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Semi-Infinite Programming in Orthogonal Wavelet Filter Design

  • K. O. Kortanek
  • Pierre Moulin
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 25)

Abstract

Quadrature mirror filters can be introduced by studying algebraic properties of discrete input signals viewed as square summable infinite sequences together with linear operators, termed filters, which act linearly through convolution on the input signals. The algebra is readily revealed through operations on complex variable transfer functions which are formally associated with the filters. By using the space of transfer functions, one can define how a signal can be decomposed into two subsidiary signals which can then be combined, using only certain linear mappings all along throughout the entire process. Ideally, the original signal may be recovered, and in this case the term perfect reconstruction is used. In addition to ideal recovery, it is desired to impose orthogonality conditions on the subsidiary signals. These constraints imply that one need only concentrate on the optimality of one of the subsidiary signals to guarantee the optimality of the other. The orthogonality properties are associated with the term quadrature mirror filters.

We describe what optimality can mean within this special structure by making a transition to a description of statistical properties which are reasonably representative of the actual transmission of discrete signals and their recovery after transmission. The concept of coding gain is reviewed, and we show how this objective function may be combined with constraining relations on the coefficients of the (chosen) primary filter necessary to formally guarantee perfect reconstruction. The constraints lead to a nonlinear transformation of the original filter coefficients to variables that appear in an equivalent linear semi-infinite programming problem developed by the second author. We show how an optimal solution in the original filter-variables may be obtained from an LSIP optimal solution by spectral decomposition. Finally, we review some elementary duality-based sensitivity analysis and present some previously published numerical results (by us with other co-authors).

