Design of Biorthogonal Wavelet Filters of DTCWT Using Factorization of Halfband Polynomials

Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 841)


In this paper, we propose a new approach for designing the biorthogonal wavelet filters (BWFs) of Dual-Tree Complex Wavelet Transform (DTCWT). Proposed approach provides an effective way to handle the frequency response characteristics of these filters. This is done by optimizing the free variables obtained using factorization of generalized halfband polynomial (GHBP). The designed filters using proposed approach have better frequency response characteristics than those obtained by using binomial spectral factorization approach. Also, their associated wavelets show improved analyticity in terms of qualitative and quantitative measures. Transform-based image denoising using the proposed filters shows better visual as well as quantitative performance.


Wavelet transform Complex wavelet Spectral factorization 


  1. 1.
    Fierro, M., Ha, H.G., Ha, Y.H.: Noise reduction based on partial-reference, dual-tree complex wavelet transform Shrinkage. IEEE Trans. Image Process. 22(5), 1859–1872 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Rabbani, H., Gazor, S.: Video denoising in three-dimensional complex wavelet domain using a doubly stochastic modelling. IET Image Process. 6(9), 1262–1274 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anantrasirichai, N., Achim, A., Kingsbury, N.G., Bull, D.R.: Atmospheric turbulence mitigation using complex wavelet-based fusion. IEEE Trans. Image Process. 22(6), 2398–2408 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Asikuzzaman, M., Alam, M.J., Lambert, A.J., Pickering, M.R.: Robust DT-CWT based DIBR 3D video watermarking using chrominance embedding. IEEE Trans. Multimedia 18(9), 1733–1748 (2016)CrossRefGoogle Scholar
  5. 5.
    Kingsbury, N.: Image processing with complex wavelets. Philos. Trans. R. Soc. London A: Math. Phy. Eng. Sci. 357(1760), 2543–2560 (1999)CrossRefGoogle Scholar
  6. 6.
    Kingsbury, N.: Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmonic Anal. 10(3), 234–253 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Selesnick, I.W.: Hilbert transform pairs of wavelet bases. IEEE Sig. Process. Lett. 8(6), 170–173 (2001)CrossRefGoogle Scholar
  8. 8.
    Selesnick, I.W.: The design of approximate Hilbert transform pairs of wavelet bases. IEEE Trans. Sig. Process. 50(5), 1144–1152 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Selesnick, I.W., Baraniuk, R.G., Kingsbury, N.C.: The dual-tree complex wavelet transform. IEEE Sig. Process. Mag. 22(6), 123–151 (2005)CrossRefGoogle Scholar
  10. 10.
    Tay, D.B.H.: Designing Hilbert-pair of wavelets: recent progress and future trends. In: 6th International Conference on Information Communication & Signal Processing, pp. 1–5. IEEE (2007)Google Scholar
  11. 11.
    Chaux, C., Duval, L., Pesquet, J.C.: Image analysis using a dual-tree M-band wavelet transform. IEEE Trans. Image Process. 15(8), 2397–2412 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chaux, C., Pesquet, J.C., Duval, L.: 2D dual-tree complex biorthogonal M-band wavelet transform. In: 2007 IEEE International Conference on Acoustics, Speech and Signal Processing-ICASSP 2007, vol. 3, pp. III-845. IEEE (2007)Google Scholar
  13. 13.
    Yu, R., Ozkaramanli, H.: Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions. IEEE Trans. Sig. Process. 53(12), 4723–4725 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yu, R., Ozkaramanli, H.: Hilbert transform pairs of biorthogonal wavelet bases. IEEE Trans. Sig. Process. 54(6), 2119–2125 (2006)CrossRefGoogle Scholar
  15. 15.
    Thiran, J.P.: Recursive digital filters with maximally flat group delay. IEEE Trans. Circ. Theory 18(6), 659–664 (1971)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Patil, B.D., Patwardhan, P.G., Gadre, V.M.: On the design of FIR wavelet filter banks using factorization of a halfband polynomial. IEEE Sig. Process. Lett. 15, 485–488 (2008)CrossRefGoogle Scholar
  17. 17.
    Daubechies, I., et al.: Ten Lectures on Wavelets, vol. 61. SIAM, Philadelphia (1992)CrossRefGoogle Scholar
  18. 18.
    Tay, D.B., Kingsbury, N.G., Palaniswami, M.: Orthonormal Hilbert-pair of wavelets with (almost) maximum vanishing moments. IEEE Sig. Process. Lett. 13(9), 533–536 (2006)CrossRefGoogle Scholar
  19. 19.
    Lightstone, M., Majani, E., Mitra, S.K.: Low bit-rate design considerations for wavelet-based image coding. Multidimension. Syst. Sig. Process. 8(1–2), 111–128 (1997)CrossRefGoogle Scholar
  20. 20.
    Rahulkar, A.D., Patil, B.D., Holambe, R.S.: A new approach to the design of biorthogonal triplet half-band filter banks using generalized half-band polynomials. Signal Image Video Process. 8(8), 1451–1457 (2014)CrossRefGoogle Scholar
  21. 21.
    Selesnick, I.W.: Accessed 04 Aug 2014
  22. 22.
    Sendur, L., Selesnick, I.W.: Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Sig. Process. 50(11), 2744–2756 (2002)CrossRefGoogle Scholar
  23. 23.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar
  24. 24.
    Zhang, L., Zhang, L., Mou, X., Zhang, D.: FSIM: a feature similarity index for image quality assessment. IEEE Trans. Image Process. 20(8), 2378–2386 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.DA-IICTGandhinagarIndia

Personalised recommendations