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Analytical Solutions for the Minkowski Addition Equation

  • Joel Edu Sánchez Castro
  • Ronaldo Fumio Hashimoto
  • Junior Barrera
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7883)

Abstract

This paper presents the formulation of a discrete equation whose solutions have a strong combinatory nature. More formally, given two subsets Y and C, we are interested in finding all subsets X that satisfy the equation (called Minkowski Addition Equation) X ⊕ C = Y. One direct application of the solutions of this equation is that they can be used to find best representations for fast computation of erosions and dilations. The main (and original) result presented in this paper (which is a theoretical result) is an analytical solution formula for this equation. One important characteristic of this analytical formula is that all solutions (which can be in worst case exponential) are expressed in a compact representation.

Keywords

Minkowski Addition Minkowski Subtraction Estructuring Element Decomposition Dilation Erosion 

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References

  1. 1.
    Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hörmander, L.: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Arkiv för Matematik 3(2), 181–186 (1955)CrossRefGoogle Scholar
  3. 3.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New York (1982)zbMATHGoogle Scholar
  4. 4.
    Maragos, P.A.: A Unified Theory of Translation-invariant Systems with Applications to Morphological Analysis and Coding of Images. PhD thesis, School of Elect. Eng. - Georgia Inst. Tech. (1985)Google Scholar
  5. 5.
    Hashimoto, R.F., Barrera, J., Ferreira, C.E.: A Combinatorial Optimization Technique for the Sequential Decomposition of Erosions and Dilations. Journal of Mathematical Imaging and Vision 13(1), 17–33 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hashimoto, R.F., Barrera, J.: A Note on Park and Chin’s Algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence 24(1), 139–144 (2002)CrossRefGoogle Scholar
  7. 7.
    Hashimoto, R.F., Barrera, J.: A Greedy Algorithm for Decomposing Convex Structuring Elements. Journal of Mathematical Imaging and Vision 18(3), 269–289 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Park, H., Chin, R.T.: Decomposition of Arbitrarily Shaped Morphological Structuring Elements. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(1), 2–15 (1995)CrossRefGoogle Scholar
  9. 9.
    Shih, F.Y., Wu, Y.T.: Decomposition of binary morphological structuring elements based on genetic algorithms. Computer Vision and Image Understanding 99(2), 291–302 (2005)CrossRefGoogle Scholar
  10. 10.
    Zhang, C.B., Wang, K.F., Shen, X.L.: Algorithm for decomposition of binary morphological convex structuring elements based on periodic lines. Imaging Science Journal (May 2012)Google Scholar
  11. 11.
    Zhuang, X., Haralick, R.: Morphological Structuring Element Decomposition. Computer Vision, Graphics, and Image Processing 35, 370–382 (1986)zbMATHCrossRefGoogle Scholar
  12. 12.
    Babu, G., Srikrishna, A., Challa, K., Babu, B.: An error free compression algorithm using morphological decomposition. In: 2012 International Conference on Recent Advances in Computing and Software Systems (RACSS), pp. 33–36 (April 2012)Google Scholar
  13. 13.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Joel Edu Sánchez Castro
    • 1
  • Ronaldo Fumio Hashimoto
    • 1
  • Junior Barrera
    • 1
  1. 1.Institute of Mathematics and StatisticsUniversity of Sao PauloBrazil

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