Keywords

Filter Bank Dual Solution Simplex Algorithm Optimal Filter Perfect Reconstruction 
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]
    K. C. Aas, K. A. Duell, and C. T. Mullis. Synthesis of extremal wavelet-generating filters using Gaussian quadrature. IEEE Trans. Sig. Proc, 43:1045–1057, 1995.zbMATHCrossRefGoogle Scholar
  2. [2]
    P. N. Brown and Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sc. Comput., 11:450–481, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    H. Caglar, Y. Liu, and A. N. Akansu. Statistically optimized PR-QMF design. SPIE, 1605:86–94, 1991.CrossRefGoogle Scholar
  4. [4]
    I. Daubechies. Ten Lectures on Wavelets. Number 61 in CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA., 1992.zbMATHCrossRefGoogle Scholar
  5. [5]
    P. Delsarte, B. Macq, and D. T. M. Slock. Signal-adapted multiresolution transform for image coding. IEEE Trans. Info. Theory, 38:897–904, 1992.CrossRefGoogle Scholar
  6. [6]
    J. Elzinga and T. G. Moore. A central cutting plane algorithm for the convex programming problem. Math. Programming, 8:134–145, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    K. Georg and R. Hettich. On the numerical stability of the simplex algorithm: The package linop. Technical report, The University of Trier, Trier, Germany, April 1985.Google Scholar
  8. [8]
    K. Glashoff and S.-A. Gustafson. Linear Optimization and Approximation. Number 45 in Applied Mathematical Sciences. Springer-Verlag, Berlin-Heidelberg-New York, 1983.zbMATHCrossRefGoogle Scholar
  9. [9]
    P. R. Gribik. A central cutting plane algorithm for semi-infinite programming problem. In R. Hettich, editor, Semi-Infinite Programming, number 15 in Lecture Notes in Control and Information Sciences, pages 66–82. Springer-Verlag, 1979.CrossRefGoogle Scholar
  10. [10]
    S. Gustafson and K. O. Kortanek. Numerical treatment of a class of semi-infinite programming problems. Naval Res. Logistics Quart., 20:477–504, 1973.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    R. Hettich and K. O. Kortanek. Semi-infinite programming: theory, methods, and applications. SIAM Review, 35:380–429, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    N. S. Jayant and P. Noll. Digital Coding of Waveforms. Prentice-Hall, 1984.Google Scholar
  13. [13]
    K. O. Kortanek. Vector-supercomputer experiments with the linear programming primal affine scaling algorithm. SIAM J. Scientific and Statistical Computing, 14:279–294, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    K. O. Kortanek and H. No. A central cutting plane algorithm for convex semi-infinite programming problems. SIAM J. Optimization, 3:901–918, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    M. Lang and B.-C. Frenzel. Software available by anonymous ftp from cml.rice.edu:/pub/markus/software, 1992. ©1992–4 LNT.Google Scholar
  16. [16]
    M. Lang and B.-C. Frenzel. Polynomial root finding. IEEE Sig. Proc. Lett., 1:141–143, 1994.CrossRefGoogle Scholar
  17. [17]
    S. G. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 11:674–693, 1989.zbMATHCrossRefGoogle Scholar
  18. [18]
    Y. Meyer. Wavelets Algorithms & Applications. SIAM Society for Industrial and Applied Mathematics, Philadelphia, PA, 1993. Translated and Revised by Robert D. Ryan,.Google Scholar
  19. [19]
    P. Moulin, M. Anitescu, K. O. Kortanek, and F. Potra. Design of signal-adapted FIR paraunitary filter banks. In Proc. ICASSP, volume 3, pages 1519–1522, Atlanta, GA, 1996.Google Scholar
  20. [20]
    P. Moulin, M. Anitescu, K. O. Kortanek, and F. Potra. The role of linear semi-infinite programming in signal-adapted QMF bank design. IEEE Transactions on Signal Processing, 45:2160–2174, 1997.CrossRefGoogle Scholar
  21. [21]
    P. Moulin and K. M. Mihcak. Theory and design of signal-adapted FIR paraunitary filter banks. Technical report, The University of Illinois Beckmann Institute, Champaign/Urbana, IL, 1997. to appear in IEEE Transactions on Signal Processing, Special Issue on Applications of Wavelets and Filter Banks, 1998.Google Scholar
  22. [22]
    T. W. Parks and C. S. Burrus. Digital Filter Design. J. Wiley & Sons, 1987.zbMATHGoogle Scholar
  23. [23]
    W. Press, B. Flannery, S. Teukolsky, and W. Vetterling. Numerical Recipes in C: The Art of Scientific Computing. Cambridge: Cambridge University Press, 1988.zbMATHGoogle Scholar
  24. [24]
    R. Reemtsen. Discretization methods for the solution of semi-infinite programming problems. J. Opt. Theory and Appl., 71:85–103, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    O. Rioul and P. Duhamel. A Remez exchange algorithm for orthonormal wavelets. IEEE Trans. Circ. and Syst. IL An. and Dig. Sig. Proc, 41:550–560, 1994.zbMATHGoogle Scholar
  26. [26]
    M. J. T. Smith and T. P. B. III. Exact reconstruction techniques for tree-structured subband coders. IEEE Trans. ASSP, 34:434–441, 1986.CrossRefGoogle Scholar
  27. [27]
    G. Strang and T. Nyugen. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA, 1996.zbMATHGoogle Scholar
  28. [28]
    M. Unser. An extension of the Karhunen-Loève transform for wavelets and perfect-reconstruction filterbanks. SPIE, 2034:45–56, 1883.CrossRefGoogle Scholar
  29. [29]
    B. Usevitch and M. T. Orchard. Smooth wavelets, transform coding, and Markov-1 processes. In Proc. ISCAS’93, pages 527–530, 1993.Google Scholar
  30. [30]
    P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice-Hall, 1993.zbMATHGoogle Scholar
  31. [31]
    P. P. Vaidyanathan and P.-Q. Hoang. Lattice structures for optimal design and robust implementation of two-channel perfect-reconstruction QMF banks. IEEE Trans. ASSP, 36:81–94, 1988.CrossRefGoogle Scholar
  32. [32]
    L. Vandendorpe. CQF filter banks matched to signal statistics. Signal Processing, 29:237–249, 1992.zbMATHCrossRefGoogle Scholar
  33. [33]
    M. Vetterli and J. Kovacevic. Wavelets and Subband Coding. Prentice-Hall, 1995.zbMATHGoogle Scholar
  34. [34]
    B. Xuan and R. H. Bamberger. Multi-dimensional, paraunitary principal component filter banks. In Proc. ICASSP’95, pages 1488–1491, Detroit, 1995.Google Scholar
  35. [35]
    A. Zygmund. Trigonometric Series. Cambridge University Press, 1959.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • K. O. Kortanek
    • 1
  • Pierre Moulin
    • 2
  1. 1.College of Business AdministrationUniversity of IowaIowa CityUSA
  2. 2.Beckman InstituteUniversity of IllinoisUrbanaUSA

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