Skip to main content
SpringerLink
Log in

A Lattice Approach to Image Segmentation

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

After a formal definition of segmentation as the largest partition of the space according to a criterion σ and a function f, the notion of a morphological connection is reminded. It is used as an input to a central theorem of the paper (Theorem 8), that identifies segmentation with the connections that are based on connective criteria. Just as connections, the segmentations can then be regrouped by suprema and infima. The generality of the theorem makes it valid for functions from any space to any other one. Two propositions make precise the AND and OR combinations of connective criteria.

The soundness of the approach is demonstrated by listing a series of segmentation techniques. One considers first the cases when the segmentation under study does not involve initial seeds. Various modes of regularity are discussed, which all derive from Lipschitz functions. A second category of examples involves the presence of seeds around which the partition of the space is organized. An overall proposition shows that these examples are a matter for the central theorem. Watershed and jump connection based segmentations illustrate this type of situation. The third and last category of examples deals with cases when the segmentation occurs in an indirect space, such as an histogram, and is then projected back on the actual space under study.

The relationships between filtering and segmentation are then investigated. A theoretical chapter introduces and studies the two notions of a pulse opening and of a connected operator. The conditions under which a family of pulse openings can yield a connected filter are clarified. The ability of segmentations to generate pyramids, or hierarchies, is analyzed. A distinction is made between weak hierarchies where the partitions increase when going up in the pyramid, and the strong hierarchies where the various levels are structured as semi-groups, and particularly as granulometric semi-groups.

The last section is based on one example, and goes back over the controversy about “lattice” versus “functional” optimization. The problem is now tackled via a case of colour segmentation, where the saturation serves as a cursor between luminance and hue. The emphasis is put on the difficulty of grouping the various necessary optimizations into a single one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Alvarez, P.-L. Lions, and J.M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion (II),” SIAM Journal on Numerical Analysis, Vol. 29, pp. 845–866, 1992.

    Article  MathSciNet  Google Scholar 

  2. L. Alvarez, F. Guichard, P.-L. Lions, and J.M. Morel,“Axiomatisation et nouveaux opérateurs de la morphologie mathématique,” C.R. Acad. Sci. Paris, 1992, pp. 265–268, t. 315, Série I.

  3. J. Angulo, J. Serra, and G. Flandrin, “Quantitative descriptors of the lymphocytes,” Analytical Cellular Pathology, Vol. 22, Nos. 1/2, pp. 69–70, 2001.

    Google Scholar 

  4. J. Angulo and J. Serra, “Colour feature extraction from luminance/saturation histogram in L 1 representation,” Tech. Report Ecole des Mines, March 2003.

  5. J. Angulo and J. Serra, “Colour segmentation by ordered mergings,” ICIP 2003, Barcelona Sept. 14–17.

  6. G. Bertrand, “Simple points, topological numbers and geodesic neighborhoods in cubic grids,” Pattern Rec. Letters, Vol. 15, pp. 1003–1012, 1994.

    Google Scholar 

  7. S. Beucher and C. Lantuejoul, “Use of watersheds in contour detection,” in Proc. Int. Workshop on Image Processing, Real-time Edge and Motion Detection/Estimation, Rennes (France), Sept. 1979.

  8. S. Beucher, Segmentation d'Images et Morphologie Mathématique, Thése de Doctorat, Ecole des Mines de Paris, 1990.

  9. G. Birkhoff, Lattice Theory, 3rd edition, A.M.S. Colloq. Publ., Vol. 25, 1983.

  10. R. van den Boomgaard, “Mathematical morphology: extensions towards computer vision,” Ph.D. Thesis, Univ. of Amsterdam, The Netherlands, 1992

  11. R. van den Boomgaard and A. Smeulders, “The morphological structure of the images: The differential equations of morphological scale-space,” IEEE Trans. Pattern. Anal. Mach. Intellig, Vol. 16, pp. 1101–1113, 1994.

