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Novel Neural Network Models for Computing Homothetic Invariances: An Image Algebra Notation

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Abstract

In this paper we propose a theoretical approach toinvariant perception. Invariant perception is an importantaspect in both natural and artificial perception systems, and itremains an important unsolved problem in heuristically basedpattern recognition. Our approach is based on a general theoryof neural networks and studies of invariant perception by thecortex. The neural structures that we propose uphold both thearchitecture and functionality of the cortex as currentlyunderstood.

The formulation of the proposed neural structuresis in the language of image algebra, a mathematical environmentfor expressing image processing algorithms. Thus, an additionalbenefit of our study is the implication that image algebraprovides an excellent environment for expressing and developingartificial perception systems.

The focus of our study is oninvariances that are expressible in terms of affinetransformations, specifically, homothetic transformations. Ourdiscussion will include both one-dimensional andtwo-dimensional signal patterns. The main contribution of thispaper is the formulation of several novel morphological neuralnetworks that compute homothetic auditory and visualinvariances. With respect to the latter, we employ the theoryand trends of currently popular artificial vision systems.

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References

  1. J. Dayhoff, Neural Network Architectures. An Introduction, Van Nostrand Reinhold: New York, pp. 115–135, 1990.

    Google Scholar 

  2. D.H. Ballard, “Cortical connections and parallel processing: Structure and function,” Vision, Brain, and Cooperative Computation, M. Arbib and A. Hanson (Eds.), M.I.T. Press, pp. 563–622, 1987.

  3. C.P. Suárez Araujo, “Contribuciones a la Integración Multisensorial y Computación Neuronal Paralela. Aplicaciones,” Doctoral Thesis, University of Las Palmas de Gran Canaria, 1990.

  4. C.P. Suárez Araujo, R. Moreno-Díaz, and M. González Rodríguez, “Computational method to obtain visual invariances in artificial vision,” in Proc. of VI Mediterranean Conference on Medical and Biological Engineering '92, Capri, Italy, 1992, Vol. 2, pp. 1277–1282.

  5. A. Trehub, “Visual-cognitive neuronal networks,” Vision, Brain, and Cooperative Computation, M. Arbib and A. Hanson (Eds.), M.I.T. Press, pp. 623–664, 1987.

  6. R. Moreno-Díaz, J. Mira Mira, C.P. Suárez Araujo, and A. Delgado, “Neuronal net to compute homothetic auditive invariances,” in Proc. V Medit. Conference on Medical and Biological Engineering, Patras, Greece, 1989, pp. 302–303.

  7. C.P. Suárez Araujo and R. Moreno-Díaz, “Neural structures to compute homothetic invariances for artificial perception systems,” Lecture Notes in Comp. Science, Springer-Verlag, Vol. 585, pp. 525–539, 1992.

  8. G.X. Ritter, D. Li, and J.N. Wilson, “Image algebra and its relationships to neural networks,” in Proc. of SPIE Tech. Symp. Southeast on Optics, Elec.-Optics, and Sensors, Orlando, 1989, Vol. 1098, pp. 90–101.

  9. G.X. Ritter, “Recent developments in image algebra,” Advance in Electronics and Electron Physics, Vol. 80, pp. 243–308, 1991.

    Google Scholar 

  10. G.X. Ritter, J.N. Wilson, and J.L. Davidson, “Image algebra: An overview,” Computer Vision, Graphics, and Image Processing, Vol. 49, No.3, pp. 297–331, 1990.

    Google Scholar 

  11. J.L. Davidson, “Classification of lattice transformations in image processing,” CVGIP: Image Understanding, Vol. 57, No.3, pp. 283–306, 1993.

    Google Scholar 

  12. G.X. Ritter and P.D. Gader, “Image algebra techniques for parallel image processing,” J. Parallel Distrib. Comput., Vol. 4, No. 5, pp. 7–44, 1987.

