Skip to main content
SpringerLink
Log in

Shape Connectivity: Multiscale Analysis and Application to Generalized Granulometries

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

Abstract

This paper develops a multiscale connectivity theory for shapes based on the axiomatic definition of new generalized connectivity measures, which are obtained using morphology-based nonlinear scale-space operators. The concept of connectivity-tree for hierarchical image representation is introduced and used to define generalized connected morphological operators. This theoretical framework is then applied to establish a class of generalized granulometries, implemented at a particular problem concerning soilsection image analysis and evaluation of morphological properties such as size distributions. Comparative results demonstrate the power and versatility of the proposed methodology with respect to the application of typical connected operators (such as reconstruction openings). This multiscale connectivity analysis framework aims at a more reliable evaluation of shape/size information within complex images, with particular applications to generalized granulometries, connected operators, and segmentation.

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. U.M. Braga-Neto and J. Goutsias, “Multiresolution connectivity: An axiomatic approach,” in Mathematical Morphology and its Applications to Image and Signal Processing, J. Goutsias, L. Vincent, and D.S. Bloomberg (Eds.), Kluwer Academic Publishers: Dordrecht, 2000. (Proceedings of the ISMM'2000).

    Google Scholar 

  2. U.M. Braga-Neto and J. Goutsias, “A complete lattice approach to connectivity in image analysis,” Tech. Report JHU/ECE 00-05, Dept. ECE, Johns Hopkins University, Baltimore, MD, Nov. 2000.

    Google Scholar 

  3. H.J.A.M. Heijmans, Morphological Image Operators, Academic Press: San Diego, 1994.

    Google Scholar 

  4. H.J.A.M. Heijmans, “Connected morphological operators for binary images,” Computer Vision and Image Understanding, Vol. 73, No. 1, pp. 99–120, 1999.

    Google Scholar 

  5. B.B. Kimia, A.R. Tannenbaum, and S.W. Zucker, “Shapes, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space,” Int. J. Computer Vision, Vol. 15, pp. 189–224, 1995.

    Google Scholar 

  6. J.J. Koenderink and A.J. van Doorn, “Dynamic shape,” Biological Cybernetics, Vol. 53, pp. 383–396, 1986.

    Google Scholar 

  7. B.S. Manjuna and R. Chellappa, “Unsupervised texture segmentation using Markov random field models,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 13, No. 5, pp. 478–482, 1991.

    Google Scholar 

  8. P. Maragos, “Pattern spectrum and multiscale shape representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 11, No. 7, pp. 701–716, 1989.

    Google Scholar 

  9. P. Maragos, “Morphological signal and image processing,” in The Digital Signal Processing Handbook, V.K. Madisetti and D.B Williams (Eds.), CRC Press/IEEE Press: Bocaraton, New York, Ch. 74, 1999, pp. 1–30.

    Google Scholar 

  10. G. Matheron, Random Sets and Integral Geometry, Wiley: New York, 1975.

    Google Scholar 

  11. C. Ronse, “Set-theoretical algebraic approaches to connectivity in continuous or digital spaces,” Journal of Mathematical Imaging and Vision, Vol. 8, pp. 41–58, 1998.

    Google Scholar 

  12. P. Salembier, A. Oliveras, and L. Garrido, “Antiextensive connected operators for image and sequence processing,” IEEE Trans. Image Processing, Vol. 7, No. 4, pp. 555–570, 1998.

    Google Scholar 

  13. P. Salembier and J. Serra, “Flat zones filtering, connected operators, and filters by reconstruction,” IEEE Trans. Image Processing, Vol. 4, No. 8, pp. 1153–1160, 1995.

    Google Scholar 

  14. J. Serra, Image Analysis and Mathematical Morphology: Theoretical Advances, Academic Press: New York, 1988.

    Google Scholar 

  15. J. Serra, “Connections for sets and functions,” Fundamenta Informaticae, Vol. 41, Nos. 1/2, pp. 147–186, 2000.

    Google Scholar 

  16. K. Sivakumar and J. Goutsias, “Morphologically constrained GRFs: Applications to texture synthesis and analysis,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 21, No. 2, pp. 99–113, 1999.

    Google Scholar 

  17. A. Sofou, C. Tzafestas, and P. Maragos, “Segmentation of soilsection images using connected operators,” in Proc. IEEE Intern. Conf. on Image Processing (ICIP'2001), Thessaloniki, Greece, Sep. 2001.

  18. L. Vincent, “Morphological grayscale reconstruction in image analysis: Applications and efficient algorithms,” IEEE Trans. Image Processing, Vol. 2, No. 2, pp. 176–201, 1993.

    Google Scholar 

  19. L. Vincent, “Granulometries and opening trees,” Fundamenta Informaticae, Vol. 41, No. 1/2, pp. 57–90, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical and Computer Engineering, Division of Signals, Control and Robotics, National Technical University of Athens, Zografou Campus, Athens, 15773, Greece

    Costas S. Tzafestas & Petros Maragos

Authors
  1. Costas S. Tzafestas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petros Maragos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tzafestas, C.S., Maragos, P. Shape Connectivity: Multiscale Analysis and Application to Generalized Granulometries. Journal of Mathematical Imaging and Vision 17, 109–129 (2002). https://doi.org/10.1023/A:1020629402912

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020629402912

Search

Navigation