The aim of this paper is to study an optimal opening in the sense of minimize the relationship perimeter over area. We analyze theoretical properties of this opening by means of classical results from variational calculus. Firstly, we explore the optimal radius as attribute in morphological attribute filtering for grey scale images. Secondly, an application of this optimal opening that yields a decomposition into meaningful parts in the case of binary image is explored. We provide different examples of 2D, 3D images and mesh-points datasets.


Attribute filter 3D Shape 2D Shape Cheeger Set 


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  1. 1.
    Najman, L., Talbot, H.: Mathematical morphology: from theory to applications p–2010. ISTE-Wiley (2010)Google Scholar
  2. 2.
    Angulo, J., Velasco-Forero, S.: Structurally adaptive mathematical morphology based on nonlinear scale-space decompositions. Image Analysis & Stereology 30(2), 111–122 (2011)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Ćurić, V., Landström, A., Thurley, M.J., Hendriks, C.L.L.: Adaptive mathematical morphology–a survey of the field. Pattern Recognition Letters 47, 18–28 (2014)CrossRefGoogle Scholar
  4. 4.
    Vincent, L.: Grayscale area openings and closings, their efficient implementation and applications. In: Proceedings of the Conference on Mathematical Morphology and its Applications to Signal Processing, pp. 22–27 (May 1993)Google Scholar
  5. 5.
    Breen, E.J., Jones, R.: Attribute openings, thinnings, and granulometries. Computer Vision and Image Understanding 64(3), 377–389 (1996)CrossRefGoogle Scholar
  6. 6.
    Talbot, H., Appleton, B.: Efficient complete and incomplete path openings and closings. Image Vision Comput. 25(4), 416–425 (2007)CrossRefGoogle Scholar
  7. 7.
    Serna, A., Marcotegui, B.: Attribute controlled reconstruction and adaptive mathematical morphology. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 207–218. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Morard, V., Decencière, E., Dokládal, P.: Efficient geodesic attribute thinnings based on the barycentric diameter. Journal of Mathematical Imaging and Vision 46(1), 128–142 (2013)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Maggi, F.: Set of Finite Perimeter and Geometric Variational Problems. Cambridge University Press (2012)Google Scholar
  10. 10.
    Mehtre, B.M., Kankanhalli, M.S., Lee, W.F.: Shape measures for content based image retrieval: a comparison. Information Processing & Management 33(3), 319–337 (1997)CrossRefGoogle Scholar
  11. 11.
    Cerri, A., Biasotti, S., Abdelrahman, M., Angulo, J., Berger, K., Chevallier, L., El-Melegy, M., Farag, A., Lefebvre, F., Andrea, Giachetti, et al.: Shrec’13 track: retrieval on textured 3d models. In: Proceedings of the Sixth Eurographics Workshop on 3D Object Retrieval, pp. 73–80. Eurographics Association (2013)Google Scholar
  12. 12.
    Velasco-Forero, S., Angulo, J.: Statistical shape modeling using morphological representations. In: 2010 20th International Conference on Pattern Recognition (ICPR), pp. 3537–3540. IEEE (2010)Google Scholar
  13. 13.
    Gueguen, L.: Classifying compound structures in satellite images: A compressed representation for fast queries. Transactions on Geoscience and Remote Sensing, 1–16 (2014)Google Scholar
  14. 14.
    Younes, L.: Spaces and manifolds of shapes in computer vision: An overview. Image and Vision Computing 30(6), 389–397 (2012)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kawohl, B., Lachand-Robert, T.: Characterization of Cheeger sets for convex subsets of the plane. Pac. J. Math. 225(1), 103–118 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Alter, F., Caselles, V.: Uniqueness of the cheeger set of a convex body. Nonlinear Analysis 70(1), 32–44 (2009)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Stredulinsky, E., Ziemer, W.: Area minimizing sets subject to a volume constraint in a convex set. The Journal of Geometric Analysis 7(4), 653–677 (1997), http://dx.doi.org/10.1007/BF02921639 CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Leonardi, G.P., Pratelli, A.: On the cheeger sets in strips and non-convex domains. arXiv preprint arXiv:1409.1376 (2014)Google Scholar
