Skip to main content
SpringerLink
Log in

Hilbert Transform Pairs of Tight Frame Wavelets with Integer Dilation Factor

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

Abstract

This paper aims to design a pair of tight frame wavelets with dilation \(M\), where all the generators form Hilbert transform pairs. To ensure the primal and dual filterbanks satisfy the perfect reconstruction condition simultaneously, the periods of phase functions should be \(\frac{2\pi }{M}\) based on our results. According to the sufficient and necessary condition of phase functions, we not only give a linear phase solution for them but also establish their formulae. Moreover, by analyzing the symmetric properties of the dual tight frame, it is shown that as long as their primal tight frames are symmetric those duals are symmetric as well. Finally, we construct an optimization model with minimizing the error to search for the optimal filter coefficients. A design example is given to illustrate the results.

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

Similar content being viewed by others

References

  1. A.F. Abdelnour, Dual-tree tight frame wavelets with symmetric envelope, in 3rd International Conference on ICTTA, pp. 1–6. IEEE (2008)

  2. A.F. Abdelnour, Symmetric tight frame wavelets with dilation factor \(M=4\). Signal Process. 91, 2852–2863 (2011)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. J.J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5(4), 389–427 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Chaux, L. Duval, J.C. Pesquet, Hilbert pairs of \(M\)-band orthonormal wavelet bases, in Proceedings European Signal and Image Proceedings Conference, pp. 6–10 (2004)

  6. C. Chaux, L. Duval, J.C. Pesquet, Image analysis using a dual-tree \(M\)-band wavelet transform. IEEE Trans. Image Process. 15(8), 2397–2412 (2006)

    Article  MathSciNet  Google Scholar 

  7. C. Chaux, J.C. Pesquet, L. Duval, \(2D\) dual-tree complex biorthogonal \(M\)-band wavelet transform, in IEEE International Conference on Acoustics, Speech and Signal Processing, 2007. ICASSP 2007, vol. 3, pp. III-845. IEEE (2007)

  8. C.K. Chui, W. He, Compactly supported tight frames associated with refinable functions. Applied and Computational Harmonic Analysis 8(3), 293–319 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. I. Daubechies, Ten lectures on wavelets. SIAM. 61, (1992). doi:10.1137/1.9781611970104

  11. F.C. Fernandes, R.L. van Spaendonck, C.S. Burrus, A new framework for complex wavelet transforms. IEEE Trans. Signal Process. 51(7), 1825–1837 (2003)

    Article  MathSciNet  Google Scholar 

  12. B. Han, H. Ji, Compactly supported orthonormal complex wavelets with dilation \(4\) and symmetry. Appl. Comput. Harmon. Anal. 26(3), 422–431 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. J.C. Han, Z.X. Cheng, The construction of \(M\)-band tight wavelet frames, in Proceedings of 2004 International Conference on Machine Learning and Cybernetics, 2004. vol. 6, pp. 3924–3927. IEEE (2004)

  14. Y. Huang, Z. Cheng, Minimum-energy frames associated with refinable function of arbitrary integer dilation factor. Chaos, Solitons & Fractals 32(2), 503–515 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. N.G. Kingsbury, The dual-tree complex wavelet transform: a new efficient tool for image restoration and enhancement. Proc. EUSIPCO 98, 319–322 (1998)

    Google Scholar 

  16. N.G. Kingsbury, The dual-tree complex wavelet transform: a new technique for shift invariance and directional filters, in Proceedings 8th IEEE DSP Workshop, vol. 8, p. 86. Citeseer (1998)

  17. N.G. Kingsbury, Image processing with complex wavelets. Philos. Trans. R. Soc. Lond. Ser. A 357(1760), 2543–2560 (1999)

    Article  MATH  Google Scholar 

  18. H. Ozkaramanli, R. Yu, On the phase condition and its solution for Hilbert transform pairs of wavelet bases. IEEE Trans. Signal Process. 51(12), 3293–3294 (2003)

    Article  MathSciNet  Google Scholar 

  19. A. Petukhov, Symmetric framelets. Constr. Approx. 19(2), 309–328 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Petukhov, Construction of symmetric orthogonal bases of wavelets and tight wavelet frames with integer dilation factor. Appl. Comput. Harmon. Anal. 17(2), 198–210 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Ron, Z. Shen, Affine systems in \(L_2({\mathbb{R}}^d)\): the analysis of the analysis operator. J. Funct. Anal. 148(2), 408–447 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. I.W. Selesnick, Hilbert transform pairs of wavelet bases. IEEE Signal Process. Lett. 8(6), 170–173 (2001)

    Article  Google Scholar 

  23. I.W. Selesnick, The design of approximate Hilbert transform pairs of wavelet bases. IEEE Trans. Signal Process. 50(5), 1144–1152 (2002)

    Article  MathSciNet  Google Scholar 

  24. I.W. Selesnick, A.F. Abdelnour, Symmetric wavelet tight frames with two generators. Appl. Comput. Harmon. Anal. 17(2), 211–225 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. I.W. Selesnick, R.G. Baraniuk, N.C. Kingsbury, The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22(6), 123–151 (2005)

    Article  Google Scholar 

  26. R. Yu, Theory of dual-tree complex wavelets. IEEE Trans. Signal Process. 56(9), 4263–4273 (2008)

    Article  MathSciNet  Google Scholar 

  27. R. Yu, H. Ozkaramanli, Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions. IEEE Trans. Signal Process. 53(12), 4723–4725 (2005)

    Article  MathSciNet  Google Scholar 

  28. R. Yu, H. Ozkaramanli, Hilbert transform pairs of biorthogonal wavelet bases. IEEE Trans. Signal Process. 54(6), 2119–2125 (2006)

    Article  Google Scholar 

  29. P. Zhao, C. Zhao, Four-channel tight wavelet frames design using Bernstein polynomial. Circuits Syst. Signal Process. 31(5), 1847–1861 (2012)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by state scholarship fund, the China Scholarship Council (CSC) and the China National Natural Science Foundation under Contract 60972089.

Author information

Authors and Affiliations

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

    Ziquan Lin & Ping Zhao

  2. School of Mathematical Sciences, Beijing Normal University, Beijing,  100875, China

    Xiaojun Zheng

Authors
  1. Ziquan Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ping Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaojun Zheng
    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

Lin, Z., Zhao, P. & Zheng, X. Hilbert Transform Pairs of Tight Frame Wavelets with Integer Dilation Factor. Circuits Syst Signal Process 33, 2917–2934 (2014). https://doi.org/10.1007/s00034-014-9777-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-014-9777-6

Keywords

Search

Navigation