On Generalizations of Weighted Finite Automata and Graphics Applications

  • Jürgen Albert
  • German Tischler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4728)


Already computations of ordinary finite automata can be interpreted as discrete grayscale or colour images. Input words are treated as addresses of pixel-components in a very natural way. In this well understood context already meaningful operations on images like zooming or self-similarity can be formally introduced. We will turn then to finite automata with states and transitions labeled by real numbers as weights. These Weighted Finite Automata (WFA), as introduced by Culik II, Karhumäki and Kari, have turned out to be powerful tools for image- and video-compression. The recursive inference-algorithm for WFA can exploit self-similarities within single pictures, between colour components and also in sequences of pictures. We will generalize WFA further to Parametric WFA by allowing different interpretations of the computed real vectors. These vector-components can be chosen as grayscale or colour intensities or e.g. as 3D-coordinates. Applications will be provided including well-known fractal sets and 3D polynomial spline-patches with textures.


Image Compression Finite Automaton Iterate Function System Inference Algorithm Input String 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, New York (1974)zbMATHGoogle Scholar
  2. 2.
    Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Springer, Heidelberg (1978)zbMATHGoogle Scholar
  3. 3.
    Berstel, J., Nait Abdullah, A.: Quadtrees generated by finite automata. In: AFCET 61-62, pp. 167–175 (1989)Google Scholar
  4. 4.
    Culik, K., Dube, S.: L-systems and mutually recursive function systems. Acta Informatica 30, 279–302 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Culik, K., Kari, J.: Image compression using weighted finite automata. Computers and Graphics 17(3), 305–313 (1993)CrossRefGoogle Scholar
  6. 6.
    Culik II, K., Kari, J.: Image-data compression using edge-optimizing algorithm for wfa inference. Journal of Information Processing and Management 30, 829–838 (1994)CrossRefGoogle Scholar
  7. 7.
    Culik II, K., Valenta, V.: Finite automata based compression of bi-level and simple color images. Computers and Graphics 21, 61–68 (1997)CrossRefGoogle Scholar
  8. 8.
    Shapiro, J.M.: Embedded image coding using zerotrees of wavelet coefficients. IEEE Transactions on Signal Processing 41(12), 3445–3462 (1993)zbMATHCrossRefGoogle Scholar
  9. 9.
    Kari, J., Fränti, P.: Arithmetic coding of weighted finite automata. Theoretical Informatics and Applications 28(3-4), 343–360 (1994)zbMATHGoogle Scholar
  10. 10.
    Hafner, U.: Refining image compression with weighted finite automata. In: Storer, J.A., Cohn, M. (eds.) Proc. Data Compression Conference, pp. 359–368 (1996)Google Scholar
  11. 11.
    Hafner, U.: Image and video coding with weighted finite automata. In: Proc. of the IEEE International Conference on Image Processing, pp. 326–329. IEEE Computer Society Press, Los Alamitos (1997)CrossRefGoogle Scholar
  12. 12.
    Hafner, U., Albert, J., Frank, S., Unger, M.: Weighted finite automata for video compression. IEEE Journal on Selected Areas in Communications 16(1), 108–119 (1998)CrossRefGoogle Scholar
  13. 13.
    Culik, K., Karhumäki, J.: Finite automata computing real functions. SIAM J. Comput. 23(4), 789–814 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fisher, Y.: Fractal image compression with quadtrees. In: Fisher, Y. (ed.) Fractal Image Compression, pp. 55–77. Springer, Heidelberg (1995)Google Scholar
  15. 15.
    Albert, J., Kari, J.: Parametric Weighted Finite Automata and Iterated Function Systems. In: Proc. Fractals in Engineering, Delft (1999)Google Scholar
  16. 16.
    Tischler, G., Albert, J., Kari, J.: Parametric Weighted Finite Automata and Multidimensional Dyadic Wavelets. In: Proc. Fractals in Engineering, Tours, France (2005)Google Scholar
  17. 17.
    Tischler, G.: Properties and Applications of Parametric Weighted Finite Automata. JALC 10, 2/3, 347–365 (2005)MathSciNetGoogle Scholar
