Neuro-Fuzzy Computing: Structure, Performance Measure and Applications

  • P. A. Stadterl
  • N. K. Bose
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 42)


Nonrecurrent and recurrent neural network structures are used, respectively, for pattern classification and image processing. An approach for automated pattern classification is developed based on a Fuzzy Voronoi Neural Network (FVNet) architecture. The FVNet training, based on spatial tessellation, results in a flexible, generic, modular structure that is subsequently refined by locally tuning the initial class decision surfaces, which are generated from the Voronoi diagram. The FVNet has subnetworks that are capable of learning fuzzy logic constructs from empirical data. A new performance measure, called the modified fuzzy divergence, which is particularly suited to evaluating neuro-fuzzy classifi-ers, has been developed. Processing of images by a Hopfield network and a Boltzmann machine is discussed with particular attention to the consequences of applying deterministic and stochastic learning rules and the different needs for neurocomputing in contrast to classical computing.


Hide Layer Feature Space Voronoi Diagram Image Restoration Fuzzy Divergence 


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  1. 1.
    Bhandari D., Pal, N. R. (1993): Some new information measures for fuzzy sets. Information Sciences 67: 209–228.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bishop, C. M. (1995): Neural Networks for Pattern Recognition. Oxford University Press, Oxford.Google Scholar
  3. 3.
    Blekas, K., Likas, A., Stafylopatis, A. (1995): A fuzzy neural network approach based on Dirichlettesselations for nearest neighbor classification of patterns. Neural Net. for Sig. Processing - Proc. IEEE Workshop, IEEE Press, 153–161.Google Scholar
  4. 4.
    Bose, N. K., Garga, A. K. (1993): Neural network design using Voronoi diagrams. IEEE Trans. Neural Net. 5(4): 778–787.CrossRefGoogle Scholar
  5. 5.
    Bose, N. K., Liang, P. (1996): Neural Network Fundamentals with Graphs, Algorithms, and Applications. McGraw-Hill, Inc., New York.Google Scholar
  6. 6.
    Bose, N. K., Rumancik, W. P. (1996): Different roles of Toeplitz and circulant structures in classical and neurobased image processing. Proc. Conf. on Info. Sci. and Syst., Princeton, NJ, 1202–1205.Google Scholar
  7. 7.
    Bose, N. K., Boo, K. J. (1998): Asymptotic eigenvalue distribution of block-Toeplitz matricies. IEEE Trans. on Information Theory 44(2): 858–861.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bounds, D. G. (1987): New optimization methods from physics and biology. Nature 329: 215–218.CrossRefGoogle Scholar
  9. 9.
    Chen Y. Q., Damper, R. I. (1997): On neural-network implementations of k-nearest neighbor pattern classifiers. IEEE Trans. on Circuits and Syst. 44(7): 622–629.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chien,Y.(1978): Interactive Pattern Recognition. Marcel Dekker, New York.Google Scholar
  11. 11.
    Cooley, W. W., Lohhnes, P. R. (1971): Multivariate Data Analysis. John Wiley, New York.MATHGoogle Scholar
  12. 12.
    Cover, T. M., Thomas, J. A. (1991): Elements of Information Theory. John Wiley, New York.CrossRefMATHGoogle Scholar
  13. 13.
    De Luca, A., Termini, S. (1972): A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Information and Control 20: 301–312.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Deco, G., Obradovic, D. (1996): An Information-Theoretic Approach to Neural Computing. Springer, New York.CrossRefMATHGoogle Scholar
  15. 15.