    Google Scholar 

  12. G. Borgefors, G. Ramella, and G. Sanniti di Baja, “Hierarchical decomposition of multi-scale skeletons,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 23, No. 11, pp. 1296–1312, 2001.

    Article  Google Scholar 

  13. M. Born and E. Wolf, Principles of Optics, Pergamon Press: Oxford, England, 1987.

    Google Scholar 

  14. U.M. Braga-Neto and J. Goutsias, “Multiresolution connectivity: An axiomatic approach,” in Mathematical Morphology and its Applications to Image Processing, J. Goustias, L. Vincent, and D.S. Bloomberg (Eds.), Kluwer, 2000, pp. 159–168.

  15. U.M. Braga-Neto and J. Goutsias, “Connectivity on complete lattices: New results,” Computer Vision and Image Understanding, Vol. 85, pp. 22–53, 2002.

    Article  Google Scholar 

  16. U.M. Braga-Neto and J. Goutsias, “A Multiscale Approach to Connectivity,” Computer Vision and Image Understanding, Vol. 89, pp. 70–107, 2003.

    Article  Google Scholar 

  17. U.M. Braga-Neto, “Multiscale connected operators,” submitted to JMIV, Special Issue “Mathematical Morphology after 40 years”.

  18. R.W. Brockett and P. Maragos, “Evolution equation for ccontinuous-scale morphology,” in Proc. IEEE Int. Conf. Acoust. Speech, Signal Processing, San Francisco: CA, 1992.

  19. V. Caselles, B. Coll, and J.M. Morel, “Geometry and color in natural images,” JMIV 16, 89–105, 2002.

    MathSciNet  Google Scholar 

  20. V. Caselles, F. Catte, T. Coll, and F. Dibos, “A geometric model for active contours,” Numerische Mathematik, Vol. 66, pp. 1–31, 1993.

    Article  MathSciNet  Google Scholar 

  21. V. Caselles, R. Kimmel, and G. Shapiro, “Geodesic active contours,” IJCV, Vol. 22, No. 1, pp. 61–79, 1997.

    Article  Google Scholar 

  22. J.M. Chassery and C. Garbay, “An iterative segmentation method based on a contextual color and shape criterion,” IEEE Transactions on Pattern Analysis, Machine Intelligence, Vol. 6, No. 6, pp. 794–800, 1984.

    Google Scholar 

  23. G. Choquet, Topology, Academic Press: N.Y., 1966.

    Google Scholar 

  24. J. Crespo, J. Serra, and R.W. Schafer, “Theoretical aspects of morphological filters by reconstruction,” Signal Processing, Vol. 47, No. 2, pp. 201–225, 1995

    Article  Google Scholar 

  25. J. Crespo and R.W. Schafer, “Locality and adjacency stability constraints for morphological connected operators,” Journal of Mathematical Imaging and Vision, Vol. 7, No. 1, pp. 85–102, 1997.

    Article  MathSciNet  Google Scholar 

  26. J. Crespo, R.W. Schafer, J. Serra, Ch. Gratin, and F. Meyer, “The flat zone approach: A general low-level region merging segmentation method,” Signal Processing, Vol. 62, No. 1, pp. 37–60, 1997.

    Article  Google Scholar 

  27. E. Diday, J. Lemaire, J. Pouget, and F. Testu, Eléments d'Analyse de Donnéés, Dunod, 1982.

  28. L. Garrido, Ph. Salembier, and D. Garcia, “Extensive operators in partition lattices for image sequence analysis,” Signal Processing, Vol. 66, No. 2, pp. 157–180, 1998.

    Article  Google Scholar 

  29. D. Gatica-Perez, C. Gu, M.-T. Sun, and S. Ruiz-Correa, “Extensive partition operators, gray level connected operators, and region merging/classification segmentation algorithms: theoretical links,” IEEE Transactions on Image Processing, Vol. 10, No. 9, pp. 1332–1345, 2001.

    Article  MathSciNet  Google Scholar 

  30. A. Hanbury Morphologie mathématique sur le cercle unité, avec application aux teintes et aux textures orientées. Thèse de Doctorat, Ecole des Mines de Paris, Mars 2002.

