[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Sparsely Connected Autoassociative Lattice Memories with an Application for the Reconstruction of Color Images

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Sparsely connected autoassociative lattice memories (SCALMs) are very general models defined on complete lattices, a mathematical structure which is obtained by imposing some ordering on a set. They are computationally cheaper and mathematically simpler than “traditional” models and other memories such as the original autoassociative morphological memories (AMMs) of Ritter and Sussner because they only compute maximums and minimums. This paper provides theoretical results on SCALMs defined on a general complete lattice as well as an application of these memories for the storage and recall of color images. Precisely, we characterize the recall phase of SCALMs in terms of their fixed points. Then, we show that any endomorphic lattice polynomial—a concept that generalizes the notion of lattice polynomial of Birkhoff—on the fundamental memory set represents a fixed point of the SCALMs. Also, we discuss the relationship between SCALMs and the original AMMs. Finally, we provide some experimental results on the performance of SCALMs, defined on different color lattices, for the reconstruction of color images corrupted by either Gaussian or impulsive noise.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acharya, T., Ray, A.: Image Processing: Principles and Applications. Wiley, Hoboken (2005)

    Book  Google Scholar 

  2. Agoston, M.K.: Computer Graphics and Geometric Modelling: Implementation & Algorithms, vol. 1. Springer, New York (2005)

    Google Scholar 

  3. Anderson, J., Silverstein, J., Ritz, S., Jones, R.: Distinctive features, categorical perception, and probability learning: Some applications of a neural model. Psychol. Rev. 84, 413–415 (1977)

    Article  Google Scholar 

  4. Angulo, J.: Morphological colour operators in totally ordered lattices based on distances: Application to image filtering, enhancement and analysis. Comput. Vis. Image Underst. 107(1–2), 56–73 (2007). Special issue on color image processing

    Article  Google Scholar 

  5. Angulo, J., Serra, J.: Morphological coding of color images by vector connected filters. In: Proceedings of the Seventh International Symposium on Signal Processing and Its Applications, July 2003, vol. 1, pp. 69–72 (2003)

    Chapter  Google Scholar 

  6. Aptoula, E., Lefèvre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognit. 40(11), 2914–2929 (2007)

    Article  MATH  Google Scholar 

  7. Aptoula, E., Lefèvre, S.: On lexicographical ordering in multivariate mathematical morphology. Pattern Recognit. Lett. 29(2), 109–118 (2008)

    Article  Google Scholar 

  8. Austin, J.: Associative memory. In: Fiesler, E., Beale, R. (eds.) Handbook of Neural Computation, pp. F1.4:1–F1.4:7. Oxford University Press, London (1997)

    Google Scholar 

  9. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1993)

    Google Scholar 

  10. Blyth, T.S.: Lattices and Ordered Algebraic Structures, 1st edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  11. Cuninghame-Green, R.: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, New York (1979)

    Book  MATH  Google Scholar 

  12. Davidson, J.: Minimax techniques for non-linear image processing transforms. In: Technical Symposium on Optics, Electro-Optics, and Sensors, Orlando, FL, March 1989. Proceedings of SPIE, vol. 1098 (1989)

    Google Scholar 

  13. Fairchild, M.D.: Color Appearance Models, 2nd edn. Wiley, New York (2005)

    Google Scholar 

  14. Foley, J.D., Dam, A.V., Huges, J.F., Feiner, S.K.: Computer Graphics: Principles and Practice, 2nd edn. Addison-Wesley, Reading (1990)

    Google Scholar 

  15. Gawne, T.J., Martin, J.M.: Responses of primate visual cortical v4 neurons to simultaneously presented stimuli. J. Neurophysiol. 88(3), 1128–1135 (2002)

    Google Scholar 

  16. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Prentice-Hall, Upper Saddle River (2002)

    Google Scholar 

  17. Graña, M.: A brief review of lattice computing. In: Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2008), Hong Kong, China, June 2008, pp. 1777–1781 (2008)

    Google Scholar 

  18. Graña, M., Villaverde, I., Maldonado, J., Hernandez, C.: Two lattice computing approaches for the unsupervised segmentation of hyperspectral images. Neurocomputing 72(10–12), 2111–2120 (2009)

