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Morphological Amoebas Are Self-snakes


This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We establish this correspondence by analysing a space-continuous formulation of iterated amoeba median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates the partial differential equation of self-snakes. This result holds true under very general assumptions on the metric used to construct the amoebas. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.

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  1. 1.

    Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal. 29, 845–866 (1992)

  2. 2.

    Barash, D.: Bilateral filtering and anisotropic diffusion: towards a unified viewpoint. In: Kerckhove, M. (ed.) Scale-Space and Morphology in Computer Vision. Lecture Notes in Computer Science, vol. 2106, pp. 273–280. Springer, Berlin (2001)

  3. 3.

    Borgefors, G.: Distance transformations in digital images. Comput. Vis. Graph. Image Process. 34, 344–371 (1986)

  4. 4.

    Borgefors, G.: On digital distance transforms in three dimensions. Comput. Vis. Image Underst. 64(3), 368–376 (1996)

  5. 5.

    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 Academic, Dordrecht (1996)

  6. 6.

    Carmona, R., Zhong, S.: Adaptive smoothing respecting feature directions. IEEE Trans. Image Process. 7(3), 353–358 (1998)

  7. 7.

    Caselles, V., Morel, J.-M., Sbert, C.: An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7(3), 376–386 (1998)

  8. 8.

    Chui, C.K., Wang, J.: PDE models associated with the bilateral filter. Adv. Comput. Math. (2008). doi:10.1007/s10444-008-9095-2

  9. 9.

    Didas, S., Weickert, J.: Combining curvature motion and edge-preserving denoising. In: Sgallari, F., Murli, F., Paragios, N. (eds.) Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 4485, pp. 568–579. Springer, Berlin (2007)

  10. 10.

    Dougherty, E.R., Astola, J. (eds.) Nonlinear Filters for Image Processing. SPIE, Bellingham (1999)

  11. 11.

    Fabbri, R., Da F. Costa, L., Torelli, J.C., Bruno, O.M.: 2D Euclidean distance transform algorithms: a comparative survey. ACM Comput. Surv. 40(1), 2 (2008)

  12. 12.

    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, Oxford (1997)

  13. 13.

    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)

  14. 14.

    Hyman, J.M., Shashkov, M.: Natural discretizations of the divergence, gradient and curl on logically rectangular grids. Int. J. Comput. Math. Appl. 33(4), 81–104 (1997)

  15. 15.

    Hyman, J., Morel, J., Shashkov, M., Steinberg, S.: Mimetic finite difference methods for diffusion equations. Comput. Geosci. 6, 333–352 (2002)

  16. 16.

    Ikonen, L.: Priority pixel queue algorithm for geodesic distance transforms. Image Vis. Comput. 25(10), 1520–1529 (2007)

  17. 17.

    Ikonen, L., Toivanen, P.: Shortest routes on varying height surfaces using gray-level distance transforms. Image Vis. Comput. 23(2), 133–141 (2005)

  18. 18.

    Jalba, A.C., Roerdink, J.B.T.M.: An efficient morphological active surface model for volumetric image segmentation. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing. Lecture Notes in Computer Science, vol. 5720, pp. 193–204. Springer, Berlin (2009)

  19. 19.

    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, San Juan, Puerto Rico, June 1997, pp. 350–355. IEEE Computer Society, Los Alamitos (1997)

  20. 20.

    Klette, R., Zamperoni, P.: Handbook of Image Processing Operators. Wiley, New York (1996)

  21. 21.

    Kuwahara, M., Hachimura, K., Eiho, S., Kinoshita, M.: Processing of RI-angiocardiographic images. In: Preston, J.K., Onoe, M. (eds.) Digital Processing of Biomedical Images, pp. 187–202. Plenum, New York (1976)

  22. 22.

    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)

  23. 23.

    Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image Vis. Comput. 25(4), 395–404 (2007)

  24. 24.

    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)

  25. 25.

    Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)

  26. 26.

    Nagao, M., Matsuyama, T.: Edge preserving smoothing. Comput. Graph. Image Process. 9(4), 394–407 (1979)

  27. 27.

    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, Sept. 1996, vol. 1, pp. 817–820 (1996)

  28. 28.

    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)

  29. 29.

    Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic Press, London (1988)

  30. 30.

    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proc. Sixth International Conference on Computer Vision, Bombay, India, Jan. 1998, pp. 839–846. Narosa Publishing House, New Delhi (1998)

  31. 31.

    Tukey, J.W.: Exploratory Data Analysis. Addison–Wesley, Menlo Park (1971)

  32. 32.

    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, Sydney, Australia, Apr. 2002, pp. 283–292. CSIRO Publishing, Collingwood (2002)

  33. 33.

    Verly, J.G., Delanoy, R.L.: Adaptive mathematical morphology for range imagery. IEEE Trans. Image Process. 2(2), 272–275 (1993)

  34. 34.

    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

  35. 35.

    Welk, M., Breuß, M., Vogel, O.: Differential equations for morphological amoebas. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing. Lecture Notes in Computer Science, vol. 5720, pp. 104–114. Springer, Berlin (2009). Erratum: http://www.mia.uni-saarland.de/publications

  36. 36.

    Whitaker, R.T., Xue, X.: Variable-conductance, level-set curvature for image denoising. In: Proc. 2001 IEEE International Conference on Image Processing, Thessaloniki, Greece, Oct. 2001, pp. 142–145 (2001)

  37. 37.

    Yezzi, A. Jr.: Modified curvature motion for image smoothing and enhancement. IEEE Trans. Image Process. 7(3), 345–352 (1998)

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Correspondence to Martin Welk.

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Welk, M., Breuß, M. & Vogel, O. Morphological Amoebas Are Self-snakes. J Math Imaging Vis 39, 87–99 (2011). https://doi.org/10.1007/s10851-010-0228-0

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  • Morphological amoebas
  • Self-snakes
  • Median filtering
  • Mathematical morphology
  • Partial differential equations