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Generalizing Operations of Binary Autoassociative Morphological Memories Using Fuzzy Set Theory

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Abstract

Morphological neural networks (MNNs) are a class of artificial neural networks whose operations can be expressed in the mathematical theory of minimax algebra. In a morphological neural net, the usual sum of weighted inputs is replaced by a maximum or minimum of weighted inputs (in this context, the weighting is performed by summing the weight and the input). We speak of a max product, a min product respectively.

In recent years, a number of different MNN models and applications have emerged. The emphasis of this paper is on morphological associative memories (MAMs), in particular on binary autoassociative morphological memories (AMMs). We give a new set theoretic interpretation of recording and recall in binary AMMs and provide a generalization using fuzzy set theory.

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References

  1. J.A. Anderson, “A simple neural network generating interactive memory,” Mathematical Biosciences, Vol. 14, pp. 197–220, 1972.

    Google Scholar 

  2. W.W. Armstrong and M.M. Thomas, “Adaptive logic networks,” in Handbook of Neural Computation. E. Fiesler and R. Beale (Eds.), IOP Publishing and Oxford University Press, 1997, pp. C1.8:1–14.

  3. G. Birkhoff, Lattice Theory, 3rd edition, Providence, RI: American Mathematical Society, 1993.

    Google Scholar 

  4. R. Cuninghame-Green, “Minimax algebra,” Lecture Notes in Economics and Mathematical Systems, Springer-Verlag: Berlin, Heidelberg, New York, 1979.

    Google Scholar 

  5. R. Cuninghame-Green, “Minimax algebra and applications,” in Advances in Electronics and Electron Physics, P. Hawkes (Ed.), Academic Press: New York, NY, 1995, Vol. 90, pp. 1–121.

    Google Scholar 

  6. T.D. Chuieh and R.M. Goodman, “Recurrent correlation associative memories,” IEEE Transactions on Neural Networks, Vol. 2, pp. 275–284, Feb. 1991.

  7. J.L. Davidson, “Foundation and applications of lattice transforms in image processing,” in Advances in Electronics and Electron Physics, P. Hawkes (Ed.), Academic Press: New York, NY, 1992, Vol. 84, pp. 61–130.

    Google Scholar 

  8. J.L. Davidson and F. Hummer, “Morphology neural networks: An introduction with applications,” Circuits, Systems, and Signal Processing, Vol. 12, No. 2, pp. 177–210, 1993.

    Google Scholar 

  9. J.L. Davidson and G.X. Ritter, “A theory of morphological neural networks,” in Digital Optical Computing II, volume 1215 of Proceedings of SPIE, San Diego, CA, July 1990, pp. 378–388.

  10. P.D. Gader, M. Khabou, and A. Koldobsky, “Morphological regularization neural networks,” Pattern Recognition, Special Issue on Mathematical Morphology and Its Applications, Vol. 30, No. 6, pp. 935–945, 2000.

    Google Scholar 

  11. P.D. Gader, Y. Won, and M. Khabou, “Image algebra networks for pattern classification,” in Image Algebra and Morphological Image Processing V, volume 2300 of Proceedings of SPIE, San Diego, CA, 1994, pp. 157–168.

    Google Scholar 

  12. M. Graña and B. Raducanu, “On the application of morphological heteroassociative neural networks,” in Proc. Int. Conf. on Image Processing (ICIP), I. Pitas (Ed.), Thessaloniki, Greece, Oct. 2001, pp. 501–504.

  13. M.H. Hassoun, “Dynamic associative neural memories,” in Associative Neural Memories: Theory and Implemantation, M.H. Hassoun (Ed.), Oxford University Press: Oxford, UK, 1993.

    Google Scholar 

  14. H.J.A.M. Heijmans, Morphological Image Operators, Academic Press: New York, NY, 1994.

    Google Scholar 

  15. J.J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” in Proceedings of the National Academy of Sciences, Vol. 79, pp. 2554–2558, 1982.

    Google Scholar 

  16. T. Kohonen, “Correlation matrix memory,” IEEE Transactions on Computers, Vol. 21, pp. 353–359, 1972.

    Google Scholar 

  17. B. Kosko, “Optical bidirectional associative memories,” in Proceedings of SPIE, No. 758, Jan. 1987.

  18. B. Kosko, “Bidirectional associative memories,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 18, pp. 49–60, 1988.

    Google Scholar 

  19. B. Kosko, Neural Networks and Fuzzy Systems, Prentice Hall: Englewood Cliffs, NJ, 1992.

    Google Scholar 

  20. B. Kosko, Fuzzy Engineering, Prentice Hall: Upper Saddle River, NJ, 1997.

    Google Scholar 

  21. R.J. McEliece, E.C. Posner, E.R. Rodemich, and S.S. Venkatesh, “The capacity of the Hopfield associative memory,” IEEE Transactions on Information Theory, Vol. 1, pp. 33–45, 1987.

    Google Scholar 

  22. K. Nakano, “Associatron: A model of associative memory,” IEEE Transactions on Systems, Man, Cybernetics, Vol. SMC-2, pp. 380–388, 1972.