    Article  Google Scholar 

  19. Grätzer, G., et al.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  20. Hanbury, A., Serra, J.: Mathematical morphology in the HLS colour space. In: Proceedings of the 12th British Machine Vision Conference, pp. 451–460 (2001)

    Google Scholar 

  21. Hanbury, A., Serra, J.: Mathematical morphology in the L ∗ a ∗ b ∗ colour space. Tech. rep., Centre de Morphologie Mathématique, École des Mines de Paris, August 2001

  22. Hassoun, M.H. (ed.): Associative Neural Memories: Theory and Implementation. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  23. Heijmans, H.: Morphological Image Operators. Academic Press, New York (1994)

    MATH  Google Scholar 

  24. Heijmans, H.J.A.M.: Mathematical morphology: A modern approach in image processing based on algebra and geometry. SIAM Rev. 37(1), 1–36 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982)

    Article  MathSciNet  Google Scholar 

  26. Kohonen, T.: Self-Organization and Associative Memory, 3rd edn. Springer, New York (1989)

    Book  Google Scholar 

  27. Kong, S.-G., Kosko, B.: Adaptive fuzzy systems for backing up a truck-and-trailer. IEEE Trans. Neural Netw. 3(2), 211–223 (1992)

    Article  Google Scholar 

  28. Kosko, B.: Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence. Prentice Hall, Englewood Cliffs (1992)

    MATH  Google Scholar 

  29. Levkowitz, H.: Color Theory and Modeling for Computer Graphics, Visualization, and Multimedia Applications. Kluwer Academic, Norwell (1997)

    Book  Google Scholar 

  30. Levkowitz, H., Herman, G.T.: Glhs: A generalized lightness, hue, and saturation color model. CVGIP, Graph. Models Image Process. 55(4), 271–285 (1993)

    Article  Google Scholar 

  31. Oh, C., Zak, S.H.: Image recall using a large scale generalized Brain-State-in-a-Box neural network. Int. J. Appl. Math. Comput. Sci. 15(1), 99–114 (2005)

    MATH  Google Scholar 

  32. Plataniotis, K., Androutsos, D., Venetsanopoulos, A.: Adaptive fuzzy systems for multichannel signal processing. Proc. IEEE 87(9), 1601–1622 (1999)

    Article  Google Scholar 

  33. Poynton, C.: Frequently asked questions about color (1997)

  34. Pratt, W.: Digital Image Processing. Wiley, New York (1978)

    Google Scholar 

  35. Ritter, G.X.: Image algebra. Unpublished manuscript. Available at: http://www.cise.ufl.edu/~jnw/CVAIIA/ (1997)

  36. Ritter, G.X., Gader, P.: Fixed points of lattice transforms and lattice associative memories. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 144. Academic Press, New York (2006)

    Google Scholar 

  37. Ritter, G.X., Sussner, P.: An introduction to morphological neural networks. In: Proceedings of the 13th International Conference on Pattern Recognition Vienna, Austria, pp. 709–717 (1996)

    Chapter  Google Scholar 

  38. Ritter, G.X., Sussner, P.: Morphological neural networks. In: Intelligent Systems: A Semiotic Perspective; Proceedings of the 1996 International Multidisciplinary Conference Gaithersburg, Maryland, pp. 221–226 (1996)

    Google Scholar 

  39. Ritter, G.X., Sussner, P.: Associative memories based on lattice algebra. In: Computational Cybernetics and Simulation, Orlando, Florida, 1997, IEEE International Conference on Systems, Man, and Cybernetics (1997)

    Google Scholar 

  40. Ritter, G.X., Urcid, G.: A lattice matrix method for hyperspectral image unmixing. Inf. Sci. 181(10), 1787–1803 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ritter, G.X., Wilson, J.N.: Handbook of Computer Vision Algorithms in Image Algebra, 2nd edn. CRC Press, Boca Raton (2001)

    MATH  Google Scholar 

  42. Ritter, G.X., Wilson, J.N., Davidson, J.L.: Image algebra: An overview. Comput. Vis. Graph. Image Process. 49(3), 297–331 (1990)

    Article  Google Scholar 

  43. Ritter, G.X., Sussner, P., de Leon, J.L.D.: Morphological associative memories. IEEE Trans. Neural Netw. 9(2), 281–293 (1998)

