, J. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42:577–685.MathSciNetzbMATHGoogle Scholar
  28. Neuberger, J.W. 1997. Sobolev Gradients and Differential Equations. Lecture Notes in Mathematics #1670. Springer.Google Scholar
  29. Osher, S. and Sethian, J. 1988. Fronts propagating with curvature-dependent speed: algorithms based on the Hamilton-Jacobi equations. J. Comp. Phys., 79:12–49.CrossRefMathSciNetzbMATHGoogle Scholar
  30. Paragios, N. and Deriche, R. 2002a. Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision. International Journal of Visual Communication and Image Representation, Special Issue on Partial Differential Equations in Image Processing, Computer Vision and Computer Graphics, 13(2):249–268.Google Scholar
  31. Paragios, N. and Deriche, R. 2002b. Geodesic active regions and level set methods for supervised texture segmentation. Int. J. Comput. Vision, 46(3):223.CrossRefzbMATHGoogle Scholar
  32. Raviv, T.R., Kiryati, N., and Sochen, N. 2004. Unlevel-set: Geometry and prior-based segmentation. In: Proc. European Conf. on Computer Vision.Google Scholar
  33. Ronfard, R. 1994. Region based strategies for active contour models. Int. J. Comput. Vision, 13(2):229–251.CrossRefGoogle Scholar
  34. Rousson, M. and Paragios, N. 2002. Shape Priors for Level Set Representations. In: Proc. European Conf. Computer Vision, vol. 2, pp. 78–93.Google Scholar
  35. Rouy, E. and Tourin, A. 1992. A viscosity solutions approach to shape-from-shading. SIAM J. Numerical Anal., 29(3):867–884.CrossRefMathSciNetzbMATHGoogle Scholar
  36. Siddiqi, K., Lauzière, Y.B., Tannenbaum, A., and Zucker, S. 1998. Area and length minimizing flows for shape segmentation. IEEE Transactions on Image Processing, 3(7):433–443.CrossRefGoogle Scholar
  37. Soatto, S. and Yezzi, A.J. 2002. DEFORMOTION: Deforming motion, shape average and the joint registration and segmentation of images. In: ECCV, vol. 3, pp. 32–57.Google Scholar
  38. Sundaramoorthi, G., Yezzi, A., and Mennucci, A. 2005. Sobolev active contours. In: VLSM, pp. 109–120.Google Scholar
  39. Tsai, A., Yezzi, A., and Willsky, A.S. 2001a. Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Transactions on Image Processing, 10(8):1169–1186.CrossRefzbMATHGoogle Scholar
  40. Tsai, A., Yezzi, A.J., III, W.M.W., Tempany, C., Tucker, D., Fan, A., Grimson, W.E.L., and Willsky, A.S. 2001b. Model-based curve evolution technique for image segmentation. In: CVPR, vol. 1, pp. 463–468.Google Scholar
  41. Vese, L.A. and Chan, T.F. 2002. A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. Comput. Vision, 50(3):271–293.CrossRefzbMATHGoogle Scholar
  42. Xu, C. and Prince, J.L. 1998. Snakes, shapes, and gradient vector flow. IEEE Transactions on Image Processing, 7(3):359–369.CrossRefMathSciNetzbMATHGoogle Scholar
  43. Yezzi, A. and Mennucci, A. 2005a. Metrics in the space of curves. Preprint, arXiv:math.DG /0412454.Google Scholar
  44. Yezzi, A., Tsai, A., and Willsky, A. 1999. A statistical approach to snakes for bimodal and trimodal imagery. In: Int. Conf. on Comput. Vision, pp. 898–903.Google Scholar
  45. Yezzi, A.J. and Mennucci, A. 2005b. Conformal metrics and true “Gradient flows” for curves. In: ICCV, pp. 913–919.Google Scholar
  46. Younes, L. 1998. Computable elastic distances between shapes. SIAM J. Appl. Math., 58(2):565–586.CrossRefMathSciNetzbMATHGoogle Scholar
  47. Zhu, S.C., Lee, T.S., and Yuille, A.L. 1995. Region competition: Unifying snakes, region growing, Energy/Bayes/MDL for multi-band image segmentation. In: ICCV, pp. 416–423.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

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

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