  31. A. Hanbury and J. Serra, “Morphological operators on the unit circle,” IEEE Transactions on Image Processing, Vol. 10, No. 12, pp. 1842-1850, 2001.

  32. A. Hanbury and J. Serra, “Colour image analysis in 3D-polar coordinates,” DAGM Symposium, Vienna April 2003.

  33. R.M. Haralick and L.G. Shapiro, “Image segmentation techniques,” Computer Vision Graphics and Image Processing, Vol. 29, pp. 100–132, 1985.

    Google Scholar 

  34. H.J.A.M. Heijmans, Connected Morphological Operators for Binary Images. Computer Vision and Image Understanding, Vol. 73, pp. 99–120, 1999.

    Article  MATH  Google Scholar 

  35. H.J.A.M. Heijmans, “Morphological scale-spaces and lie groups,” in Mathematical Morphology, H. Talbot and R. Beare, (Eds.), CSIRO Sydney, pp. 253–264, 2002.

  36. S.L. Horowitz and T. Pavlidis, “Picture segmentation by a tree traversal algorithm,” Journal of the ACM, Vol. 23, pp. 368–388, 1976.

    Google Scholar 

  37. D. Jeulin (Ed) Advances in Theory and Applications of Random Sets World Scientific, 1997, 326p.

  38. D. Jeulin, “Random texture models for material structures,” Statistics and Computing on Image Analysis, Vol. 10, No. 2, pp. 121–132, 2000.

    Google Scholar 

  39. M. Kass, A. Within and D. Terzopoulos, “Snakes active contour models,” International Journal of Computer Vision, Vol. 1, pp. 321–331, 1988.

    Google Scholar 

  40. Ch. O. Kiselman, “Digital geometry and mathematical morphology,” Lecture Notes Univ. of Uppsala, April 2004, 89p.

  41. J.J. Koenderink, “The structure of images,” Biol. Cybern, Vol. 50, pp. 363–370, 1984.

    MATH  MathSciNet  Google Scholar 

  42. C. Lantuejoul and S. Beucher, “On the use of the geodesic metric in image analysis,” J. of Microscopy Vol. 121, pp. 39–49, 1981.

    Google Scholar 

  43. M. Leventon, W. Grimson, and O. Faugeras, “Statistical shape influence on geodesic active contours,” CVPR 2000.

  44. R. Malladi, J.A. Sethian, and B.C. Vemuri, “Shape modeling with front propagation — a level set approach,” IEEE Trans. on PAMI, Vol. 17, pp 158–175, 1995.

    Google Scholar 

  45. P. Maragos, “Differential morphology and image processing,” IEEE Trans. Image Processing, Vol. 78, pp. 922–937, 1996.

    Google Scholar 

  46. P. Maragos and F. Meyer, “Nonlinear PDEs and numerical algorithms for modeling levelings and reconstructions filters,” Scale-Space Theories in Computer Vision, N. Mads et al. (Eds.) Lecture Notes in Computer Science, no. 1682, 1999, pp. 363–374.

  47. B. Marcotegui and F. Meyer, “Morphological segmentation of image sequences,” in Mathematical Morphology and its Applications to Image Processing, Serra J. and Soille P. (Eds.), Kluwer, 1994, pp. 101–108.

  48. D. Marr and E. Hildreth, “Theory of edge detection,” Proc. Roy. Soc. Lond. Vol. B207, pp. 187–217, 1980.

  49. D. Marr, Vision, Freeman and Co., 1982.

  50. G. Matheron, Random Sets and Integral Geometry, Wiley: New-York, 1975.

    Google Scholar 

  51. G. Matheron, Estimating and Choosing, Springer-Verlag, 1989, p. 141.

  52. G. Matheron, Les treillis compacts. Tech. rep. N-23/90/G, Ecole des Mines de Paris, Part 1, 1990, part 2, 1996.

  53. G. Matheron, Les nivellements. Tech. rep. N-07/97/MM Ecole des Mines de Paris, 1997.

  54. F. Meyer and P. Maragos, “Nonlinear scale-space representation with morphological levellings,” JVCIR, Vol. 11, pp. 245–265, 2000.