    Article  Google Scholar 

  44. Ritter, G.X., Urcid, G., Iancu, L.: Reconstruction of patterns from noisy inputs using morphological associative memories. J. Math. Imaging Vis. 19(2), 95–111 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ritter, G.X., Iancu, L., Schmalz, M.S.: A new auto-associative memory based on lattice algebra. In: Progress in Pattern Recognition, Image Analysis and Applications, 9th Iberoamerican Congress on Pattern Recognition, CIARP 2004. Lecture Notes in Computer Science, vol. 3287, pp. 148–155. Springer, Berlin (2004)

    Google Scholar 

  46. Ritter, G.X., Urcid, G., Schmalz, M.S.: Lattice associative memories that are robust in the presence of noise. In: Mathematical Methods in Pattern and Image Analysis. Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 5916, pp. 255–260. SPIE, Bellingham (2005)

    Google Scholar 

  47. Ritter, G.X., Urcid, G., Schmalz, M.S.: Autonomous single-pass endmember approximation using lattice auto-associative memories. Neurocomputing 72(10–12), 2101–2110 (2009)

    Article  Google Scholar 

  48. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  49. Serra, J.: Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances. Academic Press, New York (1988)

    Google Scholar 

  50. Sharma, G., Trussell, H.J.: Digital color imaging. IEEE Trans. Image Process. 6(7), 901–932 (1997)

    Article  Google Scholar 

  51. Soille, P.: Morphological Image Analysis. Springer, Berlin (1999)

    MATH  Google Scholar 

  52. Sussner, P.: Fixed points of autoassociative morphological memories. In: Proceedings of the International ICSA/IFAC Symposium on Neural Computation, Berlin (2000)

    Google Scholar 

  53. Sussner, P.: Associative morphological memories based on variations of the kernel and dual kernel methods. Neural Netw. 16(5), 625–632 (2003)

    Article  Google Scholar 

  54. Sussner, P., Esmi, E.L.: Morphological perceptrons with competitive learning: Lattice-theoretical framework and constructive learning algorithm. Inf. Sci. 181(10), 1929–1950 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sussner, P., Valle, M.E.: Grayscale morphological associative memories. IEEE Trans. Neural Netw. 17(3), 559–570 (2006)

    Article  Google Scholar 

  56. Sussner, P., Valle, M.E.: Implicative fuzzy associative memories. IEEE Trans. Fuzzy Syst. 14(6), 793–807 (2006)

    Article  Google Scholar 

  57. Sussner, P., Valle, M.E.: Recall of patterns using morphological and certain fuzzy morphological associative memories. In: Proceedings of the IEEE World Conference on Computational Intelligence 2006, pp. 209–216, Vancouver, Canada (2006)

    Google Scholar 

  58. Sussner, P., Valle, M.E.: Fuzzy associative memories and their relationship to mathematical morphology. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Handbook of Granular Computing, pp. 733–754. Wiley, New York (2008), Chap. 33

    Chapter  Google Scholar 

  59. Sussner, P., Miyasaki, R., Valle, M.E.: An introduction to parameterized IFAM models with applications in prediction. In: Proceedings of the 2009 IFSA World Congress and 2009 EUSFLAT Conference, Lisbon, Portugal, July 2009, pp. 247–252 (2009)

    Google Scholar 

  60. Sussner, P., Esmi, E.L., Villaverde, I., Graña, M.: The Kosko subsethood fuzzy associative memory (KS-FAM): Mathematical background and applications in computer vision. J. Math. Imaging Vis. (2011). doi:10.1007/s10851-011-0292-0

    Google Scholar 

  61. Tang, K., Astola, J., Neuvo, Y.: Nonlinear multivariate image filtering techniques. IEEE Trans. Image Process. 4(6), 788–798 (1995)

    Article  Google Scholar 

  62. Urcid, G., Ritter, G.X., Iancu, L.: Kernel computation in morphological bidirectional associative memories. In: Progress in Pattern Recognition, Speech and Image Analysis, 8th Iberoamerican Congress on Pattern Recognition, CIARP 2003. Lecture Notes in Computer Science, vol. 2095, pp. 563–570. Springer, Berlin (2003)

