International Journal of Computer Vision

, Volume 84, Issue 2, pp 113–129 | Cite as

New Possibilities with Sobolev Active Contours

  • Ganesh Sundaramoorthi
  • Anthony Yezzi
  • Andrea C. Mennucci
  • Guillermo Sapiro


Recently, the Sobolev metric was introduced to define gradient flows of various geometric active contour energies. It was shown that the Sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important. Some interesting properties of Sobolev gradient flows include that they stabilize certain unstable traditional flows, and the order of the evolution PDEs are reduced when compared with traditional gradient flows of the same energies. In this paper, we explore new possibilities for active contours made possible by Sobolev metrics. The Sobolev method allows one to implement new energy-based active contour models that were not otherwise considered because the traditional minimizing method render them ill-posed or numerically infeasible. In particular, we exploit the stabilizing and the order reducing properties of Sobolev gradients to implement the gradient descent of these new energies. We give examples of this class of energies, which include some simple geometric priors and new edge-based energies. We also show that these energies can be quite useful for segmentation and tracking. We also show that the gradient flows using the traditional metric are either ill-posed or numerically difficult to implement, and then show that the flows can be implemented in a stable and numerically feasible manner using the Sobolev gradient.


Active contours Gradient flows Sobolev norm Global flows Shape optimization Shape priors Ill-posed flows 