  18. 18.
    Tischler, G.: Parametric Weighted Finite Automata for Figure Drawing. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 259–268. Springer, Heidelberg (2005)Google Scholar
  19. 19.
    Tischler, G.: Refinement of Near Random Access Video Coding with Weighted Finite Automata. In: Ibarra, O.H., Yen, H.-C. (eds.) CIAA 2006. LNCS, vol. 4094, pp. 46–57. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    De Boor, C.: A practical guide to splines. Springer, Heidelberg (1978)zbMATHGoogle Scholar
  21. 21.
    Catmull, E., Rom, R.: A class of local interpolating splines, Computer Aided Geometric Design, pp. 317–326. Academic Press, London (1974)Google Scholar
  22. 22.
    Pavlidis, T.: Algorithms for Graphics and Image Processing. Computer Science Press (1982)Google Scholar
  23. 23.
    Droste, M., Kari, J., Steinby, P.: Observations on the Smoothness Properties of Real Functions Computed by Weighted Finite Automata. Fundamenta Informaticae 73, 99–106 (2006)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Culik II, K., Dube, S.: Implementing Daubechies Wavelet Transform with Weighted Finite Automata. Acta Informatica 34(5), 347–366 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Culik II, K., Kari, J.J.: Inference Algorithms for WFA and Image Compression. In: Fisher, Y. (ed.) Fractal Image Compression: Theory and Application, Springer, Heidelberg (1995)Google Scholar
  26. 26.
    Derencourt, D., Karhumäki, J., Latteux, M., Terlutte, A.: On Computational Power of Weighted Finite Automata. In: Havel, I.M., Koubek, V. (eds.) Mathematical Foundations of Computer Science 1992. LNCS, vol. 629, pp. 236–245. Springer, Heidelberg (1992)Google Scholar
  27. 27.
    ITU-T: Recommendation T.82 - Coded representation of picture and audio information - Progressive bi-level image compression (1993)Google Scholar
  28. 28.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  29. 29.
    Halava, V., Harju, T.: Undecidability in Integer Weighted Finite Automata. Fundamenta Informaticae 38(1-2), 189–200 (1999)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Halava, V., Harju, T.: Languages Accepted by Integer Weighted Finite Automata. In: Karhumäki, J., Maurer, H.A., Paun, G., Rozenberg, G. (eds.) Jewels are Forever, pp. 123–134. Springer, Heidelberg (1999)Google Scholar
  31. 31.
    Culik II, K., Dube, S.: Affine Automata: A Technique to Generate Complex Images. In: Rovan, B. (ed.) Mathematical Foundations of Computer Science 1990. LNCS, vol. 452, pp. 224–231. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  32. 32.
    Shallit, J., Stolfi, J.: Two methods for generating fractals. Computers and Graphics 13(2), 185–191 (1989)CrossRefGoogle Scholar
  33. 33.
    Culik II, K., Valenta, V., Kari, J.: Compression of Silhouette-like Images based on WFA. Journal of Universal Computer Science 3(10), 1100–1113 (1997)MathSciNetGoogle Scholar
  34. 34.
    Culik II, K., Kari, J.: Computational Fractal Geometry with WFA. Acta Informatica 34(2), 151–166 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Wallace, G.K.: The JPEG still picture compression standard. Communications of the ACM 34(4), 30–44 (1991)CrossRefGoogle Scholar
  36. 36.
    International Organization for Standardization: ISO 15444: Information Technology — JPEG 2000 Image Coding System (2002)Google Scholar
  37. 37.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana University Mathematics Journal 30(5), 713–747 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Watt, A.: Computer Graphics. Addison-Wesley, Reading (2000)Google Scholar
  39. 39.
    Barnsley, M.: Fractals everywhere, 2nd edn. Academic Press, London (1993)zbMATHGoogle Scholar
  40. 40.
    Farin, G.: Curves and surfaces for computer aided geometric design, 2nd edn. Academic Press, London (1990)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jürgen Albert
    • 1
  • German Tischler
    • 1
  1. 1.Department of Computer Science, University of Würzburg, Am Hubland, D-97074 WürzburgGermany

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