Skip to main content
SpringerLink
Log in

Four-Channel Tight Wavelet Frames Design Using Bernstein Polynomial

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The paper provides a novel technique for designing tight frame wavelet filters through the use of Bernstein polynomials. The perfect-reconstruction conditions of tight wavelet frame filters are established by using parameters of the Bernstein polynomials. The desired number of vanishing moments can be easily achieved by setting the appropriate parameters of the Bernstein polynomial to zero. The filters are obtained by the spectral factorization method and constructed by the appropriate parameters. The design technique is flexible in that it allows low-pass filters and high-pass filters with different characteristics to be designed easily.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. A.F. Abdelnour, I.W. Selesnick, Symmetric nearly shift-invariant tight frame wavelets. IEEE Trans. Signal Process. 53, 231–239 (2005)

    Article  MathSciNet  Google Scholar 

  2. L.T. Bruton, Low-sensitivity digital ladder filters. IEEE Trans. Circuits Syst. CAS-22, 168–176 (1975)

    Article  Google Scholar 

  3. H. Caglar, A.N. Akansu, A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation. IEEE Trans. Signal Process. 41, 2314–2321 (1993)

    Article  MATH  Google Scholar 

  4. K.N. Chaudhury, M. Unser, On the shiftability of dual-tree complex wavelet transforms. IEEE Trans. Signal Process. 58, 221–232 (2010)

    Article  MathSciNet  Google Scholar 

  5. C.K. Chui, W. He, Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8, 293–319 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. C.K. Chui, W. He, J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal. 13, 224–262 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61 (SIAM, Philadelphia, 1992)

    Book  MATH  Google Scholar 

  8. I. Daubechies, B. Han, A. Ron, Z.W. Shen, Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. P.J. Davis, Interpolation and Approximation (Ginn-Blaisdell, Waltham, 1963)

    MATH  Google Scholar 

  10. B. Gold, C.M. Rader, Digital Processing of Signals (Lincoln Lab, Massachusetts Institute of Technology, Boston, 1965), p. 95

    Google Scholar 

  11. B. Han, Symmetric orthonormal scaling functions and wavelets with dilation factor 4. Adv. Comput. Math. 8, 221–247 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Han, Matrix extension with symmetry and applications to symmetric orthonormal complex M-wavelets. J. Fourier Anal. Appl. 15, 684–705 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Han, Compactly supported orthonormal complex wavelets with dilation 4 and symmetry. Appl. Comput. Harmon. Anal. 26, 422–431 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Han, Q. Mo, Tight wavelet frames generated by three symmetric B-spline functions with high vanishing moments. Proc. Am. Math. Soc. 132, 77–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Han, Q. Mo, Symmetric MRA tight wavelet frames with three generators and high vanishing moments. Appl. Comput. Harmon. Anal. 18, 67–93 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Q.T. Jiang, Parameterizations of masks for tight affine frames with two symmetric/antisymmetric generators. Adv. Comput. Math. 18, 247–268 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmon. Anal. 10, 234–253 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Lawton, S.L. Lee, Z.W. Shen, Stability and orthonormality of multivariate refinable functions. SIAM J. Math. Anal. 28, 999–1014 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Petukhov, Explicit construction of framelets. Appl. Comput. Harmon. Anal. 11, 313–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Petukhov, Symmetric framelets. Constr. Approx. 19, 309–328 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. L.R. Rajagopal, S.C. Dutta Roy, Design of maximally flat FIR filters using the Bernstein polynomial. IEEE Trans. Circuits Syst. 34, 1587–1590 (1987)

    Article  Google Scholar 

  22. A. Ron, Z.W. Shen, Affine systems in L 2(R d): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. I.W. Selesnick, Smooth wavelet tight frames with zero moments. Appl. Comput. Harmon. Anal. 10, 163–181 (2000)

    Article  MathSciNet  Google Scholar 

  24. I.W. Selesnick, A.F. Abdelnour, Symmetric wavelet frames with two generators. Appl. Comput. Harmon. Anal. 17, 211–225 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. I.W. Selesnick, R.G. Baraniuk, N.G. Kingsbury, The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22, 123–151 (2005)

    Article  Google Scholar 

  26. D.B.H. Tay, Zero-pinning the Bernstein polynomial: a simple design technique for orthonormal wavelets. IEEE Signal Process. Lett. 12, 835–838 (2005)

    Article  Google Scholar 

  27. D.B.H. Tay, ETHFB: a new class of even-length biorthogonal wavelet filters for Hilbert pair design. IEEE Trans. Circuits Syst. I 55, 1580–1588 (2008)

    Article  MathSciNet  Google Scholar 

  28. D.B.H. Tay, Design of orthonormal Hilbert-pair of wavelets using zero-pinning. Signal Process. 90, 866–873 (2010)

    Article  MATH  Google Scholar 

  29. H. Wang, L. Peng, Parameterizations of univariate wavelet tight frames with short support. Commun. Nonlinear Sci. Numer. Simul. 11, 663–677 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the China National Natural Science Foundation under Contract 60972089. The authors would like to express our deep thanks to Nick Kingsbury and the reviewers for their very helpful suggestions, which greatly improved the quality of this paper.

Author information

Authors and Affiliations

  1. School of Science, Beijing Jiaotong University, Beijing, 100044, China

    Ping Zhao

  2. Faculty of Mathematics Science, Tianjin Normal University, Tianjin, 300074, China

    Chun Zhao

Authors
  1. Ping Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ping Zhao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, P., Zhao, C. Four-Channel Tight Wavelet Frames Design Using Bernstein Polynomial. Circuits Syst Signal Process 31, 1847–1861 (2012). https://doi.org/10.1007/s00034-012-9412-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-012-9412-3

Keywords

Search

Navigation