Digital Representation of Signals

  • Leonid Yaroslavsky


As it was discussed in Sect. 2.1, signal digitization can be treated in general terms as determination, for each particular signal, of an index of the equivalency cell to which the signal belongs in the signal space and signal reconstruction can be treated as generating a representative signal of the cell from its index. This is, for instance, what we do when we describe everything in our life with words in speaking or writing. In this case this is our brain that makes the job of subdividing “signal space” into the “equivalency cells” of notions and recognizing which word (cell index) corresponds to what we want to describe. The volume of our vocabulary is about 10 5 ÷ 10 6 words. The variety of signals we have to deal with in signal processing and especially in optics and holography is immeasurably larger. One can see this from a simple example of the number of different images of, for instance, 500 × 500 pixels with 256 gray levels. This number is 256 500×500 . No technical device will ever be capable of storing so many images for comparing them with input images to be digitized.


Point Spread Function Quantization Level Quantization Error Digital Representation Digital Holography 
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  1. 1.
    E.C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford, New York, 1948Google Scholar
  2. 2.
    H. F. Harmuth, Transmission of Information by Orthogonal Functions, Springer Verlag, Berlin, 1970zbMATHCrossRefGoogle Scholar
  3. 3.
    S. Mallat, A Wavelet Tour of Signal Processing, 2nd edition, Academic Press, 1999Google Scholar
  4. 4.
    N. Ahmed, K.R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer Verlag, Berlin, Heidelberg,New York, 1975Google Scholar
  5. 5.
    L. Yaroslaysky, Digital Picture Processing, An Introduction, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1985Google Scholar
  6. 6.
    K. G. Beauchamp, Transforms for Engineers, A Guide to Signal Processing, Clarendon Press, Oxford, 1987.zbMATHGoogle Scholar
  7. 7.
    O.J. Burt, E.A. Adelson, The Laplacian Pyramid as a compact Image Code, IEEE Trans. on Com., Vol. 31, No. 4, pp. 5320540, Apr., 1983Google Scholar
  8. 8.
    Y. Meyer, Wavelets, Algorithms and Applications, SIAM, Philadelphia, 1993zbMATHGoogle Scholar
  9. 9.
    M. Vetterli, J. Kovacevié, Wavelets and Subband Coding, Prentice Hall PTR, Englewwod Cliffs, New Jersey, 199Google Scholar
  10. 10.
    H. Karhunen, Über Lineare Methoden in der Wahscheinlichkeitsrechnung“, Ann. Acad. Science Finn., Ser. A.I.37, Helsinki, 1947Google Scholar
  11. 11.
    M. Loeve, Fonctions Aleatoires de Seconde Ordre, in.: P. Levy, Processes Stochastiques et Movement Brownien, Paris, France: Hermann, 1948Google Scholar
  12. 12.
    H. Hottelling, Analysis of a Complex of Statistical Variables into Principal Components, J. Educ. Psychology 24 (1933), 417: 441, 498–520CrossRefGoogle Scholar
  13. 13.
    R. Dawkins, Climbing Mount Unprobable, W.W.Norton Co, N.Y.,1998Google Scholar
  14. 14.
    H. Hofer, D. R. Willams, The Eye’s Mechanisms for Autocalibration, Optics & Photonics News, Jan. 2002Google Scholar
  15. 15.
    L. Yaroslaysky, Digital Picture Processing, An Introduction, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1985Google Scholar
  16. 16.
    E. Caroli, J. B. Stephen, G. Di Cocco, L. Natalucci, and A. Spizzichino, “Coded Aperture Imaging in X- and Gamma-ray astronomy”, Space Science Reviews, vol. 45, 1987, pp. 349–403ADSCrossRefGoogle Scholar
  17. 17.
    G. K. Skinner, “Imaging with coded aperture masks”, Nuclear Instruments and Methods in Physics Research, vol. 221, 1984, pp. 33–40ADSCrossRefGoogle Scholar
  18. 18.
    W.S. Hinshaw, A.H. Lent, An introduction to NMR imaging, Proc. IEEE, Special issue on Computerized Tomography, 71, No. 3, March 1983Google Scholar
  19. 19.
    A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, 1989zbMATHGoogle Scholar
  20. 20.
    C. Shannon, A Mathematical Theory of Communication, Bell System Techn. J., vol. 27, No. 3, 379–423; No. 4, 623–656, 194Google Scholar
  21. 21.
    R.M. Fano, Technical Report No. 65, The Research Laboratory of Electronics, MIT, March 17, 1949Google Scholar
  22. 22.
    D. A. Huffman, A Method for the Construction of Minimum Redundancy Codes, Prioc. IRE, vol. 40, No. 10, pp. 1098–1101Google Scholar
  23. 22.
    R. Gonzalez, R. E. Woods, Digital Image Processing, Prentice Hall, Upper Saddle River, N.J., 2002Google Scholar
  24. 23.
    H. S. Malvar, Signal Processing with Lapped Transforms, Artech House, Norwood, MA, 1992Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Leonid Yaroslavsky
    • 1
  1. 1.Tel Aviv UniversityIsrael

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