    Dony, R. D., Haykin, S. (1995): Neural network approaches to image compression. Proc. of the IEEE 8(2): 288–303.CrossRefGoogle Scholar
  16. 16.
    Draper, N. R., Smith, H. (1981): Applied Regression Analysis, 2nd Edition. John Wiley, New York.MATHGoogle Scholar
  17. 17.
    Garga, A. K. (1994): Design and training of neural networks using computational geometry. PhD Thesis, Department of Electrical Engineering, The Pennsylvania State University, University Park, PA.Google Scholar
  18. 18.
    Garga, A. K., Bose, N. K. (1994): A neural network approach to the construction of Delaunay tessellation of points in Rd. IEEE Trans. Cir. and Syst. 41(9): 611–613.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Geman, S., Geman, D. (1984): Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. and Machine Intell. 6: 721–741.CrossRefMATHGoogle Scholar
  20. 20.
    Hagan, M. T., Menhaj, M. B. (1994): Training feedforward networks with the Marquardt algorithm. IEEE Trans. Neural Net. 5(6): 989–993.CrossRefGoogle Scholar
  21. 21.
    Hinton, G. E.,(1989): Connectionist learning procedures. Artificial Intelligence 40: 1185–234.CrossRefGoogle Scholar
  22. 22.
    Holte, R. C., (1993): Very simple classification rules perform well on most commonly used datasets. Machine Learning 11: 63–90.CrossRefMATHGoogle Scholar
  23. 23.
    Hopfield, J. (1982): Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79: 2554–2558.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hunt, B. R., Kubler, O. (1984): Karhunen-Loeve multispectral image restoration, part I: Theory. IEEE Trans. Acoust., Speech, Signal Processing ASSP-32(3): 592–600.CrossRefGoogle Scholar
  25. 25.
    Jefferys H.(1946): An invariant form of the prior probability in estimation problems. Proc. Roy. Soc., Series A 186: 453–461.CrossRefGoogle Scholar
  26. 26.
    Joshi, A., Ramakrishman, N., Houstis, E. N., Rice, J. R. (1997): On neurobiological, neuro-fuzzy, machine learning, and statistical pattern recognition techniques. IEEE Trans. on Neural Net. 8(1): 18–31.CrossRefGoogle Scholar
  27. 27.
    Kirkpatrick, S., Gelatt, C. D., Vecchi, M. P. (1983): Optimization by simulated annealing. Science 220: 671–680.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Klir, G. J., Folger, T. A. (1988): Fuzzy Sets, Uncertainty, and Information. Prentice Hall.MATHGoogle Scholar
  29. 29.
    Kohavi, R. (1994): Bottom-up induction of oblivious read-once decision graphs: Strengths and limitations. Proc. 12th Nat. Conf. Artificial Intel. 613–618.Google Scholar
  30. 30.
    Kohonen, T. (1988): Self-Organization and Associative Memories (2nd edition). Springer-Verlag, New York.CrossRefGoogle Scholar
  31. 31.
    Kosko, B. (1993): Addition as fuzzy mutual entropy. Information Sciences 73: 273–284.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kulkarni, A. D. (1994): Artificial Neural Networks for Image Understanding, Van Nostrand Reinhold, New York, NY.Google Scholar
  33. 33.
    Kulkarni, A. D. (1994): Neural network for image restoration. Proc. of ACM 18th Annual Comp. Sci. Conf., Washington, DC, 373–378.Google Scholar
  34. 34.
    Kullback, S. (1959): Information Theory and Statistics. John Wiley, New York.MATHGoogle Scholar
  35. 35.
    Okabe, A., Boots, B., Sugihara, K. (1992): Spatial Tessellations. John Wiley, New York.MATHGoogle Scholar
  36. 36.
    Paik, J. K., Katsaggelos, A. K. (1992): Image restoration using a modified Hopfield network. IEEE Trans. Image Proc. 1(1): 49–63.CrossRefGoogle Scholar
  37. 37.
    Pal, S. K., Mitra, S. (1992): Multilayer perceptron, fuzzy sets, and classification. IEEE Trans. Neural Net. 3(5): 683–697.CrossRefGoogle Scholar
