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Vector Quantization

  • Peter F. Swaszek

Abstract

Quantization is of ubiquitous need in information-processing systems. We typically model information-bearing signals as being analog in nature, while most of our processing of the signals is digital. As examples, consider the following. Digital communication systems encode samples of an input signal into a bit stream of finite length: finite due to bandwidth constraints. Various signal-processing techniques have their mathematical operations implemented on digital computers, finite binary representations being used for the data values. Hence an efficient method of transforming continuously valued signals into discretely valued signals is both desirable and necessary to ensure good system performance. Often such signals exist on a continuum of time. Here, however, we will assume that a sampling of the signal has already occurred and will employ the information-theoretic model of a discrete-time, continuously valued vector random process. With this assumption, the problem of interest is to in some way decide how to “round” a vector observation to one of N values.

Keywords

Vector Quantizer Output Point Scalar Quantization Optimum Quantizer Quantization Region 
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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References

  1. [1]
    A. Gersho, Principles of quantization, IEEE Trans. Circ. Sys., Vol. CAS-25, pp. 427–437, 1978.CrossRefGoogle Scholar
  2. [2]
    J. Max, Quantizing for minimum distortion, IEEE Trans. Inform. Theory, Vol. IT-6, pp. 7–12, 1960.MathSciNetCrossRefGoogle Scholar
  3. [3]
    E.F. Abaya and G.L. Wise, Convergence of vector quantizers with applications to the design of optimal quantizers, Proc. Allerton Conf. Comm. Cont. Comp., University of Illinois, pp. 79–88, 1981.Google Scholar
  4. [4]
    E.F. Abaya and G.L. Wise, On the existence of optimal quantizers, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 937–940, 1982.MathSciNetCrossRefGoogle Scholar
  5. [5]
    D.T.S. Chen, On two or more dimensional optimum quantizers, IEEE Int’l Conf ASSP, pp. 540–643, 1977.Google Scholar
  6. [6]
    N.C. Gallagher and J.A. Bucklew, Properties of minimum mean square error block quantizers, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 105–107, 1982.CrossRefGoogle Scholar
  7. [7]
    Gersho, On the structure of vector quantizers, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 157–166, 1982.Google Scholar
  8. [8]
    R.M. Gray, J.C. Kieffer and Y. Linde, Locally optimum block quantizer design, Inform. Cont.,Vol. 45, pp. 178–198, 1980.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Y. Linde, A. Buzo and R.M. Gray, An algorithm for vector quantizer design, IEEE Trans. Comm., Vol. COM-28, pp. 84–95, 1980.CrossRefGoogle Scholar
  10. [10]
    J. Menez, F. Boeri and D.J. Esteban, Optimum quantizer algorithm for real time block quantizing, Proc. Int’l Conf. ASSP, pp. 980–984, 1979.Google Scholar
  11. [11]
    J.A. Bucklew, Upper bounds to the asymptotic performance of block quantizers, IEEE Trans. Inform. Theory, Vol. IT-27, pp. 577–581, 1981.MathSciNetCrossRefGoogle Scholar
  12. [12]
    J.A. Bucklew and G.L. Wise, Multidimensional asymptotic quantization theory with r-th power distortion measures, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 239–247, 1982.MathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Elias, Bounds and asymptotes for the performance of multivariate quantizers, Ann. Math. Statist., Vol. 41, pp. 1249–1259, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Gersho, Asymptotically optimal block quantization, IEEE Trans. Inform. Theory, Vol. IT-25, pp. 373–380, 1979.MathSciNetCrossRefGoogle Scholar
  15. [15]
    M.P. Schiitzenberger, On the quantization of finite dimensional messages, Inform. Cont., Vol. 1, pp. 153–158, 1958.CrossRefGoogle Scholar
  16. [16]
    Y. Yamada, S. Tazaki and R.M. Gray, Asymptotic performance of block quantizers with difference distortion measures, IEEE Trans. Inform. Theory, Vol. IT-26, pp. 6–14, 1980.MathSciNetCrossRefGoogle Scholar
  17. [17]
    P. Zador, Development and Evaluation of Procedures for Quantizing Multivariate Distributions, Stanford University Dissertation, Department of Statistics, 1963.Google Scholar
  18. [18]
    P.L. Zador, Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 139–149, 1982.MathSciNetCrossRefGoogle Scholar
  19. [19]
    W.R. Bennett, Spectra of quantized signals, Bell Sys. Tech. Jour., Vol. 27 pp. 446–472, 1948.Google Scholar
  20. [20]
    J.A. Bucklew, Companding and random quantization in several dimensions, IEEE Trans. Inform. Theory, Vol. IT-27, pp. 207–211, 1981.MathSciNetCrossRefGoogle Scholar
  21. [21]
    J.A. Bucklew, A note on optimal multidimensional compandors, IEEE Trans. Inform. Theory, Vol. IT-29, pp. 279, 1983.CrossRefGoogle Scholar
  22. [[22]
    J.H. Conway and N.J.A. Sloane, Voronoi regions of lattices, second moments of polytopes, and quantization, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 211–226, 1982.MathSciNetCrossRefGoogle Scholar
  23. [23]