    Google Scholar 

  13. G.X. Ritter, J.L. Davidson, and J.N. Wilson, “Beyond mathematical morphology,” in Proc. of SPIE Conf. Visual Communication and Image Processing II, Cambridge, MA, 1987, Vol. 845, pp. 260–269.

  14. J. Davidson and G. Ritter, “Theory of morphological neural networks,” in Proc. of SPIE Optics, Elec.-Optics, and Laser, Appl. in Sci. and Eng., 1990, Vol. 1215, pp. 378–388.

  15. G.X. Ritter and J.L. Davidson, “Recursion and feedback in image algebra,” in Proc. of SPIE's 19th AIPR Workshop on Image Understanding, Wash., D.C., 1990, Vol. 1406, pp. 74–86.

  16. G.X. Ritter, “Heterogeneous matrix products,” in Proc. of SPIE's Image Algebra and Morphological Image Processing II, San Diego, CA, 1991, Vol. 1568, pp. 92–100.

  17. J.L. Davidson and K. Sun, “Opening template learning in morphological neural nets,” The Journal of Knowledge Engineering, Vol. 5, No.2, pp. 28–36, 1992.

    Google Scholar 

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

    Google Scholar 

  19. P. Maragos, “Affine morphology and affine signals models,” in Proc. of SPIE Image Algebra and Morphological Image Processing, San Diego, CA, 1990, Vol. 1350, pp. 31–43.

    Google Scholar 

  20. S. Ullman, “An approach to object recognition: Aligning pictorial descriptions,” in M.I.T. Artif. Intell. Lab., Massachusetts Inst. Technol., Cambridge, MA, A.I. Memo 931, 1986.

  21. C.P. Suárez Araujo and R. Moreno-Díaz, “Modelo para una computación neuronal de invarianzas auditivas,” in Proc. of III Int. Symp. Biomedical, Madrid, Spain, 1987, pp. 689–694.

  22. W. Pitts and W. McCulloch, “How we know universals the perception of auditory and visual forms,” Bull. Math. Biophys., Vol. 9, pp. 127–147, 1947.

    Google Scholar 

  23. C. Gasquet and P. Witomski, Analyse de Fourier et Applications: Filtrage, Calcul Numérique, Ondelettes, Masson, Paris, 1990.

  24. D. Marr and Hildreth, “Theory of edge detection,” in Proc. R. Soc. Lond., 1980, Vol. B207, pp. 187–217.

  25. J. Davidson, “Lattices structures in the image algebra and applications to image processing,” Ph.D. Thesis, Department of Mathematics, University of Florida, Gainsville, Fl, 1989.

  26. Bartlett W. Mel, “Information processing in dendritic tree,” Neural Computation, Vol. 6, pp. 1031–1085, 1994.

    Google Scholar 

  27. S.S. Wilson, “Morphological networks,” in Proc. of SPIE Visual Comm. and Image Proc. IV, Phila., PA, 1989, Vol. 1199, pp. 483–493.

  28. C.P. Suárez Araujo and G.X. Ritter, “Morphological neural networks and image algebra in artificial perception systems,” in Proc. of SPIE Image Algebra and Morphological Image Processing III, San Diego, CA, 1992, Vol. 1769, pp. 128–142.

  29. DARPA, DARPA Neural Network Study, AFCEA International Press, 1988.

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Authors and Affiliations

  1. Department of Computer Sciences and Systems, University of Las Palmas de Gran Canaria, Campus Universitario de Tafira, 35017, Las Palmas de Gran Canaria, Canary Islands, Spain

    Carmen Paz SuáREZ Araujo

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  1. Carmen Paz SuáREZ Araujo
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Araujo, C.P.S. Novel Neural Network Models for Computing Homothetic Invariances: An Image Algebra Notation. Journal of Mathematical Imaging and Vision 7, 69–83 (1997). https://doi.org/10.1023/A:1008218108171

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