  19. 19.
    Caselles, V., Alter, A.C.F.: Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. Interfaces and Free Boundaries 7 (2005)Google Scholar
  20. 20.
    Caselles, V., Chambolle, A., Novaga, M.: Some remarks on uniqueness and regularity of cheeger sets. Rend. Semin. Math. Univ. Padova 123, 191–201 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Proceedings of a Symposium on Models for the Perception of Speech and Visual Forms. MIT, Boston (November 1967)Google Scholar
  22. 22.
    Calabi, L.: A study of the skeleton of plane figures. Parke Mathematical Laboratories (1965)Google Scholar
  23. 23.
    Durand, G.: Théprie des ensembles. points ordinaires et point singuliers des enveloppes de sphères. Comptes-rendus de l’Acad’emie de Sciences 190, 571–573 (1930)MATHGoogle Scholar
  24. 24.
    Bouligand, G.: Introduction à la gémétrie infinitésimale directe. Vuibert (1932)Google Scholar
  25. 25.
    Malandain, G., Fernández-Vidal, S.: Euclidean skeletons. Image and Vision Computing 16(5), 317–327 (1998)CrossRefGoogle Scholar
  26. 26.
    Lantuejoul, C.: La squelettisation et son application aux mesures topologiques des mosaiques polycristallines. Ph.D. dissertation, Ecole des Mines de Paris (1978)Google Scholar
  27. 27.
    Lantuejoul, C.: Skeletonization in quantitative metallography. Issues of Digital Image Processing 34(107-135), 109 (1980)Google Scholar
  28. 28.
    Salembier, P., Serra, J.: Flat zones filtering, connected operators, and filters by reconstruction. IEEE Transactions on Image Processing 4(8), 1153–1160 (1995)CrossRefGoogle Scholar
  29. 29.
    Carlinet, E., Géraud, T.: A comparison of many max-tree computation algorithms. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 73–85. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  30. 30.
    Hoffman, D.D., Richards, W.A.: Parts of recognition. Cognition 18(1), 65–96 (1984)CrossRefGoogle Scholar
  31. 31.
    Xu, J.: Morphological decomposition of 2-d binary shapes into convex polygons: A heuristic algorithm. IEEE Transactions on Image Processing 10(1), 61–71 (2001)CrossRefMATHGoogle Scholar
  32. 32.
    Yu, L., Wang, R.: Shape representation based on mathematical morphology. Pattern Recognition Letters 26(9), 1354–1362 (2005)CrossRefGoogle Scholar
  33. 33.
    Kim, D.H., Yun, I.D., Lee, S.U.: A new shape decomposition scheme for graph-based representation. Pattern Recognition 38(5), 673–689 (2005)CrossRefGoogle Scholar
  34. 34.
    Liu, G., Xi, Z., Lien, J.-M.: Dual-space decomposition of 2d complex shapes. In: 27th IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Columbus, OH. IEEE (June 2014)Google Scholar
  35. 35.
    Xu, Y., Géraud, T., Najman, L.: Morphological filtering in shape spaces: Applications using tree-based image representations. In: 2012 21st International Conference on Pattern Recognition (ICPR), pp. 485–488. IEEE (2012)Google Scholar
  36. 36.
    Jeulin, D.: Random structures in physics. In: Space, Structure and Randomness, Contributions in Honor of Georges Matheron in the Fields of Geostatistics, Random Sets, and Mathematical Morphology. Lecture Notes in Statistics, vol. 183, pp. 183–222. Springer, Heidelberg (2005)Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Centre of Mathematical MorphologyMines Paris TechParisFrance

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