Journal of Mathematical Imaging and Vision

, Volume 51, Issue 2, pp 229–247 | Cite as

Active Contour with a Tangential Component

  • Junyan WangEmail author
  • Kap Luk Chan


Conventional edge-based active contours often require the normal component of an edge indicator function on the optimal contours to approach zero, while the tangential component can still be significant. In real images, the full gradients of the edge indicator function along the object boundaries are often small. Hence, the curve evolution of edge-based active contours can terminate early before converging to the object boundaries with a careless contour initialization. We propose a novel Geodesic Snakes (GeoSnakes) active contour that requires the full gradients of the edge indicator to vanish at the optimal solution. Besides, the conventional curve evolution approach for minimizing active contour energy cannot fully solve the Euler–Lagrange equation of our GeoSnakes active contour, causing a pseudo stationary phenomenon (PSP). To address the PSP problem, we propose an auxiliary curve evolution equation, named the equilibrium flow (EF) equation. Based on the EF and the conventional curve evolution, we obtain a solution to the full Euler–Lagrange equation of GeoSnakes active contour. Experimental results validate the proposed geometrical interpretation of the early termination problem, and they also show that the proposed method is able to overcome the problem.


Active contour model Curve evolution Level set method Euler–Lagrange equation Pseudo stationary phenomenon 


  1. 1.
    Paragios, N., Deriche, R.: Geodesic active contours and level sets for the detection and tracking of moving objects. IEEE Trans. Pattern Anal. Mach. Intell. 22(3), 266–280 (2000)CrossRefGoogle Scholar
  2. 2.
    Schoenemann, T., Cremers, D.: A combinatorial solution for model-based image segmentation and real-time tracking. IEEE Trans. Pattern Anal. Mach. Intell. 32, 1153–1164 (2010)CrossRefGoogle Scholar
  3. 3.
    Yezzi, A.J., Kichenassamy, S., Kumar, A., Olver, P.J., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE Trans. Med. Imaging 16(2), 199–209 (1997)CrossRefGoogle Scholar
  4. 4.
    Leventon, M., Grimson, W., Faugeras, O.: Statistical shape influence in geodesic active contours. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2000)Google Scholar
  5. 5.
    Malladi, R., Sethian, J.A., Vemuri, B.C.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17(2), 158–175 (1995)CrossRefGoogle Scholar
  6. 6.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contour. Int. J. Comput. Vis. 22(1), 61–79 (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Kimmel, R., Bruckstein, A.M.: Regularized laplacian zero crossings as optimal edge integrators. Int. J. Comput. Vis. 53(3), 225–243 (2003)CrossRefGoogle Scholar
  8. 8.
    Corsaro, S., Mikula, K., Sarti, A., Sgallari, F.: Semi-implicit covolume method in 3D image segmentation. SIAM J. Sci. Comput. 28, 2248–2265 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Zhu, S.C., Yuille, A.: Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18(9), 884–900 (1996)CrossRefGoogle Scholar
  10. 10.
    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Sumengen, B., Manjunath, B.: Graph partitioning active contours (gpac) for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28, 509–521 (2006)CrossRefGoogle Scholar
  12. 12.
    Brox, T., Cremers, D.: On local region models and a statistical interpretation of the piecewise smooth Mumford–Shah functional. Int. J. Comput. Vis. 84(2), 184–193 (2009)CrossRefGoogle Scholar
  13. 13.
    Sagiv, C., Sochen, N., Zeevi, Y.: Integrated active contours for texture segmentation. IEEE Trans. Image Process. 16(6), 1633–1646 (2006)CrossRefGoogle Scholar
  14. 14.
    Lankton, S., Tannenbaum, A.: Localizing region-based active contours. IEEE Trans. Image Process. 17(11), 2029–2039 (2008)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Paragios, N., Mellina-Gottardo, O., Ramesh, V.: Gradient vector flow fast geometric active contours. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 402–407 (2004)CrossRefGoogle Scholar
  16. 16.
    Xie, X., Mirmehdi, M.: Mac: magnetostatic active contour model. IEEE Trans. Pattern Anal. Mach. Intell. 30(4), 632–646 (2008)CrossRefGoogle Scholar
  17. 17.
    Cohen, L.D.: On active contour models and balloons. CVGIP Image Underst. 53(2), 211–218 (1991)CrossRefzbMATHGoogle Scholar
  18. 18.
    Xu, C., Prince, J.L.: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Process. 7(3), 359–369 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Li, C., Liu, J., Fox, M.D.: Segmentation of edge preserving gradient vector flow: an approach toward automatically initializing and splitting of snakes. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2005)Google Scholar
  20. 20.
    Appleton, B., Talbot, H.: Globally optimal geodesic active contours. J. Math. Imaging Vis. 23, 67–86 (2005)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Benmansour, F., Cohen, L.D.: Fast object segmentation by growing minimal paths from a single point on 2D or 3D images. J. Math. Imaging Vis. 33(2), 209–221 (2009)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Wang, J., Chan, K.L., Wang, Y.: On the stationary solution of PDE based curve evolution. In: Proceedings of the 19th British Machine Vision Conference (2008)Google Scholar
  23. 23.
    Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numer. Math. 66(1), 1–31 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Haralick, R.M.: Digital step edges from zero crossing of second directional derivatives. IEEE Trans. Pattern Anal. Mach. Intell. 6(1), 58–68 (1984)CrossRefGoogle Scholar
  25. 25.
    McInerney, T., Terzopoulos, D.: T-snakes: topology adaptive snakes. Med. Image Anal. 4(2), 73–91 (2000)CrossRefGoogle Scholar
  26. 26.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of the Ninth IEEE International Conference on Computer Vision, vol. 2 (2003)Google Scholar
  27. 27.
    Marr, D., Hildreth, E.: Theory of edge detection. In: Proceedings of the Royal Society of London. Series B, Biological Sciences, vol. 207, pp. 187–217 (1980)Google Scholar
  28. 28.
    Vovk, U., Pernus, F., Likar, B.: A review of methods for correction of intensity inhomogeneity in MRI. IEEE Trans. Med. Imaging 26(3), 405–421 (2007)CrossRefGoogle Scholar
  29. 29.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)CrossRefGoogle Scholar
  30. 30.
    Xu, C., Yezzi, A. Jr, Prince, J.: On the relationship between parametric and geometric active contours. In: Proccedings of 34th Asilomar Conference on Signals, Systems, and Computers (2000)Google Scholar
  31. 31.
    Epstein, C.L., Gage, M.: The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modeling, and Computation. Springer, New York (1987)Google Scholar
  32. 32.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, New York (2006)zbMATHGoogle Scholar
  33. 33.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Khalil, H.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)zbMATHGoogle Scholar
  35. 35.
    Osher, S., Fedkiw, R.: Level-Set Methods and Dynamic Implicit Surfacesp. Springer, New York (2003)CrossRefGoogle Scholar
  36. 36.
    Paragios, N., Mellina-Gottardo, O., Ramesh, V.: Gradient vector flow fast geodesic active contours. In: Proceedings of the 11th IEEE International Conference on Computer Vision (2001)Google Scholar
  37. 37.
    Chan, T., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J.-P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28(2), 151–167 (2007)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. J. Comput. Phys. 118(2), 269–277 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  41. 41.
    Shi, Y., Karl, W.C.: A real-time algorithm for the approximation of level-set-based curve evolution. IEEE Trans. Image Process. 17(5), 645–656 (2008)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Electrical and Electronic EngineeringNanyang Technological UniversitySingapore Singapore

Personalised recommendations