On Morphological Hierarchical Representations for Image Processing and Spatial Data Clustering

  • Pierre Soille
  • Laurent Najman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7346)

Abstract

Hierarchical data representations in the context of classification and data clustering were put forward during the fifties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satisfied. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing.

Keywords

image representation segmentation clustering ultrametric hierarchy graphs connected components constrained connectivity watersheds min-tree alpha-tree 

References

  1. 1.
    Köthe, U., Montanvert, A., Soille, P. (eds.): Proc. of ICPR Workshop on Applications of Discrete Geometry and Mathematical Morphology. IAPR, Istanbul (2010)Google Scholar
  2. 2.
    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  3. 3.
    Meyer, F., Maragos, P.: Nonlinear scale-space representation with morphological levelings. Journal of Visual Communication and Image Representation 11, 245–265 (2000)CrossRefGoogle Scholar
  4. 4.
    Cormack, R.: A review of classification (with discussion). Journal of the Royal Statistical Society A 134, 321–367 (1971)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Estabrook, G.: A mathematical model in graph theory for biological applications. Journal of Theoretical Biology 12, 297–310 (1966)CrossRefGoogle Scholar
  6. 6.
    Matula, D.: Cluster analysis via graph theoretic techniques. In: Mulin, R., Reid, K., Roselle, P. (eds.) Proc. Louisiana Conference on Combinatorics, Graph Theory, and Computing, Winnipeg, University of Manitoba, pp. 199–212 (1970)Google Scholar
  7. 7.
    Zahn, C.: Graph-theoretical methods for detecting and describing gestalt clusters. IEEE Transactions on Computers C-20, 68–86 (1971)Google Scholar
  8. 8.
    Hubert, L.: Some applications of graph theory to clustering. Psychometrika 39(3), 283–309 (1974)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hubert, L.: Min and max hierarchical clustering using asymetric similaritly measures. Psychometrika 38, 63–72 (1973)CrossRefMATHGoogle Scholar
  10. 10.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer (1997)Google Scholar
  11. 11.
    Kong, T., Rosenfeld, A.: Digital topology: Introduction and survey. Comput. Vision Graph. Image Process. 48, 357–393 (1989)CrossRefGoogle Scholar
  12. 12.
    Spärck Jones, K.: Some thoughts on classification for retrieval. Journal of Documentation 26(2), 571–581 (1970)Google Scholar
  13. 13.
    Jardine, N., Sibson, R.: A model for taxonomy. Mathematical Biosciences 2(3-4), 465–482 (1968)CrossRefGoogle Scholar
  14. 14.
    Barthélemy, J.P., Brucker, F., Osswald, C.: Combinatorial optimization and hierarchical classifications. 4OR: A Quaterly Journal of Operations Research 2(3), 179–219 (2004)CrossRefMATHGoogle Scholar
  15. 15.
    Johnson, S.: Hierarchical clustering schemes. Psychometrika 32(3), 241–254 (1967)CrossRefGoogle Scholar
  16. 16.
    Sokal, R., Sneath, P.: Principles of Numerical Taxonomy. W.H. Freeman and Company, San Fransisco and London (1963)Google Scholar
  17. 17.
    Sneath, P.: The application of computers in taxonomy. Journal of General Microbiology 17, 201–226 (1957)CrossRefGoogle Scholar
  18. 18.
    Hartigan, J.: Representation of similarity matrices by trees. American Statistical Association Journal, 1140–1158 (1967)Google Scholar
  19. 19.
    Jardine, C., Jardine, N., Sibson, R.: The structure and construction of taxonomic hierarchies. Mathematical Biosciences 1(2), 173–179 (1967)CrossRefMATHGoogle Scholar
  20. 20.
    Benzécri, J.P.: L’analyse des données. La taxinomie, vol. 1. Dunod, Paris (1973)Google Scholar
  21. 21.
    Sørensen, T.: A method of establishing groups of equal amplitude in plant sociology based on similarity of species content and its applications to analyses of the vegetation of Danish commons. Biologiske Skrifter 5(4), 1–34 (1948)Google Scholar
  22. 22.
    Florek, K., Łukaszewicz, J., Perkal, J., Steinhaus, H., Zubrzycki, S.: Sur la liaison et la division des points d’un ensemble fini. Colloquium Mathematicum 2, 282–285 (1951)Google Scholar
  23. 23.
    Gower, J., Ross, G.: Minimum spanning trees and single linkage cluster analysis. Applied Statistics 18(1), 54–64 (1969)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kruskal, J.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society 7(1), 48–50 (1956)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Borůvka, O.: O jistém problému minimálním (On a certain minimal problem). Acta Societatis Scientiarum Naturalium Moravicae III(3), 37–58 (1926)Google Scholar
