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Algebraic and PDE Approaches for Multiscale Image Operators with Global Constraints: Reference Semilattice Erosions and Levelings

  • Petros Maragos
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)

Abstract

This paper begins with analyzing the theoretical connections between levelings on lattices and scale-space erosions on reference semilattices. They both represent large classes of self-dual morphological operators that exhibit both local computation and global constraints. Such operators are useful in numerous image analysis and vision tasks ranging from simplification, to geometric feature detection, to segmentation. Previous definitions and constructions of levelings were either discrete or continuous using a PDE. We bridge this gap by introducing generalized levelings based on triphase operators that switch among three phases, one of which is a global constraint. The triphase operators include as special cases reference semilattice erosions. Algebraically, levelings are created as limits of iterated or multiscale triphase operators. The sub-class of multiscale geodesic triphase operators obeys a semigroup, which we exploit to find a PDE that generates geodesic levelings. Further, we develop PDEs that can model and generate continuous-scale semilattice erosions, as a special case of the leveling PDE. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution which are proved to converge as iterations of triphase operators, and provide insights via image experiments.

Keywords

Complete Lattice Global Constraint Mathematical Morphology Discrete Algorithm Signal Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    R. Brockett and P. Maragos, “Evolution Equations for Continuous-Scale Morphological Filtering”, IEEE Trans. Signal Process., vol. 42, pp. 3377–3386, Dec. 1994.CrossRefGoogle Scholar
  2. 2.
    H.J.A.M. Heijmans, Morphological Image Operators, Acad. Press, Boston, 1994.zbMATHGoogle Scholar
  3. 3.
    H.J.A.M. Heijmans and R. Keshet, “First Steps Towards A Self-DualMorphology”, in Proc. Int’l Conf. Image Processing, Vancouver, Canada, Sep. 2000.Google Scholar
  4. 4.
    H.J.A.M. Heijmans and R. Keshet (Kresch), “Inf-Semilattice Approach to Self-Dual Morphology”, Report PNA-R0101, CWI, Amsterdam, Jan. 2001.Google Scholar
  5. 5.
    R. Keshet (Kresch), “Mathematical Morphology On Complete Semilattices and Its Applications to Image Processing”, Fundamentae Informatica 41, pp. 33–56, 2000.Google Scholar
  6. 6.
    P. Maragos and F. Meyer, “Nonlinear PDEs and Numerical Algorithms for Modeling Levelings and Reconstruction Filters”, in Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science 1682, pp. 363–374, Springer, 1999.CrossRefGoogle Scholar
  7. 7.
    G. Matheron, “Les Nivellements”, Technical Report, Centre de Morphologie Mathematique, 1997.Google Scholar
  8. 8.
    F. Meyer, “The Levelings”, in Mathematical Morphology and Its Applications to Image and Signal Processing, H. Heijmans & J. Roerdink, Eds, Kluwer Acad., 1998.Google Scholar
  9. 9.
    F. Meyer and P. Maragos, “Nonlinear Scale-Space Representation with Morphological Levelings”, J. Visual Commun. & Image Representation, 11, p.245–265, 2000.CrossRefGoogle Scholar
  10. 10.
    F. Meyer and J. Serra, “Contrasts and Activity Lattice”, Signal Processing, vol. 16,no. 4, pp. 303–317, Apr. 1989.MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Osher and L.I. Rudin, “Feature-Oriented Image Enhancement Using Schock Filters”, SIAM J. Numer. Anal., vol. 27,no. 4, pp. 919–940, Aug. 1990.zbMATHCrossRefGoogle Scholar
  12. 12.
    S. Osher and J. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations”, J. Comp.. Phys. 79, 1988.Google Scholar
  13. 13.
    P. Salembier and J. Serra, “Flat Zones Filtering, Conected Operators, and Filters by Reconstruction”, IEEE Trans. Image Process., vol. 4, pp.1153–1160, Aug. 1995.CrossRefGoogle Scholar
  14. 14.
    J. Serra, “Connections for Sets and Functions”, Fundamentae Informatica 41, pp. 147–186, 2000.MathSciNetzbMATHGoogle Scholar
  15. 15.
    L. Vincent, “Morphological grayscale reconstruction in image analysis: Applications and efficient algorithms”, IEEE Trans. Image Proces., 2(2), p.176–201, 1993.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Petros Maragos
    • 1
  1. 1.Dept. of Electrical & Computer EngineeringNational Technical University of AthensAthensGreece

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