Binary Decision Diagrams as a New Paradigm for Morphological Machines

  • Junior Barrera
  • Ronaldo Fumio Hashimoto
Part of the Computational Imaging and Vision book series (CIVI, volume 30)


Mathematical Morphology (MM) is a general framework for studying mappings between complete lattices. In particular, mappings between binary images that are translation invariant and locally defined within a window are of special interest in MM. They are called W-operators. A key aspect of MM is the representation of W-operators in terms of dilations, erosions, intersection, union, complementation and composition. When W-operators are expressed in this form, they are called morphological operators. An implementation of this decomposition structure is called morphological machine (MMach). A remarkable property of this decomposition structure is that it can be represented efficiently by graphs called Binary Decision Diagrams (BDDs). In this paper, we propose a new architecture for MMachs that is based on BDDs. We also show that reduced and ordered BDDs (ROBDDs) are non-ambiguous schemes for representing W- operators and we present a method to compute them. This procedure can be applied for the automatic proof of equivalence between morphological operators, since the W-operator they represent are equal if and only if they have the same ROBDD.


Binary Decision Diagram Morphological Machine Morphological Language Morphological Operator 


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  1. [1]
    S. B. Akers. Binary Decision Diagrams. IEEE Transactions on Computers, C-27(6):509–516, June 1978.Google Scholar
  2. [2]
    G. J. F. Banon and J. Barrera. Minimal Representations for Translation-Invariant Set Mappings by Mathematical Morphology. SIAM J. Appl. Math., 51(6):1782–1798, December 1991.CrossRefGoogle Scholar
  3. [3]
    J. Barrera and G. P. Salas. Set Operations on Closed Intervals and Their Applications to the Automatic Programming of Morphological Machines. Electronic Imaging, 5(3):335–352, July 1996.Google Scholar
  4. [4]
    G. Birkhoff. Lattice Theory. American Mathematical Society Colloquium Publications, Rhode Island, 1967.Google Scholar
  5. [5]
    K. S. Brace, R. L. Rudell, and R. E. Bryant. Efficient Implementation of a BDD Package. In Proceedings of the ACM/IEEE Design Automation Conference (DAC), pages 40–45. ACM/IEEE, 1990.Google Scholar
  6. [6]
    R. E. Bryant. Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transactions on Computers, C-35(8):677–691, August 1986.Google Scholar
  7. [7]
    G. de Micheli. Synthesis and Optimization of Digital Circuits. McGraw-Hill Higher Education, 1994.Google Scholar
  8. [8]
    H. J. A. M. Heijmans. Morphological Image Operators. Academic Press, Boston, 1994.Google Scholar
  9. [9]
    C.Y. Lee. Representation of Switching Circuits by Binary-Decision Programs. Bell Systems Technical Journal, 38:985–999, July 1959.Google Scholar
  10. [10]
    H. M. F. Madeira, J. Barrera, R. Hirata Jr., and N. S. T. Hirata. A New Paradigm for the Architecture of Morphological Machines: Binary Decision Diagrams. In SIBGRAPI’99-XII Brazilian Symposium of Computer Graphic and Image Processing, pages 283–292. IEEE Computer Society, November 1999.Google Scholar
  11. [11]
    H. M. F. Madeira and J. Barrera. Incremental Evaluation of BDD-Represented Set Operators. In SIBGRAPI 2000-XIII Brazilian Symposium of Computer Graphic and Image Processing, pages 308–315. IEEE Computer Society, 2000.Google Scholar
  12. [12]
    L. Robert and G. Malandain. Fast Binary Image Processing Using Binary Decision Diagrams. Computer Vision and Image Understanding, 72(1):1–9, October 1998.CrossRefGoogle Scholar
  13. [13]
    J. Serra. Image Analysis and Mathematical Morphology. Volume 2: Theoretical Advances. Academic Press, 1988.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Junior Barrera
    • 1
  • Ronaldo Fumio Hashimoto
    • 1
  1. 1.Departamento de Ciencia da ComputacaoInstitute de Matematica e Estatistica - USPCidade Universitaria - Sao Paulo - SPBrasil

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