    J.H. Conway and N.J.A. Sloane, Fast quantizing and decoding algorithms for lattice quantizers and codes, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 221–226, 1982.Google Scholar
  24. [24]
    L. Fejes Toth, Sur la representation d’une population infinie par un nombre finie d’elements, Acta Mathematica, Magyar Tudomanyos Aka- demia Budapest, Vol. 10, pp. 299–304, 1959.MathSciNetzbMATHGoogle Scholar
  25. [25]
    D.J. Newman, The hexagon theorem, IEEE Trans. Inform. Theory, Vol. IT- 28, pp. 137–139, 1982.CrossRefGoogle Scholar
  26. [26]
    K.D. Rines and N.C. Gallagher, Jr., The design of multidimensional quantizers using prequantization, Proc. Allerton Conf Comm. Cont. Comp., pp. 446–453, 1979.Google Scholar
  27. [27]
    K.D. Rines and N.C. Gallagher Jr., The design of two dimensional quantizers using prequantization, IEEE Trans. Inform. Theory, Vol. IT-28 pp. 232–239, 1982.CrossRefGoogle Scholar
  28. [28]
    K.D. Rines, N.C. Gallagher Jr. and J.A. Bucklew, Nonuniform multidimensional quantizers, Proc. Princeton Conf. Inform. Sci. Sys., Princeton University, pp. 43–46, 1982.Google Scholar
  29. [29]
    K. Sayood and J.D. Gibson, An algorithm for designing vector quantizers, Proc. Allerton Conf. Comm., Cont., Comp., pp. 301–310, 1982.Google Scholar
  30. [30]
    J.A. Bucklew and N.C. Gallagher Jr., Quantization schemes for bivari- ate Gaussian random variables, IEEE Trans. Inform. Theory, Vol. IT- 25, pp. 537–543, 1979.MathSciNetCrossRefGoogle Scholar
  31. [31]
    J.A. Bucklew and N.C. Gallagher Jr., Two-dimensional quantization of bivariate circularly symmetric densities, IEEE Trans. Inform. Theory, Vol. IT-25, pp. 667–671, 1979.Google Scholar
  32. [32]
    W.J. Dallas, Magnitude-coupled phase quantization, Appl. Optics, Vol. 13, pp. 2274–2279, 1974.CrossRefGoogle Scholar
  33. [33]
    J.G. Dunn, The performance of a class of n dimensional quantizers for a Gaussian source, Proc. Columbia Symp. Signal Tran., New York, pp. 76–81, 1965.Google Scholar
  34. [34]
    R.M. Gray and E.D. Karnin, Multiple local optima in vector quantizers, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 256–261, 1982.MathSciNetCrossRefGoogle Scholar
  35. [35]
    J.J.Y. Huang and P.M. Schultheiss, Block quantization of correlated Gaussian random variables, IEEE Trans. Comm. Sys., Vol. CS-11, pp. 289–296, 1963.CrossRefGoogle Scholar
  36. [36]
    W.A. Pearlman, Polar quantization of a complex Gaussian random variable, IEEE Trans. Comm., Vol. COM-27, pp. 892–899, 1979.CrossRefGoogle Scholar
  37. [37]
    P.F. Swaszek, Further notes on circularly symmetric quantizers, Proc. Conf. Inform. Sci. Sys., Johns Hopkins University, pp. 794 - 800, 1983.Google Scholar
  38. [38]
    P.F. Swaszek, Uniform spherical coordinate quantizers, Proc. Allerton Conf Comm., Cont. Comp., pp. 491–500, 1983.Google Scholar
  39. [39]
    P.F. Swaszek and J.B. Thomas, Optimum circularly symmetric quantizers, J. Franklin Inst., Vol. 313, pp. 373 - 384, 1982.CrossRefGoogle Scholar
  40. [[40]
    P.F. Swaszek, Asymptotic performance of optimal circularly symmetric quantizers, submitted for publication.Google Scholar
  41. [41]
    P.F. Swaszek and J.B. Thomas, Multidimensional spherical coordinates quantization, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 570–576, 1983.CrossRefGoogle Scholar
  42. [42]
    P.F. Swaszek and T. Ku, Asymptotic performance of unrestricted polar quantizers, Proc. Princeton Conf Inform. Sci. and Sys., pp. 581–586, 1984.Google Scholar
  43. [43]
    M. Tasto and P.A. Wintz, Note on the error signal of block quantizers, IEEE Trans. Comm., Vol. COM-21, pp. 216–219, 1973.CrossRefGoogle Scholar
  44. [44]
    S.G. Wilson, Magnitude/phase quantization of independent Gaussian variates, IEEE Trans. Comm., Vol. COM-28, pp. 1924–1929, 1980.CrossRefGoogle Scholar
  45. [45]
    A. Buzo, A.H. Gray, Jr., R.M. Gray and J.D. Markel, Speech coding based upon vector quantization, IEEE Trans. A.oust., Speech, Sig. Proc., Vol. ASSP-28, pp. 562–574, 1980.MathSciNetCrossRefGoogle Scholar
  46. [46]
    N.C. Gallagher, Jr., Quantizing schemes for the discrete Fourier transform of a random time series, IEEE Trans. Inform. Theory, Vol. IT-24, pp. 156–163, 1978.MathSciNetCrossRefGoogle Scholar
  47. [47]
    A. Habibi and P.A. Wintz, Image coding by linear transformation and block quantization, IEEE Trans. Comm., Vol. COM-19, pp. 50–62, 1971.CrossRefGoogle Scholar
  48. [48]
    J. MacQueen, Some methods for classification and analysis of multivariate observations, Proc. 5th Berkeley Symp. Math. Stat, and Prob., Vol. 1, Berkeley, CA: Univ. of California Press, pp. 281–297, 1967.Google Scholar
  49. [49]
    W.A. Pearlman, Optimum fixed level quantization of the DFT of achromatic images, Proc. Allerton Conf. Comm. Cont. and Comp., pp. 313–319, 1979.Google Scholar
  50. [50]
    D. Pollard, Quantization and the method of k-means, IEEE Trans. Inform. Theory, Vol. IT-28, pp. 199–205, 1982.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Peter F. Swaszek
    • 1
  1. 1.Department of Electrical EngineeringUniversity of Rhode IslandKingstonUSA

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