  26. 26.
    Graham, R., Hell, P.: On the history of the minimum spanning tree problem. Ann. History Comput. 7(1), 43–57 (1985)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wishart, D.: Mode analysis: a generalization of nearest neighour which reduced chain effect. In: Cole, A. (ed.) Numerical Taxonomy, pp. 282–311. Academic Press, New York (1968)Google Scholar
  28. 28.
    Jardine, N., Sibson, R.: The construction of hierarchic and non-hierarchic classifications. The Computer Journal 11, 177–184 (1968)MATHGoogle Scholar
  29. 29.
    Horowitz, S., Pavlidis, T.: Picture segmentation by a directed split-and-merge procedure. In: Proc. Second Int. Joint Conf. Pattern Recognition, pp. 424–433 (1974)Google Scholar
  30. 30.
    Zucker, S.: Region growing: childhood and adolescence. Computer Graphics and Image Processing 5, 382–399 (1976)CrossRefGoogle Scholar
  31. 31.
    Jardine, N., Sibson, R.: Mathematical Taxonomy. Wiley, London (1971)MATHGoogle Scholar
  32. 32.
    Rosenfeld, A.: Fuzzy digital topology. Information and Control 40, 76–87 (1979)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Soille, P.: Morphological partitioning of multispectral images. Journal of Electronic Imaging 5(3), 252–265 (1996)CrossRefGoogle Scholar
  34. 34.
    Ahuja, N.: On detection and representation of multiscale low-level image structure. ACM Computing Surveys 27(3), 304–306 (1995)CrossRefGoogle Scholar
  35. 35.
    Serra, J.: A lattice approach to image segmentation. Journal of Mathematical Imaging and Vision 24(1), 83–130 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ronse, C.: Partial partitions, partial connections and connective segmentation. Journal of Mathematical Imaging and Vision 32(2), 97–105 (2008)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Transactions on Pattern Analysis and Machine Intelligence 30(7), 1132–1145 (2008)CrossRefGoogle Scholar
  38. 38.
    Horowitz, S., Pavlidis, T.: Picture segmentation by a tree traversal algorithm. Journal of the ACM 23(2), 368–388 (1976)CrossRefMATHGoogle Scholar
  39. 39.
    Nagao, M., Matsuyama, T., Ikeda, Y.: Region extraction and shape analysis in aerial photographs. Computer Graphics and Image Processing 10(3), 195–223 (1979)CrossRefGoogle Scholar
  40. 40.
    Nagao, M., Matsuyama, T.: A Structural Analysis of Complex Aerial Photographs. Plenum, New York (1980)CrossRefGoogle Scholar
  41. 41.
    Baraldi, A., Parmiggiani, F.: Single linkage region growing algorithms based on the vector degree of match. IEEE Transactions on Geoscience and Remote Sensing 34(1), 137–148 (1996)CrossRefGoogle Scholar
  42. 42.
    Morris, O., Lee, M., Constantinides, A.: Graph theory for image analysis: an approach based on the shortest spanning tree. IEE Proceedings 133(2), 146–152 (1986)Google Scholar
  43. 43.
    Meyer, F., Maragos, P.: Morphological Scale-Space Representation with Levelings. In: Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.) Scale-Space 1999. LNCS, vol. 1682, pp. 187–198. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  44. 44.
    Nacken, P.: Image segmentation by connectivity preserving relinking in hierarchical graph structures. Pattern Recognition 28(6), 907–920 (1995)CrossRefGoogle Scholar
  45. 45.
    Kropatsch, W., Haxhimusa, Y.: Grouping and segmentation in a hierarchy of graphs. In: Bouman, C., Miller, E. (eds.) Proc. of the 16th IS&T SPIE Annual Symposium, Computational Imaging II. SPIE, vol. 5299, pp. 193–204 (May 2004)Google Scholar
  46. 46.
    Felzenszwalb, P., Huttenlocher, D.: Image segmentation using local variations. In: Proc. of IEEE Int. Conf. on Comp. Vis. and Pat. Rec (CVPR), pp. 98–104 (1998)Google Scholar
  47. 47.
    Felzenszwalb, P., Huttenlocher, D.: Efficient graph-based segmentation. IJCV 59(2), 167–181 (2004)CrossRefGoogle Scholar
  48. 48.
    Wu, Z., Leahy, R.: An optimal graph-theoretic approach to data clustering: theory and its applications to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993)CrossRefGoogle Scholar
  49. 49.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)CrossRefGoogle Scholar
  50. 50.
    Marfil, R., Molina-Tanco, L., Bandera, A., Rodriguez, J., Sandoval, F.: Pyramid segmentation algorithms revisited. Pattern Recognition 39(8), 1430–1451 (2006)CrossRefMATHGoogle Scholar
  51. 51.
    Kropatsch, W.G., Haxhimusa, Y., Ion, A.: Multiresolution Image Segmentations in Graph Pyramids. In: Kandel, A., Bunke, H., Last, M. (eds.) Applied Graph Theory in Computer Vision and Pattern Recognition. SCI, vol. 52, pp. 3–41. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  52. 52.
    Guigues, L., Le Men, H., Cocquerez, J.P.: The hierarchy of the cocoons of a graph and its application to image segmentation. Pattern Recognition Letters 24(8), 1059–1066 (2003)CrossRefMATHGoogle Scholar
  53. 53.
    Guigues, L., Cocquerez, J.P., Le Men, H.: Scale-sets image analysis. IJCV 68(3), 289–317 (2006)CrossRefGoogle Scholar
  54. 54.
    Arbeláez, P., Cohen, L.: Energy partition and image segmentation. Journal of Mathematical Imaging and Vision 20, 43–57 (2004)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Arbeláez, P.: Boundary extraction in natural images using ultrametric contour maps. In: Proc. of Computer Vision and Pattern Recognition Workshop. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  56. 56.
    Beucher, S.: Segmentation d’images et morphologie mathématique. PhD thesis, Ecole des Mines de Paris (June 1990)Google Scholar
  57. 57.
    Beucher, S.: Watershed, hierarchical segmentation and waterfall algorithm. In: Serra, J., Soille, P. (eds.) Mathematical Morphology and its Applications to Image Processing, pp. 69–76. Kluwer Academic Publishers (1994)Google Scholar
  58. 58.
    Beucher, S., Meyer, F.: The morphological approach to segmentation: the watershed transformation. In: Dougherty, E. (ed.) Mathematical Morphology in Image Processing. Optical Engineering, vol. 34, pp. 433–481. Marcel Dekker, New York (1993)Google Scholar
  59. 59.
    Vincent, L., Soille, P.: Watersheds in digital spaces: an efficient algorithm based on immersion simulations. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(6), 583–598 (1991)CrossRefGoogle Scholar
  60. 60.
    Najman, L., Schmitt, M.: Geodesic saliency of watershed contours and hierarchical segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 18(12), 1163–1173 (1996)CrossRefGoogle Scholar
  61. 61.
    Cousty, J., Najman, L.: Incremental Algorithm for Hierarchical Minimum Spanning Forests and Saliency of Watershed Cuts. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 272–283. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  62. 62.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest-path forests and topological watersheds. IEEE Transactions on Pattern Analysis and Machine Intelligence 32(5), 925–939 (2010)CrossRefGoogle Scholar
  63. 63.
    Meyer, F.: Minimum spanning forests for morphological segmentation. In: Serra, J., Soille, P. (eds.) Mathematical Morphology and its Applications to Image Processing, pp. 77–84. Kluwer Academic Publishers (1994)Google Scholar
  64. 64.
    Salembier, P., Garrido, L.: Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval. IEEE Transactions on Image Processing 9(4), 561–576 (2000)CrossRefGoogle Scholar
  65. 65.
    Jones, R.: Component trees for image filtering and segmentation. In: Coyle, E. (ed.) Proc. of IEEE Workshop on Nonlinear Signal and Image Processing, Mackinac Island (September 1997)Google Scholar
  66. 66.
    Jones, R.: Connected filtering and segmentation using component trees. Comput. Vis. Image Underst. 75(3), 215–228 (1999)CrossRefGoogle Scholar
  67. 67.
    Salembier, P., Oliveras, A., Garrido, L.: Antiextensive connected operators for image and sequence processing. IEEE Transactions on Image Processing 7(4), 555–570 (1998)CrossRefGoogle Scholar
  68. 68.
    Meyer, F.: An overview of morphological segmentation. International Journal of Pattern Recognition and Artificial Intelligence 15(7), 1089–1118 (2001)CrossRefGoogle Scholar
  69. 69.
    Meyer, F., Najman, L.: Segmentation, minimum spanning tree and hierarchies. In: Najman, L., Talbot, H. (eds.) Mathematical Morphology: From Theory to Applications, pp. 255–287. Wiley-ISTE (2010)Google Scholar
  70. 70.
    Salembier, P., Wilkinson, M.: Connected operators: A review of region-based morphological image processing techniques. IEEE Signal Processing Magazine 26(6), 136–157 (2009)CrossRefGoogle Scholar
  71. 71.
    Salembier, P.: Connected operators based on tree pruning strategies. In: Najman, L., Talbot, H. (eds.) Mathematical Morphology: From Theory to Applications, pp. 205–221. Wiley-ISTE (2010)Google Scholar
  72. 72.
    Soille, P.: On genuine connectivity relations based on logical predicates. In: Proc. of 14th Int. Conf. on Image Analysis and Processing, Modena, Italy, pp. 487–492. IEEE Computer Society Press (2007)Google Scholar
  73. 73.
    Soille, P.: Preventing Chaining through Transitions While Favouring It within Homogeneous Regions. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 96–107. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  74. 74.
    Gueguen, L., Soille, P.: Frequent and Dependent Connectivities. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 120–131. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  75. 75.
    Soille, P.: Advances in the Analysis of Topographic Features on Discrete Images. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 175–186. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  76. 76.
    Soille, P., Grazzini, J.: Constrained Connectivity and Transition Regions. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 59–69. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  77. 77.
    Soille, P., Vincent, L.: Determining watersheds in digital pictures via flooding simulations. In: Kunt, M. (ed.) Visual Communications and Image Processing 1990, vol. 1360, pp. 240–250. Society of Photo-Instrumentation Engineers, Bellingham (1990)Google Scholar
  78. 78.
    Ouzounis, G., Soille, P.: Pattern Spectra from Partition Pyramids and Hierarchies. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 108–119. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  79. 79.
    Ouzounis, G., Soille, P.: Attribute-constrained connectivity and alpha-tree representation. IEEE Transactions on Image Processing (2011)Google Scholar
  80. 80.
    Soille, P.: Constrained connectivity for the processing of very high resolution satellite images. International Journal of Remote Sensing 31(22), 5879–5893 (2010)CrossRefGoogle Scholar
  81. 81.
    Najman, L.: Ultrametric Watersheds. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 181–192. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  82. 82.
    Najman, L.: On the equivalence between hierarchical segmentations and ultrametric watersheds. Journal of Mathematical Imaging and Vision 40(3), 231–247 (2011)MathSciNetCrossRefGoogle Scholar
  83. 83.
    Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(2-3), 217–230 (2005)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Cousty, J., Najman, L.: Incremental Algorithm for Hierarchical Minimum Spanning Forests and Saliency of Watershed Cuts. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 272–283. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  85. 85.
    Mattiussi, C.: The Finite Volume, Finite Difference, and Finite Elements Methods as Numerical Methods for Physical Field Problems. Advances in Imaging and Electron Physics 113, 1–146 (2000)CrossRefGoogle Scholar
  86. 86.
    Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE Transactions on Image Processing 15(11), 3531–3539 (2006)CrossRefGoogle Scholar
  87. 87.
    Breen, E., Jones, R.: Attribute openings, thinnings, and granulometries. Comput. Vis. Image Underst. 64(3), 377–389 (1996)CrossRefGoogle Scholar
  88. 88.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE Transactions on Pattern Analysis and Machine Intelligence 31(8), 1362–1374 (2009)CrossRefGoogle Scholar
  89. 89.
    Adams, R., Bischof, L.: Seeded region growing. IEEE Transactions on Pattern Analysis and Machine Intelligence 16(6), 641–647 (1994)CrossRefGoogle Scholar
  90. 90.
    Hubert, L.: Some extension of Johnson’s hierarchical clustering. Psychometrika 37, 261–274 (1972)MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    Cousty, J., Najman, L., Serra, J.: Raising in watershed lattices. In: 15th IEEE ICIP 2008, San Diego, USA, pp. 2196–2199 (2008)Google Scholar
  92. 92.
    Cousty, J., Najman, L., Serra, J.: Some Morphological Operators in Graph Spaces. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 149–160. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  93. 93.
    Dias, F., Cousty, J., Najman, L.: Some Morphological Operators on Simplicial Complex Spaces. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 441–452. Springer, Heidelberg (2011)Google Scholar
  94. 94.
    Levillain, R., Géraud, T., Najman, L.: Writing Reusable Digital Topology Algorithms in a Generic Image Processing Framework. In: Köthe, U., Montanvert, A., Soille, P. (eds.) WADGMM 2010. LNCS, vol. 7346, pp. 140–153. Springer, Heidelberg (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pierre Soille
    • 1
  • Laurent Najman
    • 2
  1. 1.Joint Research Centre, European CommissionInstitute for the Protection and Security of the CitizenIspraItaly
  2. 2.Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI, ESIIEUniversité Paris-EstFrance

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