Component-Trees and Multivalued Images: Structural Properties

  • Nicolas Passat
  • Benoît Naegel


Component-trees model the structure of grey-level images by considering their binary level-sets obtained from successive thresholdings. They also enable to define anti-extensive filtering procedures for such images. In order to extend this image processing approach to any (grey-level or multivalued) images, both the notion of component-tree, and its associated filtering framework, have to be generalised. In this article we deal with the generalisation of the component-tree structure. We define a new data structure, the component-graph, which extends the notion of component-tree to images taking their values in any (partially or totally) ordered set. The component-graphs are declined in three variants, of increasing richness and size, whose structural properties are studied.


Mathematical morphology Component-tree Multivalued images Anti-extensive filtering Component-graph 



The research leading to these results has received funding from the French Agence Nationale de la Recherche (Grant Agreement ANR-10-BLAN-0205).


  1. 1.
    Naegel, B., Passat, N.: Component-trees and multi-value images: a comparative study. In: ISMM, Proceedings. Lecture Notes in Computer Science, vol. 5720, pp. 261–271. Springer, Berlin (2009) Google Scholar
  2. 2.
    Passat, N., Naegel, B.: An extension of component-trees to partial orders. In: ICIP, Proceedings, pp. 3981–3984 (2009) Google Scholar
  3. 3.
    Wishart, D.: Mode analysis: a generalization of the nearest neighbor. In: Numerical Taxonomy, pp. 282–319. Academic Press, San Diego (1969) Google Scholar
  4. 4.
    Hartigan, J.A.: Statistical theory in clustering. J. Classif. 2(1), 63–76 (1985) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hanusse, P., Guillataud, P.: Sémantique des images par analyse dendronique. In: RFIA, Proceedings, pp. 577–588 (1991) Google Scholar
  6. 6.
    Chen, L., Berry, M.W., Hargrove, W.W.: Using dendronal signatures for feature extraction and retrieval. Int. J. Imaging Syst. Technol. 11(4), 243–253 (2000) CrossRefGoogle Scholar
  7. 7.
    Mattes, J., Demongeot, J.: Efficient algorithms to implement the confinement tree. In: DGCI, Proceedings. Lecture Notes in Computer Science, vol. 1953, pp. 392–405. Springer, Berlin (2000) Google Scholar
  8. 8.
    Salembier, P., Oliveras, A., Garrido, L.: Anti-extensive connected operators for image and sequence processing. IEEE Trans. Image Process. 7(4), 555–570 (1998) CrossRefGoogle Scholar
  9. 9.
    Najman, L., Talbot, H. (eds.): Mathematical Morphology: from Theory to Applications. ISTE/J. Wiley, New York (2010) Google Scholar
  10. 10.
    Salembier, P., Wilkinson, M.H.F.: Connected operators: a review of region-based morphological image processing techniques. IEEE Signal Process. Mag. 26(6), 136–157 (2009) CrossRefGoogle Scholar
  11. 11.
    Breen, E.J., Jones, R.: Attribute openings, thinnings, and granulometries. Comput. Vis. Image Underst. 64(3), 377–389 (1996) CrossRefGoogle Scholar
  12. 12.
    Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE Trans. Image Process. 15(11), 3531–3539 (2006) CrossRefGoogle Scholar
  13. 13.
    Berger, C., Géraud, T., Levillain, R., Widynski, N., Baillard, A., Bertin, E.: Effective component tree computation with application to pattern recognition in astronomical imaging. In: ICIP, Proceedings, pp. 41–44 (2007) Google Scholar
  14. 14.
    Wilkinson, M.H.F., Gao, H., Hesselink, W.H., Jonker, J.E., Meijster, A.: Concurrent computation of attribute filters on shared memory parallel machines. IEEE Trans. Pattern Anal. Mach. Intell. 30(10), 1800–1813 (2008) CrossRefGoogle Scholar
  15. 15.
    Jones, R.: Connected filtering and segmentation using component trees. Comput. Vis. Image Underst. 75(3), 215–228 (1999) CrossRefGoogle Scholar
  16. 16.
    Dokládal, P., Bloch, I., Couprie, M., Ruijters, D., Urtasun, R., Garnero, L.: Topologically controlled segmentation of 3D magnetic resonance images of the head by using morphological operators. Pattern Recognit. 36(10), 2463–2478 (2003) CrossRefGoogle Scholar
  17. 17.
    Ouzounis, G.K., Wilkinson, M.H.F.: Mask-based second-generation connectivity and attribute filters. IEEE Trans. Pattern Anal. Mach. Intell. 29(6), 990–1004 (2007) CrossRefGoogle Scholar
  18. 18.
    Passat, N., Naegel, B., Rousseau, F., Koob, M., Dietemann, J.L.: Interactive segmentation based on component-trees. Pattern Recognit. 44(10–11), 2539–2554 (2011) CrossRefMATHGoogle Scholar
  19. 19.
    Mattes, J., Richard, M., Demongeot, J.: Tree representation for image matching and object recognition. In: DGCI, Proceedings. Lecture Notes in Computer Science, vol. 1568, pp. 392–405. Springer, Berlin (1999) Google Scholar
  20. 20.