  1. Boykov, Y., & Jolly, M.-P. (2001). Interactive graph cuts for optimal boundary and region segmentation of objects in N–D images. In ICCV (pp. 105–112). Google Scholar
  2. Brook, A., Bruckstein, A. M., & Kimmel, R. (2005). On similarity-invariant fairness measures. In Scale-Space (pp. 456–467). Google Scholar
  3. Bruckstein, A. M., & Netravali, A. N. (1990). On minimal energy trajectories. Computer Vision, Graphics, and Image Processing, 49(3), 283–296. CrossRefGoogle Scholar
  4. Caselles, V., Catte, F., Coll, T., & Dibos, F. (1993). A geometric model for edge detection. Numerische Mathematik, 66, 1–31. MATHCrossRefMathSciNetGoogle Scholar
  5. Caselles, V., Kimmel, R., & Sapiro, G. (1995). Geodesic active contours. In Proceedings of the IEEE int. conf. on computer vision (pp. 694–699). Cambridge, MA, USA. Google Scholar
  6. Chan, T., & Vese, L. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277. MATHCrossRefGoogle Scholar
  7. Charpiat, G., Keriven, R., Pons, J., & Faugeras, O. (2005). Designing spatially coherent minimizing flows for variational problems based on active contours. In ICCV. Google Scholar
  8. Charpiat, G., Maurel, P., Keriven, R., Pons, J.-P., & Faugeras, O. D. (2007). Generalized gradients: priors on minimization flows. International Journal of Computer Vision, 73(3), 325–344. CrossRefGoogle Scholar
  9. Chen, Y., Tagare, H., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K., Briggs, R., & Geiser, E. (2002). Using prior shapes in geometric active contours in a variational framework. International Journal of Computer Vision, 50(3), 315–328. MATHCrossRefGoogle Scholar
  10. Cohen, L. D., & Kimmel, R. (1996). Global minimum for active contour models: a minimal path approach. In CVPR (pp. 666–673). Google Scholar
  11. Cremers, D., & Schnörr, C. (2001). Diffusion-snakes: combining statistical shape knowledge and image information in a variational framework. In Proc. IEEE workshop on variational, geometric, level set methods in computer vision (pp. 137–144). Google Scholar
  12. Cremers, D., & Soatto, S. (2003). A pseudo distance for shape priors in level set segmentation. In IEEE int. workshop on variational, geometric and level set methods (pp. 169–176). Google Scholar
  13. Delingette, H. (2001). On smoothness measures of active contours and surfaces. In VLSM ’01: Proceedings of the IEEE workshop on variational and level set methods (VLSM’01) (p. 43). Washington, DC, USA. Google Scholar
  14. Droske, M., & Rumpf, M. (2004). A level set formulation for the willmore flow. Interfaces and Boundaries, 6(3), 361–378. MATHMathSciNetCrossRefGoogle Scholar
  15. Eckstein, I., Pons, J., Tong, Y., Kuo, C., & Desbrun, M. (2007). Generalized surface flows for mesh processing. In Symposium on geometry processing (pp. 183–192). Google Scholar
  16. Foulonneau, A., Charbonnier, P., & Heitz, F. (2006). Affine-invariant geometric shape priors for region-based active contours. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(8), 1352–1357. CrossRefGoogle Scholar
  17. Fua, P., & Leclerc, Y. G. (1990). Model driven edge detection. Machine Vision and Applications, 3(1), 45–56. CrossRefGoogle Scholar
  18. Guyader, C. L., & Vese, L. (2007). Self-repelling snakes for topology segmentation models (Technical Report). UCLA. Google Scholar
  19. Horn, B. K. P. (1983). The curve of least energy. ACM Transactions on Mathematical Software, 9(4), 441–460. MATHCrossRefMathSciNetGoogle Scholar
  20. Jackson, J., Yezzi, A., & Soatto, S. (2004). Tracking deformable moving objects under severe occulsions. In IEEE conference on decision and control. Google Scholar
  21. Kass, M., Witkin, A., & Terzopoulos, D. (1987). Snakes: active contour models. International Journal of Computer Vision, 1, 321–331. CrossRefGoogle Scholar
  22. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., & Yezzi, A. (1995). Gradient flows and geometric active contour models. In Proceedings of the IEEE int. conf. on computer vision (pp. 810–815). Google Scholar
  23. Kim, J., Fisher, J., Yezzi, A., Cetin, M., & Willsky, A. (2002). Nonparametric methods for image processing using information theory and curve evolution. In IEEE international conference on image processing (Vol. 3, pp. 797–800). Google Scholar
  24. Kolmogorov, V., & Boykov, Y. (2005). What metrics can be approximated by geo-cuts, or global optimization of length/area and flux. In ICCV (pp. 564–571). Google Scholar
  25. Leventon, M., Grimson, E., & 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
  26. Ma, T., & Tagare, H. (1999). Consistency and stability of active contours with Euclidean and non-Euclidean arc lengths. IEEE Transactions on Image Processing, 8(11), 1549–1559. MATHCrossRefMathSciNetGoogle Scholar
  27. Malladi, R., Sethian, J., & Vemuri, B. (1995). Shape modeling with front propagation: a level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(2), 158–175. CrossRefGoogle Scholar
  28. Mansouri, A.-R., Mukherjee, D. P., & Acton, S. T. (2004). Constraining active contour evolution via Lie Groups of transformation. IEEE Transactions on Image Processing, 13(6), 853–863. CrossRefMathSciNetGoogle Scholar
