Skip to main content
SpringerLink
Log in

Partial Partitions, Partial Connections and Connective Segmentation

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

Abstract

In connective segmentation (Serra in J. Math. Imaging Vis. 24(1):83–130, [2006]), each image determines subsets of the space on which it is “homogeneous”, in such a way that this family of subsets always constitutes a connection (connectivity class); then the segmentation of the image is the partition of space into its connected components according to that connection.

Several concrete examples of connective segmentations or of connections on sets, indicate that the space covering requirement of the partition should be relaxed. Furthermore, morphological operations on partitions require the consideration of wider framework.

We study thus partial partitions (families of mutually disjoint non-void subsets of the space) and partial connections (where connected components of a set are mutually disjoint but do not necessarily cover the set). We describe some methods for generating partial connections. We investigate the links between the two lattices of partial connections and of partial partitions. We generalize Serra’s characterization of connective segmentation and discuss its relevance. Finally we give some ideas on how the theory of partial connections could lead to improved segmentation algorithms.

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. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society Colloquium Publications, vol. 25. American Mathematical Society, Providence (1995). 8th printing

    Google Scholar 

  2. Bloch, I., Heijmans, H., Ronse, C.: Mathematical morphology. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 857–944. Springer, Berlin (2007). Chap. 14

    Chapter  Google Scholar 

  3. Braga-Neto, U., Goutsias, J.: Connectivity on complete lattices: new results. Comput. Vis. Image Underst. 85(1), 22–53 (2002)

    Article  MATH  Google Scholar 

  4. Braga-Neto, U., Goutsias, J.: A theoretical tour of connectivity in image processing and analysis. J. Math. Imaging Vis. 19(1), 5–31 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Crespo, J., Schafer, R., Serra, J., Gratin, C., Meyer, F.: The flat zone approach: A general low-level region merging segmentation method. Signal Process. 62(1), 37–60 (1997)

    Article  MATH  Google Scholar 

  6. Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M., Scott, D.: Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  7. Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003)

    Google Scholar 

  8. Heijmans, H.: Morphological Image Operators. Advances in Electronics and Electron Physics Series. Academic Press, Boston (1994)

    MATH  Google Scholar 

  9. Heijmans, H.: Connected morphological operators for binary images. Comput. Vis. Image Underst. 73(1), 99–120 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Matheron, G., Serra, J.: Strong filters and connectivity. In: Serra, J. (ed.) Image Analysis and Mathematical Morphology. II: Theoretical Advances, pp. 141–157. Academic Press, London (1988). Chap. 7

    Google Scholar 

  11. Ore, O.: Theory of equivalence relations. Duke Math. J. 9, 573–627 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ronse, C.: Set-theoretical algebraic approaches to connectivity in continuous or digital spaces. J. Math. Imaging Vis. 8(1), 41–58 (1998)

    Article  MathSciNet  Google Scholar 

  13. Ronse, C.: Reconstructing masks from markers in non-distributive lattices. Appl. Algebra Eng. Commun. Comput. 19(1), 51–85 (2008)

    Article  MathSciNet  Google Scholar 

  14. Ronse, C., Serra, J.: Geodesy and connectivity in lattices. Fundam. Inform. 46(4), 349–395 (2001)

    MATH  MathSciNet  Google Scholar 

  15. Ronse, C., Serra, J.: Fondements algébriques de la morphologie. In: Najman L., Talbot H. (eds.) Morphologie Mathématique: état de l’art. Hermès (2008, to appear). Chap. 3

  16. Serra, J. (ed.): Image Analysis and Mathematical Morphology, II: Theoretical Advances. Academic Press, London (1988)

    Google Scholar 

  17. Serra, J.: Mathematical morphology for Boolean lattices. In: Serra, J. (ed.) Image Analysis and Mathematical Morphology, II: Theoretical Advances, pp. 37–58. Academic Press, London (1988). Chap. 2

    Google Scholar 

  18. Serra, J.: Connectivity on complete lattices. J. Math. Imaging Vis. 9(3), 231–251 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Serra, J.: Connections for sets and functions. Fundam. Inform. 41(1/2), 147–186 (2000)

    MATH  MathSciNet  Google Scholar 

  20. Serra, J.: Morphological operators for the segmentation of colour images. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds.) Space, Structure and Randomness—Contributions in Honor of Georges Matheron in the Fields of Geostatistics, Random Sets, and Mathematical Morphology. Lecture Notes in Statistics, vol. 183, pp. 223–255. Springer, Berlin (2005)

    Google Scholar 

  21. Serra, J.: Morphological segmentations of colour images. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol. 30, pp. 151–176. Springer, Dordrecht (2005)

    Chapter  Google Scholar 

  22. Serra, J.: Viscous lattices. J. Math. Imaging Vis. 22(2/3), 269–282 (2005)

    Article  MathSciNet  Google Scholar 

  23. Serra, J.: A lattice approach to image segmentation. J. Math. Imaging Vis. 24(1), 83–130 (2006)

    Article  MathSciNet  Google Scholar 

  24. Soille, P.: Morphological Image Analysis: Principles and Applications, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  25. Wilkinson, M.: Attribute-space connectivity and connected filters. Image Vis. Comput. 25(4), 426–435 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LSIIT UMR 7005 CNRS-ULP, Parc d’Innovation, Boulevard Sébastien Brant, BP 10413, 67412, Illkirch Cedex, France

    Christian Ronse

Authors
  1. Christian Ronse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Ronse.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ronse, C. Partial Partitions, Partial Connections and Connective Segmentation. J Math Imaging Vis 32, 97–125 (2008). https://doi.org/10.1007/s10851-008-0090-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-008-0090-5

Keywords

Search

Navigation