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Symbolic Computation for Moments and Filter Coefficients of Scaling Functions
Georg Regensburger1 and Otmar Scherzer2
1Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, 4040 A-Linz, Austria
2Department of Computer Science, University of Innsbruck, Techniker Str. 25, A-6020 Innsbruck, Austria
Annals of Combinatorics 9 (2) p.223-243 June, 2005
AMS Subject Classification: 42C40, 65T60, 13P10, 94A12, 05A10, 33C45
Algebraic relations between discrete and continuous moments of scaling functions are investigated based on the construction of Bell polynomials. We introduce families of scaling functions which are parametrized by moments. Filter coefficients of scaling functions and wavelets are computed with computer algebra methods (in particular Gröbner bases) using relations between moments. Moreover, we propose a novel concept for data compression based on parametrized wavelets.
Keywords: scaling functions, moments, Bell polynomials, wavelets, Gröbner bases, data compression


1. G.E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge - New York, 1998.

2. W. Bäni, Wavelets, Eine Einführung für Ingenieure, Oldenbourg Verlag, München, Wien, 2002.

3. E.T. Bell, Exponential polynomials, Ann. of Math. (2) 35 (1934) 258--277.

4. B. Buchberger, An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal, Ph.D. Thesis, University of Innsbruck, German, 1965.

5. B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970) 374--383.

6. C. Cassisa and P.E. Ricci, Orthogonal invariants and the Bell polynomials, Dedicated to the memory of Gaetano Fichera (Italian), Rend. Mat. Appl. 20 (7) (2000) 293--303.

7. A.S. Cavaretta, W. Dahmen, and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (453) (1991) vi+ 186 pp.

8. F. Chyzak, P. Paule, O. Scherzer, A. Schoisswohl, and B. Zimmermann, The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval, Experiment. Math. 10 (1) (2001) 67--86.

9. C.B. Collins, The role of Bell polynomials in integration, J. Comput. Appl. Math. 131 (1-2) (2001) 195--222.

10. L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.

11. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms, Second Edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997.

12. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (7) (1988) 909--996.

13. I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.

14. I. Daubechies, Orthonormal bases of compactly supported wavelets. II. Variations on a theme, SIAM J. Math. Anal. 24 (2) (1993) 499--519.

15. I. Daubechies and J.C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (5) (1991) 1388--1410.

16. I. Daubechies and J.C. Lagarias, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (4) (1992) 1031--1079.

17. R.A. Gopinath and C.S. Burrus, On the moments of the scaling function ψ0 , In: Proc. of the IEEE ISCAS, Vol 2, San Diego, CA, 1992, pp. 963--966.

18. J. Lebrun and I.W. Selesnick, Gröbner bases and wavelet design, J. Symbolic Comput. 37 (2) (2004) 227--259.

19. J. Lebrun and M. Vetterli, High-order balanced multiwavelets: theory, factorization, and design, IEEE Trans. Signal Process. 49 (9) (2001) 1918--1930.

20. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press Inc., San Diego, CA, 1998.

21. J. Riordan, An Introduction to Combinatorial Analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons Inc., New York, 1958.

22. O. Rioul, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal. 23 (6) (1992) 1544--1576.

23. S. Roman, The formula of Faa di Bruno, Amer. Math. Monthly 87 (10) (1980) 805--809.

24. S. Roman, The Umbral Calculus, Academic Press Inc., New York, 1984.

25. I.W. Selesnick and C.S. Burrus, Maximally flat low-pass FIR filters with reduced delay, IEEE Trans. Circuits Systems II: Analog and Digital Signal Proc. 45 (1) (1998) 53--68.

26. G. Strang, Wavelets and dilation equations: a brief introduction, SIAM Rev. 31 (4) (1989) 614--627.

27. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press,Wellesley, MA, 1996.

28. W. Sweldens and R. Piessens, Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions, SIAM J. Numer. Anal. 31 (4) (1994) 1240--1264.

29. M. Unser and T. Blu,Wavelet theory demystified, IEEE Trans. Signal Process. 51 (2) (2003) 470--483.