  29. Michor, P. W., & Mumford, D. (2006). Riemannian geometries of space of plane curves. Journal of the European Mathematical Society, 8, 1–48. MATHMathSciNetCrossRefGoogle Scholar
  30. Mio, W., Srivastava, A., & Klassen, E. (2004). Interpolations with elasticae in Euclidean spaces. Quaterly of Applied Mathematics, LXII(3), 359–378. MathSciNetGoogle Scholar
  31. Mumford, D., & Shah, J. (1985). Boundary detection by minimizing functionals. In Proc. IEEE conf. computer vision pattern recognition. Google Scholar
  32. Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42, 577–685. MATHCrossRefMathSciNetGoogle Scholar
  33. Nain, D., Yezzi, A. J., & Turk, G. (2004). Vessel segmentation using a shape driven flow. In MICCAI (1) (pp. 51–59). Google Scholar
  34. Neuberger, J. W. (1997). Sobolev gradients and differential equations. Lecture notes in mathematics, Vol. 1670. Berlin: Springer. MATHGoogle Scholar
  35. Paragios, N., & Deriche, R. (2000). Geodesic active contours and level sets for the detection and tracking of moving objects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 266–280. CrossRefGoogle Scholar
  36. Paragios, N., & Deriche, R. (2002). Geodesic active regions and level set methods for supervised texture segmentation. International Journal of Computer Vision, 46(3), 223. MATHCrossRefGoogle Scholar
  37. Polden, A. (1996). Curves and surfaces of least total curvature and fourth-order flows. Ph.D. thesis, Mathematisches Institut Unversitat Tubingen, Germany. Google Scholar
  38. Raviv, T. R., Kiryati, N., & Sochen, N. (2004). Unlevel-set: geometry and prior-based segmentation. In Proc. European conf. on computer vision. Google Scholar
  39. Rochery, M., Jermyn, I., & Zerubia, J. (2003). Higher order active contours and their application to the detection of line networks in satellite imagery. In IEEE Workshop on VLSM. Google Scholar
  40. Ronfard, R. (1994). Region based strategies for active contour models. International Journal of Computer Vision, 13(2), 229–251. CrossRefGoogle Scholar
  41. Rousson, M., & Paragios, N. (2002). Shape priors for level set representations. In Proc. European conf. computer vision (Vol. 2, pp. 78–93). Google Scholar
  42. Rudin, W. (1973). Functional analysis. New York: McGraw-Hill. MATHGoogle Scholar
  43. Sapiro, G., & Tannenbaum, A. (1995). Area and length preserving geometric invariant scale-spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(1), 67–72. CrossRefGoogle Scholar
  44. Schoenemann, T., & Cremers, D. (2007a). Globally optimal image segmentation with an elastic shape prior. In IEEE international conference on computer vision (ICCV). Rio de Janeiro, Brazil. Google Scholar
  45. Schoenemann, T., & Cremers, D. (2007b). Introducing curvature into globally optimimal image segmentation: minimum ratio cycles on product graphs. In IEEE international conference on computer vision (ICCV). Rio de Janeiro, Brazil. Google Scholar
  46. Sundaramoorthi, G., & Yezzi, A. J. (2005). More-than-topology-preserving flows for active contours and polygons. In ICCV (pp. 1276–1283). Google Scholar
  47. Sundaramoorthi, G., Yezzi, A., & Mennucci, A. (2005). Sobolev active contours. In VLSM (pp. 109–120). Google Scholar
  48. Sundaramoorthi, G., Jackson, J. D., Yezzi, A. J., & Mennucci, A. (2006). Tracking with Sobolev active contours. In CVPR (1) (pp. 674–680). Google Scholar
  49. Sundaramoorthi, G., Yezzi, A., & Mennucci, A. (2007). Sobolev active contours. International Journal of Computer Vision, 73(3), 345–366. CrossRefGoogle Scholar
  50. Sundaramoorthi, G., Yezzi, A., & Mennucci, A. (2008). Coarse-to-fine segmentation and tracking with Sobolev active contours. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(5), 851–864. CrossRefGoogle Scholar
  51. Tsai, A., Yezzi, A. J., Tempany, W. M. W. C., III, Tucker, D., Fan, A., Grimson, W. E. L., & Willsky, A. S. (2001). Model-based curve evolution technique for image segmentation. In CVPR (1) (pp. 463–468). Google Scholar
  52. Yezzi, A., & Mennucci, A. (2005a). Metrics in the space of curves. Preprint, arXiv:math.DG/0412454.
  53. Yezzi, A. J., & Mennucci, A. (2005b). Conformal metrics and true “gradient flows” for curves. In ICCV (pp. 913–919). Google Scholar
  54. Yezzi, A., Tsai, A., & Willsky, A. (1999). A statistical approach to snakes for bimodal and trimodal imagery. In Int. conf. on computer vision (pp. 898–903). Google Scholar
  55. Zhu, S. C., Lee, T. S., & Yuille, A. L. (1995). Region competition: unifying snakes, region growing, energy/bayes/MDL for multi-band image segmentation. In ICCV (pp. 416). Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Ganesh Sundaramoorthi
    • 1
  • Anthony Yezzi
    • 1
  • Andrea C. Mennucci
    • 2
  • Guillermo Sapiro
    • 3
  1. 1.School of Electrical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Scuola Normale SuperiorePisaItaly
  3. 3.Dept. of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations