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On Morphological Hierarchical Representations for Image Processing and Spatial Data Clustering

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Applications of Discrete Geometry and Mathematical Morphology (WADGMM 2010)

Part of the book series: Lecture Notes in Computer Science ((LNIP,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.

A preliminary version of this paper was presented at the workshop WADGMM 2010 [1] held in conjunction with ICPR 2010, Istanbul, August 2010.

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References

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

    Google Scholar 

  3. Meyer, F., Maragos, P.: Nonlinear scale-space representation with morphological levelings. Journal of Visual Communication and Image Representation 11, 245–265 (2000)

    Article  Google Scholar 

  4. Cormack, R.: A review of classification (with discussion). Journal of the Royal Statistical Society A 134, 321–367 (1971)

    Article  MathSciNet  Google Scholar 

  5. Estabrook, G.: A mathematical model in graph theory for biological applications. Journal of Theoretical Biology 12, 297–310 (1966)

    Article  Google Scholar 

  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. Zahn, C.: Graph-theoretical methods for detecting and describing gestalt clusters. IEEE Transactions on Computers C-20, 68–86 (1971)

    Google Scholar 

  8. Hubert, L.: Some applications of graph theory to clustering. Psychometrika 39(3), 283–309 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hubert, L.: Min and max hierarchical clustering using asymetric similaritly measures. Psychometrika 38, 63–72 (1973)

    Article  MATH  Google Scholar 

  10. Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer (1997)

    Google Scholar 

  11. Kong, T., Rosenfeld, A.: Digital topology: Introduction and survey. Comput. Vision Graph. Image Process. 48, 357–393 (1989)

    Article  Google Scholar 

  12. Spärck Jones, K.: Some thoughts on classification for retrieval. Journal of Documentation 26(2), 571–581 (1970)

    Google Scholar 

  13. Jardine, N., Sibson, R.: A model for taxonomy. Mathematical Biosciences 2(3-4), 465–482 (1968)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  15. Johnson, S.: Hierarchical clustering schemes. Psychometrika 32(3), 241–254 (1967)

    Article  Google Scholar 

  16. Sokal, R., Sneath, P.: Principles of Numerical Taxonomy. W.H. Freeman and Company, San Fransisco and London (1963)

    Google Scholar 

  17. Sneath, P.: The application of computers in taxonomy. Journal of General Microbiology 17, 201–226 (1957)

    Article  Google Scholar 

  18. Hartigan, J.: Representation of similarity matrices by trees. American Statistical Association Journal, 1140–1158 (1967)

    Google Scholar 

  19. Jardine, C., Jardine, N., Sibson, R.: The structure and construction of taxonomic hierarchies. Mathematical Biosciences 1(2), 173–179 (1967)

    Article  MATH  Google Scholar 

  20. Benzécri, J.P.: L’analyse des données. La taxinomie, vol. 1. Dunod, Paris (1973)

    Google Scholar 

  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. 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. Gower, J., Ross, G.: Minimum spanning trees and single linkage cluster analysis. Applied Statistics 18(1), 54–64 (1969)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Graham, R., Hell, P.: On the history of the minimum spanning tree problem. Ann. History Comput. 7(1), 43–57 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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. Jardine, N., Sibson, R.: The construction of hierarchic and non-hierarchic classifications. The Computer Journal 11, 177–184 (1968)

    MATH  Google Scholar 

  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. Zucker, S.: Region growing: childhood and adolescence. Computer Graphics and Image Processing 5, 382–399 (1976)

    Article  Google Scholar 

  31. Jardine, N., Sibson, R.: Mathematical Taxonomy. Wiley, London (1971)

    MATH  Google Scholar 

  32. Rosenfeld, A.: Fuzzy digital topology. Information and Control 40, 76–87 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  33. Soille, P.: Morphological partitioning of multispectral images. Journal of Electronic Imaging 5(3), 252–265 (1996)

    Article  Google Scholar 

  34. Ahuja, N.: On detection and representation of multiscale low-level image structure. ACM Computing Surveys 27(3), 304–306 (1995)

    Article  Google Scholar 

  35. Serra, J.: A lattice approach to image segmentation. Journal of Mathematical Imaging and Vision 24(1), 83–130 (2006)

    Article  MathSciNet  Google Scholar 

  36. Ronse, C.: Partial partitions, partial connections and connective segmentation. Journal of Mathematical Imaging and Vision 32(2), 97–105 (2008)