    Google Scholar 

  23. H. Oh and S.C. Kothari, “Adaption of the relaxation method for learning in bidirectional associative memory,” IEEE Transactions on Neural Networks, Vol. 5, No. 4, July 1994.

  24. G. Palm, “On associative memory,” Biological Cybernetics, Vol. 36, pp. 19–31, 1980.

    Google Scholar 

  25. W. Pedrycz, “Neural computations in relational systems,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 3, pp. 289–297, 1991.

    Google Scholar 

  26. L.F.C. Pessoa and P. Maragos, “Morphological regularization neural networks,” Pattern Recognition, Special Issue on Mathematical Morphology and Its Applications, Vol. 30, No. 6, pp. 945–960, 2000.

    Google Scholar 

  27. B. Raducanu and M. Graña, “Morphological neural networks for robust visual processing in mobile robotics,” in Proceedings of the International Joint Conference on Neural Networks, Como, Italy, July 2000.

  28. B. Raducanu, M. Graña, and P. Sussner, “Morphological neural networks for vision based self-localization,” in Proc. of ICRA2001, Int. Conf. on Robotics and Automation, Seoul, Korea, May 2001, pp. 2059–2064.

  29. B. Raducanu, M. Graña, and P. Sussner, “Advances in mobile robot self-localization using morphological neural networks,” in Proc. of IFAC2001, Cheju Island, South Korea, May 2001, pp. 146–151.

  30. G.X. Ritter, Computer Vision Algorithms in Image Algebra, CRC Press: Boca Raton, FL, 1996.

    Google Scholar 

  31. G.X. Ritter and J.L. Davidson, “Recusrion and feedback in image algebra,” in SPIE's 19th Workshop on Image Understanding in the 90's, Proceedings of SPIE: McLean, VA, Oct. 1990.

  32. G.X. Ritter, J.L. Diaz-de-Leon, and P. Sussner, “Morphological bidirectional associative memories,” Neural Networks, Vol. 12, pp. 851–867, 1999.

    Google Scholar 

  33. G.X. Ritter, P. Sussner, and J.L. Diaz-de-Leon, “Morphological associative memories,” IEEE Transactions on Neural Networks, Vol. 9, No. 2, pp. 281–292, 1998.

    Google Scholar 

  34. G.X. Ritter and P. Sussner, “An introduction to morphological neural networks,” in Proceedings of the 13th International Conference on Pattern Recognition, Viena, Austria, 1996, pp. 709–717.

  35. G.X. Ritter and J.N. Wilson, Handbook of Computer Vision Algorithms in Image Algebra, 2nd edition, CRC Press: Boca Raton, FL, 2001.

    Google Scholar 

  36. Ritter and Wilson, 1996.

  37. G.X. Ritter and G. Urcid, “Lattice algebra approach to single-neuron computation,” IEEE Transactions on Neural Networks, Vol. 14, No. 2, pp. 282–295, March 2003.

    Google Scholar 

  38. P. Sussner, “Associative morphological memories based on variations of the kernel and dual kernel methods,”Neural Networks, Special Issue of IJCNN'03, Vol. 16, Nos. 5–6, pp. 625–632, 2003.

    Google Scholar 

  39. P. Sussner, “Fixed points of autoassociative morphological memories,” in Proceedings of the International Joint Conference on Neural Networks, Como, Italy, July 2000.

  40. P. Sussner, “Observations on morphological associative memories and the kernel method,” Neurocomputing, Vol. 31, pp. 167–183, March 2000.

    Google Scholar 

  41. P. Sussner, “A relationship between binary autoassociative morphological memories and fuzzy set theory,” in Proceedings of the International Joint Conference on Neural Networks, Washington, DC, July 2001.

  42. Y. Won, P.D. Gader, and P.C. Coffield, “Morphological shared-weight neural network for pattern classification and automatic target detection,” in Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, Nov. 1995, Vol. 4, pp. 2134–2138.

    Google Scholar 

  43. Y. Won, P.D. Gader, and P.C. Coffield, “Morphological shared-weight neural network with applications to automatic target detection,” IEEE Transactions on Neural Networks, Vol. 8, No. 5, pp. 1195–1203, 1997.

    Google Scholar 

  44. Y. Wu and S.N. Batalama, “An efficient learning algorithm for associative memories,” IEEE Transactions on Neural Networks, Vol. 11, No. 5, pp. 1058–1066, 2000.

    Google Scholar 

  45. L.A. Zadeh, “Fuzzy sets,” Information and Control, Vol. 8, No. 3, pp. 338–353, 1965.

    Google Scholar 

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  1. Institute of Mathematics, Statistics, and Scientific Computing, State University of Campinas, Campinas, CEP13081-970, SP, Brazil

    Peter Sussner

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Sussner, P. Generalizing Operations of Binary Autoassociative Morphological Memories Using Fuzzy Set Theory. Journal of Mathematical Imaging and Vision 19, 81–93 (2003). https://doi.org/10.1023/A:1024721313295

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