We introduce an algorithm for active contour segmentation in which the level set function encoding the contour is processed by median filtering using morphological amoebas. These are adaptive structure elements introduced by Lerallut et al. which can be combined with different morphological operations. Recently it has been proven that iterated amoeba median filtering of an image approximates the well-known self-snakes partial differential equation. Following this approach we prove a partial approximation property of amoeba active contours with respect to geodesic active contours. Experiments prove the viability of the algorithm and confirm the theoretical results.


Input Image Active Contour Active Contour Model Initial Contour Geodesic Active Contour 
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  1. 1.
    Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. Journal of Computational Physics 118(2), 269–277 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM Journal on Numerical Analysis 29, 845–866 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barash, D.: Bilateral filtering and anisotropic diffusion: Towards a unified viewpoint. In: Kerckhove, M. (ed.) Scale-Space 2001. LNCS, vol. 2106, pp. 273–280. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34, 344–371 (1986)CrossRefGoogle Scholar
  5. 5.
    Borgefors, G.: On digital distance transforms in three dimensions. Computer Vision and Image Understanding 64(3), 368–376 (1996)CrossRefGoogle Scholar
  6. 6.
    Braga-Neto, U.M.: Alternating sequential filters by adaptive neighborhood structuring functions. In: Maragos, P., Schafer, R.W., Butt, M.A. (eds.) Mathematical Morphology and its Applications to Image and Signal Processing. Computational Imaging and Vision, vol. 5, pp. 139–146. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  7. 7.
    Breuß, M., Burgeth, B., Weickert, J.: Anisotropic continuous-scale morphology. In: Martí, J., Benedí, J.M., Mendonça, A.M., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4478, pp. 515–522. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numerische Mathematik 66, 1–31 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proc. Fifth International Conference on Computer Vision, pp. 694–699. IEEE Computer Society Press, Cambridge (1995)CrossRefGoogle Scholar
  10. 10.
    Cohen, L.D.: On active contour models and balloons. Computer Vision, Graphics, and Image Processing: Image Understanding 53(2), 211–218 (1991)zbMATHGoogle Scholar
  11. 11.
    Didas, S., Weickert, J.: Combining curvature motion and edge-preserving denoising. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 568–579. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Eckhardt, U.: Root images of median filters. Journal of Mathematical Imaging and Vision 19, 63–70 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Guichard, F., Morel, J.-M.: Partial differential equations and image iterative filtering. In: Duff, I.S., Watson, G.A. (eds.) The State of the Art in Numerical Analysis. IMA Conference Series (New Series), vol. 63, pp. 525–562. Clarendon Press, Oxford (1997)Google Scholar
  14. 14.
    Kimmel, R., Sochen, N., Malladi, R.: Images as embedding maps and minimal surfaces: movies, color, and volumetric medical images. In: Proc. 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 350–355. IEEE Computer Society Press, San Juan (1997)CrossRefGoogle Scholar
  15. 15.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: Proc. of Fifth International Conference on Computer Vision, pp. 810–815. IEEE Computer Society Press, Cambridge (1995)CrossRefGoogle Scholar
  16. 16.
    Klette, R., Zamperoni, P.: Handbook of Image Processing Operators. Wiley, New York (1996)zbMATHGoogle Scholar
  17. 17.
    Ikonen, L.: Priority pixel queue algorithm for geodesic distance transforms. Image and Vision Computing 25(10), 1520–1529 (2007)CrossRefGoogle Scholar
  18. 18.
    Ikonen, L., Toivanen, P.: Shortest routes on varying height surfaces using gray-level distance transforms. Image and Vision Computing 23(2), 133–141 (2005)CrossRefGoogle Scholar
  19. 19.
    Lerallut, R., Decencière, E., Meyer, F.: Image processing using morphological amoebas. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol. 30. Springer, Dordrecht (2005)Google Scholar
  20. 20.
    Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 25(4), 395–404 (2007)CrossRefGoogle Scholar
  21. 21.
    Malladi, R., Sethian, J.A., Vemuri, B.C.: A topology independent shape modeling scheme. In: Vemuri, B. (ed.) Geometric Methods in Computer Vision. Proceedings of SPIE, vol. 2031, pp. 246–258. SPIE Press, Bellingham (1993)CrossRefGoogle Scholar
  22. 22.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics 79, 12–49 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Sapiro, G.: Vector (self) snakes: a geometric framework for color, texture and multiscale image segmentation. In: Proc.1996 IEEE International Conference on Image Processing, Lausanne, Switzerland, vol. 1, pp. 817–820 (September 1996)Google Scholar
  24. 24.
    Shih, F.Y., Cheng, S.: Adaptive mathematical morphology for edge linking. Information Sciences 167(1–4), 9–21 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proc. Sixth International Conference on Computer Vision, pp. 839–846. Narosa Publishing House, Bombay (1998)Google Scholar
  26. 26.
    Tukey, J.W.: Exploratory Data Analysis. Addison–Wesley, Menlo Park (1971)zbMATHGoogle Scholar
  27. 27.
    van den Boomgaard, R.: Decomposition of the Kuwahara–Nagao operator in terms of linear smoothing and morphological sharpening. In: Talbot, H., Beare, R. (eds.) Mathematical Morphology: Proc. Sixth International Symposium, pp. 283–292. CSIRO Publishing, Sydney (2002)Google Scholar
  28. 28.
    Verly, J.G., Delanoy, R.L.: Adaptive mathematical morphology for range imagery. IEEE Transactions on Image Processing 2(2), 272–275 (1993)CrossRefGoogle Scholar
  29. 29.
    Welk, M., Breuß, M., Vogel, O.: Morphological amoebas are self-snakes. Journal of Mathematical Imaging and Vision (in press, 2011)Google Scholar
  30. 30.
    Whitaker, R.T., Xue, X.: Variable-conductance, level-set curvature for image denoising. In: Proc. 2001 IEEE International Conference on Image Processing, Thessaloniki, Greece, pp. 142–145 (October 2001)Google Scholar
  31. 31.
    Yezzi Jr., A.: Modified curvature motion for image smoothing and enhancement. IEEE Transactions on Image Processing 7(3), 345–352 (1998)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Martin Welk
    • 1
  1. 1.Institute for Biomedical Image AnalysisUniversity for Health Sciences, Medical Informatics and TechnologyHall/TyrolAustria

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