    Chapter  Google Scholar 

  63. Urcid, G., Ritter, G.X., Nieves-V., J.A.: Redundant encoding of patterns in lattice associative memories. In: Proceedings of the Lattice Based Modeling Workshop, 6th International Conference on Concept Lattices and Their Applications Olomouc, Czech Republic, pp. 45–57 (2008)

    Google Scholar 

  64. Urcid, G., Nieves-V., J.A., Garcia-A., A., Valdiviezo-N., J.: Robust image retrieval from noisy inputs using lattice associative memories. In: Proceedings of SPIE—The International Society for Optical Engineering: Image Processing: Algorithms and Systems VII San Jose, CA, USA, vol. 7245, pp. 1–12 (2009)

    Google Scholar 

  65. Valle, M.E.: A class of sparsely connected autoassociative morphological memories for large color images. IEEE Trans. Neural Netw. 20(6), 1045–1050 (2009)

    Article  Google Scholar 

  66. Valle, M.E., Sussner, P.: Fuzzy morphological associative memories based on uninorms. In: Proceedings of the IEEE World Conference on Computational Intelligence 2008 (WCCI 2008), Hong Kong, China, June 2008, pp. 1582–1589 (2008)

    Google Scholar 

  67. Valle, M.E., Sussner, P.: A general framework for fuzzy morphological associative memories. Fuzzy Sets Syst. 159(7), 747–768 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  68. Valle, M.E., Sussner, P.: Storage and recall capabilities of fuzzy morphological associative memories with adjunction-based learning. Neural Netw. 24(1), 75–90 (2011)

    Article  MATH  Google Scholar 

  69. Vandenbroucke, N., Macaire, L., Postaire, J.-G.: Color image segmentation by pixel classification in an adapted hybrid color space. Application to soccer image analysis. Comput. Vis. Image Underst. 90(2), 190–216 (2003)

    Article  Google Scholar 

  70. Vazquez, R.A., Sossa, H.: A bidirectional hetero-associative memory for true-color patterns. Neural Process. Lett. 28(3), 131–153 (2008)

    Article  Google Scholar 

  71. Vazquez, R.A., Sossa, H.: Morphological hetero-associative memories applied to restore true-color patterns. In: Proceedings of the 6th International Symposium on Neural Networks: Advances in Neural Networks—Part III. ISNN 2009, vol. 5553, pp. 520–529. Springer, Berlin (2009)

    Google Scholar 

  72. Yáñez-Márquez, C., Cruz-Meza, M.E., Sánchez-Garfias, F.A., López-Yáñez, I.: Using alpha-beta associative memories to learn and recall RGB images. In: Proceedings of the 4th international symposium on Neural Networks: Advances in Neural Networks, Part III. ISNN ’07, vol. 4493, pp. 828–833. Springer, Berlin (2007)

    Google Scholar 

  73. Zhang, D., Zuo, W.: Computational intelligence-based biometric technologies. IEEE Comput. Intell. Mag. 2(2), 26–36 (2007)

    Article  MathSciNet  Google Scholar 

  74. Zhang, B.-L., Zhang, H., Ge, S.S.: Face recognition by applying wavelet subband representation and kernel associative memory. IEEE Trans. Neural Netw. 15(1), 166–177 (2004)

    Article  MathSciNet  Google Scholar 

  75. Zhang, H., Huang, W., Huang, Z., Zhang, B.: A kernel autoassociator approach to pattern classification. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 35(3), 593–606 (2005)

    Article  Google Scholar 

  76. Zheng, P., Zhang, J., Tang, W.: Color image associative memory on a class of Cohen–Grossberg networks. Pattern Recognit. 43(10), 3255–3260 (2010)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Center for Exact Sciences, University of Londrina, CEP 86051-990, Londrina, PR, Brazil

    Marcos Eduardo Valle

  2. Center for Exact Sciences and Technology, State University of West Paraná, CEP 85819-110, Cascavel, PR, Brazil

    Daniela Maria Grande Vicente

Authors
  1. Marcos Eduardo Valle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniela Maria Grande Vicente
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcos Eduardo Valle.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Valle, M.E., Grande Vicente, D.M. Sparsely Connected Autoassociative Lattice Memories with an Application for the Reconstruction of Color Images. J Math Imaging Vis 44, 195–222 (2012). https://doi.org/10.1007/s10851-011-0322-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-011-0322-y

Keywords

Search

Navigation