  38. 38.
    Preparata, F., Shamos, M. (1985): Computational Geometry. Springer-Verlag, New York.CrossRefGoogle Scholar
  39. 39.
    Quinlan, J. R. (1986): Induction of decision trees. Machine Learning 3: 81–106.Google Scholar
  40. 40.
    Quinlan, J. R., (1993): C4.5: Programs for Machine Learning. Morgan Kaufmann, San Mateo, CA.Google Scholar
  41. 41.
    Randelman, R. E., Grest, G. S. (1986): N-city traveling salesman problem - optimization by simulated annealing. J. Stat. Phys. 45: 885–890.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Renyi, A. (1961): On measures of entropy and information. Proc. of the Fourth Berkeley Sym. on Math., Stat., and Prob. Los Angeles, CA, 1: 547–561.MathSciNetGoogle Scholar
  43. 43.
    Rumancik, W. P. (1996): Study of neural networks for image restoration. MS Thesis, Department of Electrical Engineering, The Pennsylvania State University, University Park, PA.Google Scholar
  44. 44.
    Rumancik, W. P., Bose, N. K. (1998): Improved image restoration by neurocomputing principles. Proc. IASTED Conf. on Intelligent Syst. and Control. Acta Press, Nova Scojca 152–156.Google Scholar
  45. 45.
    Shannon, C. E. (1948): A mathematical theory of communication. Bell Syst. Tech. J. 27: 379–423, 623–656.MathSciNetMATHGoogle Scholar
  46. 46.
    Simpson, P. K. (1992): Fuzzy min-max neural networks - part 1: Classification. IEEE Trans. Neural Net. 3(5): 776–786.CrossRefGoogle Scholar
  47. 47.
    Stadter, P. A., Phoha, S., Bose, N. K. (1996): A neuro-fuzzy interface for tracking warfare dynamics via interacting automata. 1996 Command and Control Research and Tech. Sym., Monterey, CA, 457–468.Google Scholar
  48. 48.
    Stadter, P. A., Garga, A. K. (1997): A neural architecture for fuzzy classification with application to complex system tracking. 1997 IEEE Int’l. Conf. Neural Net., Houston, TX, IV-2439–2444.Google Scholar
  49. 49.
    Stadter, P. A. (1997): A neural architecture for fuzzy classification with applications. PhD Thesis, Department of Electrical Engineering, The Pennsylvania State University, University Park, PA.Google Scholar
  50. 50.
    Sun, Y., Li, J., Yu, S. (1995): Improvement on performance of modified Hopfield neural network for image restoration. IEEE Trans. Image Proc. 5(5):683–692.MathSciNetGoogle Scholar
  51. 51.
    Tatsouka, M. M. (1988): Multivariate Analysis, Techniques for Educational and Psychological Research, 2nd Edition. MacMillan, New York.Google Scholar
  52. 52.
    Wang, Z., Klir, G. J. (1992): Fuzzy Measure Theory. Plenum Press, New York.CrossRefMATHGoogle Scholar
  53. 53.
    Weisberg, S. (1985): Applied Linear Regression. Wiley, New York.MATHGoogle Scholar
  54. 54.
    Yeh, S. H., Sezan, M. (1991): Hopfield-type neural networks. In: Katsaggelos, A. (Ed.): Digitial Image Restoration. Springer-Verlag, New York 57–88.Google Scholar
  55. 55.
    Zadeh, L. A. (1965): Fuzzy sets. Information and Control 8: 338–353.MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Zadeh, L. A. (1965): Fuzzy sets and systems. Proc. of the Sym. on Syst. Theory, Brooklyn, NY, 29–37.Google Scholar
  57. 57.
    Zhou, Y., Chellappa, R., Vaid, A., Jenkins, B. K. (1988): Image restoration using a neural network. IEEE Trans. Acoust., Speech, Signal Processing ASSP-36: 1141–1151.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • P. A. Stadterl
    • 1
  • N. K. Bose
    • 2
  1. 1.Applied Physics LaboratoryThe Johns Hopkins UniversityLaurelUSA
  2. 2.The Pennsylvania State University, Dept. of Electrical EngineeringUniversity ParkUSA

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