Multidimensional perfect reconstruction filter banks: an approach of algebraic geometry

Article

Abstract

In image compression and some other applications, multidimensional filter banks are gaining their popularity due to the decrease in implementation cost. In this article, we study these filter banks from a viewpoint of algebraic geometry, where some insights are emerging. Familiar properties, such as perfect reconstruction and linear phase, appear differently when studied from this new angle. For the sake of practicability and a better understanding of problems, our focus in this work is further restricted to the filter banks that can achieve linear phase and perfect reconstruction properties. According to different symmetry nature, the filter banks are categorized into two types. Filters in a multidimensional filter bank represented by multivariate polynomials have common zeros that are different in nature from their 1D counterparts. In this work, the relation between the above-mentioned properties and various zeros is investigated. A criterion to test such filter banks is proposed and this criterion is also interpreted in resultant theory. Based on these results, as well as a proposed conjecture, a factorization of lifting scheme is presented for one type of these filter banks. For the other type of filter banks, we propose a method on perfect reconstruction completion.

Keywords

Filter banks Perfect reconstruction Linear phase Common zeros Factorization Lifting scheme Groebner basis Resultant 

References

  1. Adams, W. W., & Loustaunau, P. (1996). An introduction to Gröbner bases. Providence, R.I.: American Mathematical Society.Google Scholar
  2. Basu S. (1998) Multi-dimensional filter banks and wavelets—a system theoretic perspective. Journal of the Franklin Institute 335(8): 1367–1409CrossRefGoogle Scholar
  3. Bolcskei H., Hlawatsch F., Feichtinger H.G. (1998) Frame-theoretic analysis of oversampled filter banks. IEEE Transactions on Signal Processing 46(12): 3256–3268CrossRefGoogle Scholar
  4. Bose, N. K. (2003). Multidimensional Systems Theory and Applications. Kluwer Academic Publishers.Google Scholar
  5. Bunchberger, B. (1965). An algorithm for finding the bases elements of the residue class ring modulo a zero dimensional polynomial ideal. Ph.D thesis, University of Innsbruck.Google Scholar
  6. Charoenlarpnopparut C., Bose N.K. (1999) Multidimensional FIR filter bank design using Gröbner bases. IEEE Transactions on Circuits and Systems II: Express Briefs 46(12): 1475–1486MATHCrossRefGoogle Scholar
  7. Charoenlarpnopparut C., Bose N.K. (2001) Gröbner bases for problem solving in multidimensional systems. Multidimensional Systems and Signal Processing archive 12(3–4): 345–364MathSciNetGoogle Scholar
  8. Chen T., Vaidyanathan P.P. (1993) Recent developments in multidimensional multirate systems. IEEE Transactions on Circuits and Systems for Video Technology 3(2): 116–137CrossRefGoogle Scholar
  9. Cox, D., Little, J., & O’Shea, D. (1996). Ideals, varieties, and algorithms (2nd ed.). Springer-Verlag New York Inc.Google Scholar
  10. Galkowski, K. (Ed.). (2001). Multidimensional signals, circuits and systems. Taylor and Francis.Google Scholar
  11. Greuel, G. M., Pfister, G., & Schönemann, H. (2005). “Singular 3.0.” A computer algebra system for polynomial computations, centre for computer algebra, University of Kaiserslautern. http://www.singular.uni-kl.de.
  12. Kalker T., Park H., Vetterli M. (1995) Groebner basis techniques in multidimensional multirate systems. Proceeding, ICASSP 4: 2121–2124Google Scholar
  13. Karlsson G., Vetterli M. (1990) Theory of Two-dimensional Multirate Filter Banks. IEEE Transactions on ASSP 38(6): 925–937Google Scholar
  14. Kovacevic J., Vetterli M. (1995) Nonseparable two- and three-dimensional wavelets. IEEE Transactions on Signal Processing 43(5): 1269–1273CrossRefGoogle Scholar
  15. Kurosawa K. (1994) On McClellan transform and 2-D QMF banks. Proceeding, ISCAS 2: 505–508Google Scholar
  16. Lin Z., Xu L., Wu Q. (2004) Appliation of Gröbner bases to signals and image processing: A survey. Linear algebra and its application 391: 169–202MATHCrossRefMathSciNetGoogle Scholar
  17. Lin Z., Xu L., Bose N.K. (2008) A tutorial on Gröbner bases with applications in signals and systems (Invited paper). IEEE Transactions on Circuits and Systems I 55(1): 445–461CrossRefMathSciNetGoogle Scholar
  18. Lin Y.P., Vaidyanathan P.P. (1996) Theory and design of two-dimensional filter banks: A review. Multidimensional Systems and Signal Processing 7(3–4): 263–330MATHCrossRefGoogle Scholar
  19. Mersereau R.M., Mecklenbrauker W.F.G., Quatieri T.F. Jr. (1976) McClellan transformations for two-dimensional digital filtering, I: Design. IEEE Transactions on Circuits and Systems CAS-23: 405–414CrossRefMathSciNetGoogle Scholar
  20. Nguyen T.Q., Vaidyanathan P.P. (1989) Two-channel perfect-reconstruction FIR QMF structures which yield linear-phase analysis and synthesis filters. IEEE Transactions on ASSP 37(5): 676–690CrossRefGoogle Scholar
  21. Park H. (2001) Parametrized family of 2-D non-factorable FIR lossless systems and Gröbner bases. Multidimensional Systems and Signal Processing 12(3–4): 345–364MATHCrossRefMathSciNetGoogle Scholar
  22. Park H. (2002) Optimal design of synthesis filters in multidimensional perfectreconstruction FIR filter banks using Grobner bases. IEEE Transactions on Circuits and Systems I: Regular Papers 49(6): 843–851CrossRefGoogle Scholar
  23. Park H., Kalker T., Vetterli M. (1997) Gröbner bases and multidimensional FIR multirate systems. Multidimensional Systems and Signal Processing 8(1–2): 11–30MATHGoogle Scholar
  24. Vaidyanathan P.P. (1993) Multirate systems and filter banks. Englewood Cliffs, Prentice HallMATHGoogle Scholar
  25. Viscito E., Allebach J.P. (1991) The analysis and design of multidimensional FIR perfectreconstruction filter banks for arbitrary sampling lattices. IEEE Transactions on Circuits and Systems 38(1): 29–41CrossRefGoogle Scholar
  26. Wei D., Evans B.L., Bovik A.C. (1997) Loss of perfect reconstruction in multidimensional filterbanks and wavelets designed via extended McClellan transformations. IEEE Singal Processing Letter 4(10): 295–297CrossRefGoogle Scholar
  27. Zhang L., Makur A. (2005) Structurally linear phase factorization of 2-channel filter banks based on lifting. Proceeding, ICASSP 4: 609–612Google Scholar
  28. Zhang L., Makur A. (2006) Comments on ‘multi-dimensional filter banks and wavelets-A system theoretic perspective’. Journal of the Franklin Institute 343(7): 699–704MATHCrossRefGoogle Scholar
  29. Zhang, L., & Makur, A. (2005). Structurally linear phase factorization of 2-channel filter banks based on lifting. Proceeding, ICASSP, 4, 609–612.Google Scholar
  30. Zhou J., Do M.N. (2006) Multidimensional multichannel FIR deconvolution using Gröbner bases. IEEE. Transactions on Image Processing 15(10): 2998–3007CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Media Technology Lab, Division of Information Engineering, School of Electrical & Electronic EngineeringNanyang Technological UniversitySingaporeSingapore

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