    Google Scholar 

  55. F. Meyer and S. Beucher, “Morphological Segmentation,” JVCIR, Vol. 1, No. 1, pp. 21–46, 1990.

    Google Scholar 

  56. F. Meyer, “Minimum spanning forests for morphological segmentation,” in Mathematical Morphology and its Applications to Image and Signal Processing, P. Maragos et al. (Eds.), Kluwer, 1996.

  57. F. Meyer, “The levelings,” in Mathematical Morphology and its Applications to Image and Signal Processing, Kluwer, pp. 191–206. 1998.

  58. F. Meyer, “Hierarchies of partitions and morphological segmentation,” in IEEE Workshop on Scale-Space and Morphology in Computer Vision & ICCV 2001. Vancouver, July 7–8, 2001.

  59. F. Meyer, “An overview of morphological segmentation,” International Journal of Pattern Recognition and Artificial Intelligence, Vol. 15, No. 7, pp. 1089–1118, 2001.

    Article  Google Scholar 

  60. Y. Meyer, Wavelets and Operators. Advanced Mathematics, Cambridge University Press, 1992.

  61. J.-M. Morel and S. Solimi, Variational Methods in Image Segmentation, Birkhäuser Boston 1995.

    Google Scholar 

  62. D. Mumford and J. Shah, “Boundary detection by minimizing functionals,” in IEEE Conference on Computer Vision and Pattern Recognition, San Francisco 1985.

  63. D. Mumford and J. Shah, “Boundary detection by minimizing functional,” in Image Understanding, S. Ulmann and W. Richards (Eds.), 1988.

  64. S. Osher and J.A. Sethian, “Front propagating with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comput. Physics, Vol. 79, pp. 12–49, 1988.

    Article  MathSciNet  Google Scholar 

  65. M. Pardàs, J. Serra, and L.Torrès, “Connectivity filters for image sequences,” in Image Algebra and Morphological Processing III, P.D. Gader, E.R. Dougherty, and J.C. Serra (Eds.), SPIE, Vol. 1769, pp. 318–329, 1992.

  66. P. Perona and J. Malik, “A scale space and edge detection using anisotropic diffusion,” in Proc. IEEE Computer Soc. Workshop on Computer vision, 1987.

  67. A.R. Rao, A Taxonomy For Texture Description And Identification. Springer-Verlag, 1990.

  68. C. Ronse, “Set theoretical algebraic approaches to connectivity in continuous or digital spaces,” JMIV, Vol. 8 No. 1, pp. 41–58, 1998.

    MathSciNet  Google Scholar 

  69. C. Ronse and J. Serra, “Geodesy and connectivity in lattices,” Fundamenta Informaticae, Vol 46, No. 4, pp. 349–395, 2001.

    MathSciNet  Google Scholar 

  70. A. Rosenfeld, “Sequential operations in digital picture processing,” Journal of the ACM, Vol. 13, 471–494, 1966.

    MATH  Google Scholar 

  71. A. Rosenfeld and A.C. Kak, Digital Picture Processing, Academic Press, New-York, 1982.

    Google Scholar 

  72. P. Salembier and J. Serra, “Flat zones filtering, connected operators, and filters by reconstruction,” IEEE Transactions on Image Processing, Vol. 4, No. 8, 1153–1160, 1995.

    Article  Google Scholar 

  73. P. Salembier and F. Marqués, “Region-based representations of image and video: Segmentation tools for multimedia services,” IEEE Transactions on Circuits and Systems for Video Technology, Vol. 9, No. 8, pp. 1147–1167, 1999.

    Article  Google Scholar 

  74. P. Salembier and L. Garrido, “Connected operators based on region-tree pruning,” in Mathematical Morphology and its applications to image processing, J. Goustias, L. Vincent and D.S. Bloomberg (Eds.), Kluwer, pp. 170–178, 2000.

