Journal of Mathematical Imaging and Vision

, Volume 10, Issue 3, pp 253–267 | Cite as

Markovian Analysis of Adaptive Reconstructive Multiparameter τ-Openings

  • Yidong Chen
  • Edward R. Dougherty


A classical single-parameter τ-opening is a union of openings in which each structuring element is scaled by the same parameter. Multiparameter binary τ-openings generalize the model in two ways: first, parameters for each opening are individually defined; second, a structuring element can be parameterized relative to its overall shape, not merely sized. The reconstructive filter corresponding to an opening is defined by fully passing any grain (connected component) that is not fully eliminated by the opening and deleting all other grains. Adaptive design results from treating the parameter vector of a reconstructive multiparameter τ-opening as the state space of a Markov chain. Signal and noise are modeled as unions of randomly parameterized and randomly translated primary grains, and the parameter vector is transitioned depending on whether an observed grain is correctly or incorrectly passed. Various adaptive models are considered, transition probabilities are discussed, the state-probability increment equations are deduced from the appropriate Chapman-Kolmogorov equations, and convergence of the adaptation is characterized by the steady-state distribution relating to the Markov chain.

adaptive filter granulometry Markov chain mathematical morphology opening 


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  1. 1.
    G.J.F. Banon and J. Barrera, “Minimal representation of translation invariant set mappings by mathematical morphology,” SIAM Journal on Applied Mathematics, Vol. 51, No. 6, 1991.Google Scholar
  2. 2.
    E.R. Dougherty and R.P. Loce, “Optimal mean-absolute-error hit-or-miss filters: Morphological representation and estimation of the binary conditional expectation,” Optical Engineering., Vol. 32, No. 4, pp. 815–823, 1993.Google Scholar
  3. 3.
    J. Barrera, E.R. Dougherty, and N. Tomita, “Automatic programming of binary morphological machines by design of statistically optimal operators in the context of computational learning theory,” Electronic Imaging, Vol. 6, No. 1, 1997.Google Scholar
  4. 4.
    E.R. Dougherty, “Optimal mean-square N-observation digital morphological filters-part I: Optimal binary filters,” CVGIP: Image Understanding, Vol. 55, No. 1, pp. 36–54, 1992.Google Scholar
  5. 5.
    R.P. Loce and E.R. Dougherty, “Optimal morphological restoration: The morphological filter mean-absolute-error theorem,” Visual Communication and Image Representation, Vol. 3, No. 4, 1992.Google Scholar
  6. 6.
    E.R. Dougherty, R.M. Haralick, Y. Chen, C. Agerskov, U. Jacobi, and P.H. Sloth, “Estimation of optimal τ-opening parameters based on independent observation of signal and noise pattern spectra,” Signal Processing, Vol. 29, 1992.Google Scholar
  7. 7.
    E.R. Dougherty and C. Cuciurean-Zapan, “Optimal reconstructive τ-openings for disjoint and statistically modeled nondisjoint grains,” Signal Processing, Vol. 56, pp. 45–58, 1997.Google Scholar
  8. 8.
    R.M. Haralick, P.L. Katz, and E.R. Dougherty, “Model-based morphology: The opening spectrum,” CVGIP: Graphical Models and Image Processing, Vol. 57, No. 1, pp. 1–12, 1995.Google Scholar
  9. 9.
    E.R. Dougherty, “Optimal binary morphological bandpass filters induced by granulometric spectral representation,” Mathematical Imaging and Vision, Vol. 7, No. 2, pp. 175–192, 1997.Google Scholar
  10. 10.
    P. Salembier, “Structuring element adaptation for morphological filters,” Visual Communication and Image Representation, Vol. 3, No. 2, pp. 115–136, 1992.Google Scholar
  11. 11.
    Y. Chen and E.R. Dougherty, “Adaptive reconstructive τ-openings: Convergence and the steady-state distribution,” Electronic Imaging, Vol. 5, No. 3, 1996.Google Scholar
  12. 12.
    G. Matheron, Random Sets and Integral Geometry, JohnWiley: New York, 1975.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Yidong Chen
    • 1
  • Edward R. Dougherty
    • 2
  1. 1.National Human Genome Research Institute, National Institutes of HealthUSA
  2. 2.Texas Center for Applied TechnologyDepartment of Electrical Engineering, Texas A&M UniversityUSA

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