    Article  MathSciNet  Google Scholar 

  37. Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Transactions on Pattern Analysis and Machine Intelligence 30(7), 1132–1145 (2008)

    Article  Google Scholar 

  38. Horowitz, S., Pavlidis, T.: Picture segmentation by a tree traversal algorithm. Journal of the ACM 23(2), 368–388 (1976)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  40. Nagao, M., Matsuyama, T.: A Structural Analysis of Complex Aerial Photographs. Plenum, New York (1980)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  44. Nacken, P.: Image segmentation by connectivity preserving relinking in hierarchical graph structures. Pattern Recognition 28(6), 907–920 (1995)

    Article  Google Scholar 

  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. 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. Felzenszwalb, P., Huttenlocher, D.: Efficient graph-based segmentation. IJCV 59(2), 167–181 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  49. Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)

    Article  Google Scholar 

  50. Marfil, R., Molina-Tanco, L., Bandera, A., Rodriguez, J., Sandoval, F.: Pyramid segmentation algorithms revisited. Pattern Recognition 39(8), 1430–1451 (2006)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  53. Guigues, L., Cocquerez, J.P., Le Men, H.: Scale-sets image analysis. IJCV 68(3), 289–317 (2006)

    Article  Google Scholar 

  54. Arbeláez, P., Cohen, L.: Energy partition and image segmentation. Journal of Mathematical Imaging and Vision 20, 43–57 (2004)

    Article  MathSciNet  Google Scholar 

  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. Beucher, S.: Segmentation d’images et morphologie mathématique. PhD thesis, Ecole des Mines de Paris (June 1990)

    Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. Jones, R.: Connected filtering and segmentation using component trees. Comput. Vis. Image Underst. 75(3), 215–228 (1999)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  68. Meyer, F.: An overview of morphological segmentation. International Journal of Pattern Recognition and Artificial Intelligence 15(7), 1089–1118 (2001)

    Article  Google Scholar 

  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. Salembier, P., Wilkinson, M.: Connected operators: A review of region-based morphological image processing techniques. IEEE Signal Processing Magazine 26(6), 136–157 (2009)

    Article  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  79. Ouzounis, G., Soille, P.: Attribute-constrained connectivity and alpha-tree representation. IEEE Transactions on Image Processing (2011)

    Google Scholar 

  80. Soille, P.: Constrained connectivity for the processing of very high resolution satellite images. International Journal of Remote Sensing 31(22), 5879–5893 (2010)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  82. Najman, L.: On the equivalence between hierarchical segmentations and ultrametric watersheds. Journal of Mathematical Imaging and Vision 40(3), 231–247 (2011)

    Article  MathSciNet  Google Scholar 

  83. Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(2-3), 217–230 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  86. Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE Transactions on Image Processing 15(11), 3531–3539 (2006)

    Article  Google Scholar 

  87. Breen, E., Jones, R.: Attribute openings, thinnings, and granulometries. Comput. Vis. Image Underst. 64(3), 377–389 (1996)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  89. Adams, R., Bischof, L.: Seeded region growing. IEEE Transactions on Pattern Analysis and Machine Intelligence 16(6), 641–647 (1994)

    Article  Google Scholar 

  90. Hubert, L.: Some extension of Johnson’s hierarchical clustering. Psychometrika 37, 261–274 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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. 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 

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Authors and Affiliations

  1. Joint Research Centre, European Commission, Institute for the Protection and Security of the Citizen, Via E. Fermi 2749, I-21027, Ispra, VA, Italy

    Pierre Soille

  2. Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI, ESIIE, Université Paris-Est, France

    Laurent Najman

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  1. Pierre Soille
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  2. Laurent Najman
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Editors and Affiliations

  1. HCI, University of Heidelberg, Germany, Speyerer Strasse 6, D-69115, Heidelberg, Germany

    Ullrich Köthe

  2. GIPSA-lab, 961 rue de la Houille Blanche, 38402, Saint Martin d’Hères cedex, France

    Annick Montanvert

  3. EC JRC Space Application Institute, T.P. 262, 21020, Ispra, Italy

    Pierre Soille

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Soille, P., Najman, L. (2012). On Morphological Hierarchical Representations for Image Processing and Spatial Data Clustering. In: Köthe, U., Montanvert, A., Soille, P. (eds) Applications of Discrete Geometry and Mathematical Morphology. WADGMM 2010. Lecture Notes in Computer Science, vol 7346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32313-3_4

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