Design of regular nonseparable bidimensional wavelets using gröbner basis techniques

IEEE Transactions on Signal Processing

Volumn 46, Issue 4, 1998, Pages 845-856

  Faugère, Jean Charles ^a   Moreau de Saint Martin, François Moreau ^b,c   Rouillier, Fabrice ^d  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^b ORANGE LABS   (France)

^c Ministère de l'Industrie   (France)

^d INRIA   (France)

Author keywords

Gr bner basis; Image coding; Multidimensional digital filters; Multidimensional filter banks; Multidimensional wavelets; Regular wavelets; Wavelet transforms

Indexed keywords

EID: 9144273964     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/78.668541     Document Type: Article

References (37)

1. A. Auzinger and S. Stetter, "An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations," Int. Series Numer. Math., vol. 86, pp. 11-30, 1998.
2. B. Buchberger, "Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal," Ph.D. disertation, Univ. Innsbruck, Innsbruck, Austria, 1965.
3. -, "An algorithmical criterion for the solvability of algebraic systems," Aequationes Mathematicae, vol. 4, no. 3, pp. 374-383, 1970, in German.
4. _, "Gröbner bases: An algorithmic method in polynomial ideal theory," in Recent Trends in Multidimensional Systems Theory, K. Reider, Ed. Munich, Germany: Bose, 1985.
5. A. Cohen and I. Daubechies, "Non-separable bidimensional wavelet bases," Revista Mat. Iberoame'ricana, vol. 9, no. 1, pp. 51-137, 1993.
6. _, "A new technique to estimate the regularity of refinable func-tions," preprint, 1995.
7. A. Cohen, K. Gröchenig, and L. Villemoes, "Regularity of multi-variate refinable functions," preprint, 1995.
8. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commuataive Algebra. Berlin, Germany: Springer-Verlag, 1992, undergraduate texts in mathematics.
9. D. Lazard, "Systems of algebraic equations (algorithms and complexity)," in Comput. Algebraic Geom. Commutative Algebra, Proc. Cortona Conf., D. Eisenbud and L. Robbiano, Eds. Cambridge, U.K.: Cambridge Univ. Press, Oct. 1993.
10. I. Daubechies, "Orthogonal bases of compactly supported wavelets," Commun. Pure Appl. Math., vol. XLI, pp. 909-996, 1988.
11. -, "Ten lectures on wavelets," in CBMS-NSF Series in Applied Mathematics, Philadelphia, PA: Soc. Indus. Appl. Math., 1992, no. 61.
12. P. Delogne and B. Macq, "Universal variable length coding for an integrated approach to image coding," Ann. Télécommun., vol. 46, pp. 7-8, 1991.
13. J. C. Faugère, "Résolution des systèmes d'équations polynomials," Ph.D. dissertation, Univ. Paris VI-Pierre et Marie Curie, France, 1994.
14. J. C. Faugère, P. Gianni, D. Lazard, and T. Mora, "Efficient computation of zero-dimensional Groebner basis by change of ordering," J. Symbolic Comput., 1994.
15. J. C. Faugère, "A new efficient algorithm for computing Gröbner basis (F4)," Tech. Rep., Task 3.3.2.1, 1997, preprint.
16. J. Davenport, Y. Siret, and E. Tournier, Calcul Formel, 2nd ed. Paris, France: Masson, 1993.
17. J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for K"," IEEE Trans. Inform. Theory, vol. 38, pp. 533-555, Mar. 1992.
18. _, "Nonseparable two-and three-dimensional wavelets," IEEE Trans. Signal Processing, vol. 43, pp. 1269-1272, May 1995.
19. M. Moreno Maza, "Calculs de pgcd au-dessus des tours d'extensions simples et résolution des systèmes d'équations algébriques," Ph.D. dissertation, Univ. Paris 6, Paris, France, 1997.
20. F. Moreau de Saint-Martin, "Bancs de filtres et ondelettes: Synthèses, adaptations et applications," Ph.D. dissertation, CEREMADE, Univ. Paris IX, Dauphine, France, 1997.
21. P. Onno, "Bancs de filtres et quantification vectorielle sur réseau: Étude conjointe pour la compression d'images," Ph.D. dissertation, Univ. de Rennes I, Rennes, France, 1996.
22. H. Park, T. Kalker, and M. Vetterli, "Groebner bases and multidimen-sional fir multirate systems," J. Multidimensional Syst. Signal, 1996.
23. P. Pedersen, M.-F. Roy, and A. Szpirglas, "Counting real zeros in the multivariate case," in Computational Algebraic Geometry, F. Essette and A. Galigo, Eds., 1993, pp. 203-223.
24. K. Ramchandran and M. Vetterli, "Best wavelet packet bases in a rate-distortion sense," IEEE Trans. Image Processing, vol. 2, pp. 160-175, Apr. 1993.
25. O. Rioul, "Simple regularity criteria for subdivision schemes," Soc. Ind. Appl. Math. J. Math. Anal., vol. 23, no. 6, pp. 1544-1576, Nov. 1992.
26. F. Rouillier, "Algorithmes efficaces pour l'étude des zéros réels des systèmes polynomiaux," Ph.D. thesis, Univ. de Rennes I, Rennes, France, May 1996.
27. Y. Shoham and A. Gersho, "Efficient bit allocation for an arbitrairy set of quantizers," IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, Sept. 1988.
28. D. Stanhill and Y. Y. Zeevi, "Two-dimensional linear-phase orthogonal filter-banks and wavelets," preprint, 1995.
29. _, "Two-dimensional orthogonal wavelets with vanishing moments," preprint, 1995.
30. T. Becker and V. Weispfenning, Gröbner Bases, A Computationnal Approach to Commutative Algebra. Berlin, Germany: Springer-Verlag, 1993, graduate texts in mathematics.
31. S. Venkataranam and B. C. Levy, "Nonseparable orthogonal linear phase perfect reconstruction filter banks and their application to image compression," in Proc. Int. Conf. Image Process., 1994, pp. 334-338.
32. _, "State space representations of 2-D FIR lossless matrices," IEEE Trans. Circuits Syst. II, vol. 41, pp. 117-131, Feb. 1994.
33. _, "A comparison of design methods for 2-D FIR orthogonal perfect reconstruction filter banks," IEEE Trans. Circuits Syst. II, vol. 42, pp. 525-536, Aug. 1995.
34. L. F. Villemoes, "Continuity of nonseparable quincunx wavelets," Appt. Comput. Harmon. Anal., to be published.
35. _, "Sobolev regularity of wavelets and stability of iterated filter banks," in Progress in Wavelet Analysis and Applications, Y. Meyer and S. Roques, Eds., pp. 243-251;
36. also editions frontières, in Proc. Int. Conf. Wavelets Applicat., Toulouse, France, 1992.
37. E. Viscito and J. P. Allebach, "The analysis and design of multidimensional FIR perfect reconstruction filter banks for arbitrary sampling lattices," IEEE Trans. Circuits Syst., vol. 38, pp. 29-41, Jan. 1991.