  75. J. Serra, Image analysis and mathematical morphology, part II: theoretical advances, J. Serra (Ed.), Academic Press, London, 1988.

  76. J. Serra and L. Vincent, “An overview of morphological filtering,” IEEE Trans, on Circuits, Systems and Signal Processing, Vol. 11, No. 1, pp. 47–108, 1992.

    MathSciNet  Google Scholar 

  77. J. Serra and P. Salembier, “Connected operators and pyramids,” in Non linear Algebra and Morphological Image Processing SPIE, San Diego, Vol. 2030, pp. 65–76, 1993.

  78. J. Serra, “Equicontinuous functions, a model for mathematical morphology,” in Non Linear Algebra and Morphological Image Processing SPIE, San Diego, Vol. 1769, pp. 252–263, 1992.

  79. J. Serra Cube, “Cube-octahedron or rhombododecahedron as bases for 3-D shape descriptions,” Advances in Visual Form Analysis, C. Arcelli et al. (Eds.), World Scientific, pp. 502–519, 1997.

  80. J. Serra, “Connectivity on complete lattices,” Journal of Mathematical Imaging and Vision, Vol. 9, pp. 231–251, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  81. J. Serra, “Set connections and discrete filtering,” (Proc. DGCI 1999) Lecture Notes in Computer Science, G. Bertrand, M. Couprie, and L Perroton (Eds.), Springer, Vol. 1568, pp. 191–206, 1999.

  82. J. Serra, “Connectivity for sets and functions,” Fundamenta Informaticae, Vol. 41, pp. 147–186, 2000.

    MATH  MathSciNet  Google Scholar 

  83. J. Serra, Espaces couleur et traitement d'images (tech. report Ecole des Mines N-34/02/MM), 13p. 2002.

  84. J. Serra, Viscous Lattices, JMIV, Vol. 22, Nos. 2/3, pp. 269–282, 2005

  85. J. Serra, “Morphological descriptions using three-dimensional wavefronts,” Image Anal Stereol, Vol. 21, pp. 1–9, 2002.

    Google Scholar 

  86. J. Serra, Segmentation, Tech. Rep. EMP N-06/03/MM march 2003, 25p.

  87. J. Serra, “Morphological segmentation of colour images,” in Mathematical Morphology 40 Years On, C. Ronse, L. Najman, and E. Decencière (Eds.), Springer 2005, pp. 151–176.

  88. J. A. Sethian, Level Set Methods. Cambridge Univ. Press, 1996.

  89. P. Soille, Morphologie Mathématique: du relief à la dimensionalité — Algorithmes et méthodes-, Thèse de Doctorat, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1992.

  90. S.R. Sternberg, “Morphology for grey-tone functions,” Computer Vision, Graphics and Image Processing, Vol. 35, pp. 333–355, 1986.

    Google Scholar 

  91. C. Tzafestas and P. Maragos, “Shape Connectivity,” JMIV, Vol. 17, No. 2, pp. 109–129, 2002.

    MathSciNet  Google Scholar 

  92. P. Soille,Morphological image analysis, Springer-Verlag, Berlin Heidelberg, 1999.

    Google Scholar 

  93. C. Vachier,Extraction de caractéristiques, segmentation d'images et morphologie mathématique. Thèse de Doctorat, Ecole des Mines de Paris, 1995.

  94. L. Vincent, “Morphological grayscale reconstruction in image analysis: Applications and efficient algorithms,” IEEE Transactions in Image Processing, Vol. 2, pp. 176–201, 1993.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire A2SI ESIEE, BP 99 93162 Noisy-le-Grand, Cedex, France

    Jean Serra

Authors
  1. Jean Serra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Serra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Serra, J. A Lattice Approach to Image Segmentation. J Math Imaging Vis 24, 83–130 (2006). https://doi.org/10.1007/s10851-005-3616-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-005-3616-0

Keywords

Search

Navigation