Journal of Fourier Analysis and Applications

, Volume 16, Issue 6, pp 885–900 | Cite as

Complex Wavelets and Framelets from Pseudo Splines

Article

Abstract

In this paper, we introduce complex pseudo splines that are derived from pseudo splines of type I. First, we show that the shifts of every complex pseudo spline are linearly independent. Therefore we can construct a biorthogonal wavelet system. Next, we investigate the Riesz basis property of the corresponding wavelet system generated by complex pseudo splines. The regularity of the complex pseudo splines will be analyzed. Furthermore, by using complex pseudo splines, we will construct symmetric or antisymmetric complex tight framelets with desired approximation order.

Keywords

Complex pseudo splines Complex tight frame Complex Riesz wavelets Biorthogonal complex refinable function Dual frame 

Mathematics Subject Classification (2000)

42C40 42C05 41A25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen, D.R., Han, B., Riemenschneider, S.D.: Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments. Adv. Comput. Math. 13, 131–165 (2000) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chui, C.K., Wang, J.Z.: On compactly supported spline wavelets and a duality principle. Trans. Am. Math. Soc. 330, 903–915 (1992) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Clonda, D., Lina, J.M., Goulard, B.: Complex Daubechies wavelets: properties and statistical image modelling. Signal Process 84, 1–23 (2004) MATHCrossRefGoogle Scholar
  5. 5.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Daubechies, I., Han, B., Ron, A., Shen, Z.W.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dyn, N., Hormann, K., Sabin, M.A., Shen, Z.W.: Polynomial reproduction by symmetric subdivision schemes. J. Approx. Theory. 155, 28–42 (2008) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dong, B., Shen, Z.W.: Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22, 78–104 (2007) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dong, B., Shen, Z.W.: Linear independence of pseudo-splines. Proc. Am. Math. Soc. 134, 2685–2694 (2006) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dong, B., Shen, Z.W.: Construction of biorthogonal wavelets from pseudo-splines. J. Approx. Theory 138, 211–231 (2006) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114, 185–204 (1986) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Han, B.: Analysis and construction of optimal multivariate biorthogonal wavelets with compact support. SIAM J. Math. Anal. 31, 274–304 (1999) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Han, B.: Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules. Adv. Comput. Math. (2008). doi:10.1007/s10444-008-9102-7 Google Scholar
  14. 14.
    Han, B., Hui, J.: Compactly supported orthonormal complex wavelets with dilation 4 and symmetry. Appl. Comput. Harmon. Anal. (2008). doi:10.1016/j.acha.2008.10.005 Google Scholar
  15. 15.
    Han, B., Mo, Q.: Multiwavelet frames from refinable function vectors. Adv. Comput. Math. 18, 211–245 (2003) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Han, B., Shen, Z.W.: Wavelets with short support. SIAM J. Math. Anal. 38, 530–556 (2007) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jia, R.Q.: Interpolatory subdivision schemes induced by box splines. Appl. Comput. Harmon. Anal. 8, 286–292 (2000) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Jia, R.Q.: Cascade algorithms in wavelet analysis. In: Zhou, D.X. (ed.) Wavelet Analysis: Twenty Years’ Developments, pp. 196–230. World Scientific, Singapore (2002) CrossRefGoogle Scholar
  19. 19.
    Jia, R.Q., Wang, J.Z.: Stability and linear independence associated with wavelet decompositions. Proc. Am. Math. Soc. 117, 1115–1124 (1993) MATHMathSciNetGoogle Scholar
  20. 20.
    Khare, A., Tiwary, U.S.: A new method for deblurring and denoising of medical images using complex wavelet transform. In: Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Google Scholar
  21. 21.
    Lawton, W.: Applications of complex valued wavelet transforms to subband decomposition. IEEE Trans. Signal Process. 41, 3566–3568 (2006) CrossRefGoogle Scholar
  22. 22.
    Lemarié-Rieusset, P.G.: On the existence of compactly supported dual wavelets. Appl. Comput. Harmon. Anal. 3, 117–118 (1997) CrossRefGoogle Scholar
  23. 23.
    Li, S., Shen, Y.: Pseudo box splines. Appl. Comput. Harmon. Anal. (2008). doi:10.1016/j.acha.2008.07.004 Google Scholar
  24. 24.
    Lina, J.M.: Image processing with complex Daubechies wavelets. J. Math. Imaging Vis. 7, 211–223 (1997) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Mallat, W.J.: Multiresolution approximations and wavelet orthonormal basis of L 2(ℝ). Trans. Am. Math. Soc. 315, 69–87 (1989) MATHMathSciNetGoogle Scholar
  26. 26.
    Mo, Q., Shen, Y., Li, S.: A new proof of some polynomial inequalities related to pseudo-splines. Appl. Comput. Harmon. Anal. 23, 415–418 (2007) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Portilla, J., Simoncelli, E.P.: A parametric texture model based on joint statistics of complex wavelet coefficients. Int. J. Comput. Vis. 40, 49–71 (2000) MATHCrossRefGoogle Scholar
  28. 28.
    Ron, A.: A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution. Constr. Approx. 5, 297–308 (1989) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ron, A., Shen, Z.W.: Affine systems in L 2(ℝd) I: the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Ron, A., Shen, Z.W.: Affine systems in L 2(ℝd) II: dual systems. J. Fourier Anal. Appl. 3, 617–637 (1997) CrossRefMathSciNetGoogle Scholar
  31. 31.
    Selesnick, I.W.: Smooth wavelet tight frames with zero moments. Appl. Comput. Harmon. Anal. 10, 163–181 (2000) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Zhang, X.P., Desai, M.D., Peng, Y.N.: Orthonormal complex filter banks and wavelets: some properties and design. IEEE Trans. Signal Process. 47, 1039–1048 (1999) MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouP.R. China

Personalised recommendations