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Advanced Topics on Multidimensional Signals

  • Gianfranco Cariolaro

Abstract

The theory of the previous chapters is applicable to multidimensional signals, but to arrive at their full formulation, some topics, mainly multidimensional groups, must be further developed. In this chapter this will be done for gratings and especially for lattices, first with an efficient representation of gratings and then that of lattices, and then we will consider the problem of generating all subgroups of a given group. Other topics that are further investigated are those of cells, with their multiplicity of representations, and the evaluation of sum and intersection, which is formulated in the framework of integer matrices.

The final advanced topic is change of signal dimensionality, which is encountered, e.g., in television scanning, where a time-varying image (a 3D signal) is converted to a 1D signal. This topic has not been developed in the literature in operator form and represents an original contribution of the Unified Signal Theory.

Keywords

Dimensionality Reduction Voronoi Cell Coordinate Change Elementary Operation Signal Extension 
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.

References

  1. 1.
    G. Cariolaro, Scanning theory with application to HDTV. Available at www.springer.com/978-0-85729-463-0
  2. 2.
    G. Cariolaro, E. Ruffa, Lattice operations with Mathematica. Available at www.springer.com/978-0-85729-463-0
  3. 3.
    J.W.S. Cassels, An Introduction to the Geometry of Numbers (Springer, Berlin, 1959) MATHCrossRefGoogle Scholar
  4. 4.
    T. Chen, P.P. Vaidyanathan, The role of integer matrices in multidimensional multirate systems. IEEE Trans. Signal Process. SP-41, 1035–1047 (1993) CrossRefGoogle Scholar
  5. 5.
    E. Dubois, The sampling and reconstruction of time-varying imagery with application in video systems. Proc. IEEE 73, 502–523 (1985) CrossRefGoogle Scholar
  6. 6.
    F.R. Gantmacher, The Theory of Matrices, vol. 1 (Chelsea, New York, 1977) Google Scholar
  7. 7.
    T. Kailath, Linear Systems (Prentice Hall, Englewood Cliffs, 1980) MATHGoogle Scholar
  8. 8.
    C.C. MacDuffee, The Theory of Matrices (Chelsea, New York, 1946) Google Scholar
  9. 9.
    R.M. Mersereau, D.E. Dudgeon, The representation of two-dimensional sequences as one-dimensional sequence. IEEE Trans. Acoust. Speech Signal Process. ASSP-22, 320–325 (1974) CrossRefGoogle Scholar
  10. 10.
    P. Mertz, F. Gray, A theory of scanning and its relation to the characteristics of the transmitted signal in telephotography and television. Bell Syst. Tech. J. 13, 464–515 (1934) Google Scholar
  11. 11.
    G.J. Tonge, The television scanning process. SMPTE J. 93, 657–666 (1984) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.University of PadovaPadovaItaly

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