    Mosorov, V.: A main stem concept for image matching. Pattern Recognit. Lett. 26(8), 1105–1117 (2005) CrossRefGoogle Scholar
  21. 21.
    Alajlan, N., Kamel, M.S., Freeman, G.H.: Geometry-based image retrieval in binary image databases. IEEE Trans. Pattern Anal. Mach. Intell. 30(6), 1003–1013 (2008) CrossRefGoogle Scholar
  22. 22.
    Urbach, E.R., Roerdink, J.B.T.M., Wilkinson, M.H.F.: Connected shape-size pattern spectra for rotation and scale-invariant classification of gray-scale images. IEEE Trans. Pattern Anal. Mach. Intell. 29(2), 272–285 (2007) CrossRefGoogle Scholar
  23. 23.
    Westenberg, M.A., Roerdink, J.B.T.M., Wilkinson, M.H.F.: Volumetric attribute filtering and interactive visualization using the max-tree representation. IEEE Trans. Image Process. 16(12), 2943–2952 (2007) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Menotti, D., Najman, L., de Albuquerque Araújo, A.: 1D component tree in linear time and space and its application to gray-level image multithresholding. In: ISMM, Proceedings, INPE, vol. 1, pp. 437–448 (2007) Google Scholar
  25. 25.
    Naegel, B., Wendling, L.: A document binarization method based on connected operators. Pattern Recognit. Lett. 31(11), 1251–1259 (2010) CrossRefGoogle Scholar
  26. 26.
    Urbach, E.R., Boersma, N.J., Wilkinson, M.H.F.: Vector attribute filters. In: ISMM, Proceedings. Computational Imaging and Vision, vol. 30, pp. 95–104. Springer, Berlin (2005) Google Scholar
  27. 27.
    Aptoula, E., Lefèvre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognit. 40(11), 2914–2929 (2007) CrossRefMATHGoogle Scholar
  28. 28.
    Ronse, C., Agnus, V.: Morphology on label images: flat-type operators and connections. J. Math. Imaging Vis. 22(2), 283–307 (2005) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Barnett, V.: The ordering of multivariate data. J. R. Stat. Soc., Ser. A, Stat. Soc. 139(3), 318–354 (1976) CrossRefGoogle Scholar
  30. 30.
    Goutsias, J., Heijmans, H.J.A.M., Sivakumar, K.: Morphological operators for image sequences. Comput. Vis. Image Underst. 62(3), 326–346 (1995) CrossRefGoogle Scholar
  31. 31.
    Talbot, H., Evans, C., Jones, R.: Complete ordering and multivariate mathematical morphology. In: ISMM, Proceedings, pp. 27–34. Kluwer Academic, Norwell (1998) Google Scholar
  32. 32.
    Aptoula, E., Lefèvre, S.: On lexicographical ordering in multivariate mathematical morphology. Pattern Recognit. Lett. 29(2), 109–118 (2008) CrossRefGoogle Scholar
  33. 33.
    Angulo, J.: Geometric algebra colour image representations and derived total orderings for morphological operators—Part I: Colour quaternions. J. Vis. Commun. Image Represent. 21(1), 33–48 (2010) CrossRefGoogle Scholar
  34. 34.
    Gimenez, D., Evans, A.N.: An evaluation of area morphology scale-spaces for colour images. Comput. Vis. Image Underst. 110(1), 32–42 (2008) CrossRefGoogle Scholar
  35. 35.
    Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008) CrossRefGoogle Scholar
  36. 36.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Paths, homotopy and reduction in digital images. Acta Appl. Math. 113(2), 167–193 (2011) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Maunder, C.R.F.: Algebraic Topology. Dover, New York (1996) Google Scholar
  38. 38.
    Stong, R.E.: Finite topological spaces. Trans. Am. Math. Soc. 123(25), 325–340 (1966) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17(1), 146–160 (1970) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989) CrossRefGoogle Scholar
  41. 41.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989) CrossRefGoogle Scholar
  42. 42.
    Ronse, C.: Set-theoretical algebraic approaches to connectivity in continuous or digital spaces. J. Math. Imaging Vis. 8(1), 41–58 (1998) CrossRefMathSciNetGoogle Scholar
  43. 43.
    Serra, J.: Connectivity on complete lattices. J. Math. Imaging Vis. 9(3), 231–251 (1998) CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Braga-Neto, U., Goutsias, J.: Connectivity on complete lattices: new results. Comput. Vis. Image Underst. 85(1), 22–53 (2002) CrossRefMATHGoogle Scholar
  45. 45.
    Ronse, C.: Idempotent block splitting on partial partitions, I: Isotone operators. Order 28(2), 273–306 (2011) CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Caselles, V., Monasse, P.: Geometric Description of Images as Topographic Maps. Lecture Notes in Mathematics, vol. 29. Springer, Berlin (2010) CrossRefMATHGoogle Scholar
  47. 47.
    Naegel, B., Passat, N.: Toward connected filtering based on component-graphs. In: ISMM, Proceedings. Lecture Notes in Computer Science, vol. 7883, pp. 350–361. Springer, Berlin (2013) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CReSTICUniversité de Reims Champagne-ArdenneReimsFrance
  2. 2.ICube, CNRSUniversité de StrasbourgStrasbourgFrance

Personalised recommendations