Minimal Component-Hypertrees

  • Alexandre MorimitsuEmail author
  • Wonder Alexandre Luz Alves
  • Dennis Jose Silva
  • Charles Ferreira Gobber
  • Ronaldo Fumio HashimotoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)


Component trees are interesting structures of nested connected components, efficiently represented by max-trees, used to implement fast algorithms in Image Processing. In these structures, connected components are constructed using a single neighborhood. In recent years, an extension of component trees, called component-hypertrees, was introduced. It consists of a sequence of component trees, generated from a sequence of increasing neighborhoods, in which their connected components are also hierarchically organized. Although this structure could be useful in applications dealing with clusters of objects, not much attention has been given to component-hypertrees. A naive implementation can be costly both in terms of time and memory. So, in this paper, we present algorithms and data structures to efficiently compute and store these structures without redundancy obtaining a minimal representation of component-hypertrees. Experimental results using our efficient algorithm show that the number of nodes is reduced by approximately 70% in comparison to a naive implementation.


Mathematical Morphology Component-hypertree Component tree Connected component Connected operators 



This study was financed in part by the CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Finance Code 001); FAPESP - Fundação de Amparo a Pesquisa do Estado de São Paulo (Proc. 2018/15652-7); CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico (Proc. 428720/2018-8).


  1. 1.
    Carlinet, E., Géraud, T.: A comparative review of component tree computation algorithms. IEEE Trans. Image Process. 23(9), 3885–3895 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Karatzas, D., Mestre, S.R., Mas, J., Nourbakhsh, F., Roy, P.P.: ICDAR 2011 robust reading competition-challenge 1: reading text in born-digital images (web and email). In: 2011 International Conference on Document Analysis and Recognition (ICDAR), pp. 1485–1490. IEEE (2011)Google Scholar
  3. 3.
    Morimitsu, A., Alves, W.A.L., Hashimoto, R.F.: Incremental and efficient computation of families of component trees. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) ISMM 2015. LNCS, vol. 9082, pp. 681–692. Springer, Cham (2015). Scholar
  4. 4.
    Najman, L., Couprie, M.: Building the component tree in quasi-linear time. IEEE Trans. Image Process. 15(11), 3531–3539 (2006)CrossRefGoogle Scholar
  5. 5.
    Nayef, N., et al.: ICDAR 2017 robust reading challenge on multi-lingual scene text detection and script identification-RRC-MLT. In: 2017 14th IAPR International Conference on Document Analysis and Recognition (ICDAR), vol. 1, pp. 1454–1459. IEEE (2017)Google Scholar
  6. 6.
    Ouzounis, G.K., Wilkinson, M.H.: Mask-based second-generation connectivity and attribute filters. IEEE Trans. Pattern Anal. Mach. Intell. 29(6), 990–1004 (2007)CrossRefGoogle Scholar
  7. 7.
    Ouzounis, G.K., Wilkinson, M.H.: Hyperconnected attribute filters based on k-flat zones. IEEE Trans. Pattern Anal. Mach. Intell. 33(2), 224–239 (2011)CrossRefGoogle Scholar
  8. 8.
    Passat, N., Naegel, B.: Component-hypertrees for image segmentation. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 284–295. Springer, Heidelberg (2011). Scholar
  9. 9.
    Salembier, P., Oliveras, A., Garrido, L.: Antiextensive connected operators for image and sequence processing. IEEE Trans. Image Process. 7(4), 555–570 (1998)CrossRefGoogle Scholar
  10. 10.
    Serra, J.: Connectivity on complete lattices. J. Math. Imaging Vis. 9(3), 231–251 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Silva, D.J., Alves, W.A., Morimitsu, A., Hashimoto, R.F.: Efficient incremental computation of attributes based on locally countable patterns in component trees. In: 2016 IEEE International Conference on Image Processing (ICIP), pp. 3738–3742. IEEE (2016)Google Scholar
  12. 12.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM (JACM) 22(2), 215–225 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wilkinson, M.H., 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

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexandre Morimitsu
    • 1
    Email author
  • Wonder Alexandre Luz Alves
    • 2
  • Dennis Jose Silva
    • 1
  • Charles Ferreira Gobber
    • 2
  • Ronaldo Fumio Hashimoto
    • 1
    Email author
  1. 1.Department of Computer Science, Institute of Mathematics and StatisticsUniversidade de São PauloSão PauloBrazil
  2. 2.Informatics and Knowledge Management Graduate ProgramUniversidade Nove de JulhoSão PauloBrazil

Personalised recommendations