Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T04:09:43.506Z Has data issue: false hasContentIssue false

Conjugate Symmetric Complex Tight Wavelet Frames with Two Generators

Published online by Cambridge University Press:  28 May 2015

Yanmei Xue*
Affiliation:
School of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, P.R. China
Ning Bi*
Affiliation:
Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, P.R. China
Yuan Zhang*
Affiliation:
School of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, P.R. China
*
Corresponding author.Email address:ymxuel@163.com
Corresponding author.Email address:mcsbn@mail.sysu.edu.cn
Corresponding author.Email address:zhangyuanmath@yahoo.com.en
Get access

Abstract

Two algorithms for constructing a class of compactly supported conjugate symmetric complex tight wavelet frames ψ = {ψ1, ψ2} are derived. Firstly, a necessary and sufficient condition for constructing the conjugate symmetric complex tight wavelet frames is established. Secondly, based on a given conjugate symmetric low pass filter, a description of a family of complex wavelet frame solutions is provided when the low pass filter is of even length. When one wavelet is conjugate symmetric and the other is conjugate antisymmetric, the two wavelet filters can be obtained by matching the roots of associated polynomials. Finally, two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames which have some vanishing moments.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Petukhov, A., Explicit construction offramelets, Appl. Comput. Harmon. Anal., vol. 11 (2001), pp. 313327.CrossRefGoogle Scholar
[2]Petukhov, A., Symmetric framelets, Constr. Approx., vol. 19 (2003), pp. 309328.CrossRefGoogle Scholar
[3]Chui, C. K. and He, W. J., Compactly supported tight frames associated with refinable functions, Appl. Comput. Harmon. Anal., vol. 8 (2000), pp. 293319.CrossRefGoogle Scholar
[4]Peng, L. Z. and Wang, H. H., Construction for a class of smooth wavelet tight frames, Sci. China., vol. 46 (2003), pp. 445458.Google Scholar
[5]Jiang, Q. T., Parameterizations of masks for tight affine frames with two symmetric/antisymmetric generators, Adv. Comput. Math., vol. 18 (2003), pp. 247268.CrossRefGoogle Scholar
[6]Han, B. and Mo, Q., Symmetric MRA tight wavelet frames with three generators and high vanishing moments, Appl. Comput. Harmon. Anal., vol. 18 (2005), pp. 6793.CrossRefGoogle Scholar
[7]Han, B. and Mo, Q., Splitting a matrix of Laurent polynomials with symmetry and its application to symmetric framelet filter banks, SIAM Journal on Matrix Analysis and its Appliations., vol. 26 (2004), pp. 97124.CrossRefGoogle Scholar
[8]Han, B., Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules, Adv. Comput. Math., vol. 32 (2010), pp. 209237.CrossRefGoogle Scholar
[9]Zhang, X. P., Desai, M. D. and Peng, Y. N., Orthogonal complex filter banks and wavelets: Some properties and design, IEEE Trans. Signal Processing., vol. 47 (1999), pp. 10391048.CrossRefGoogle Scholar
[10]Belzer, B., Lina, J. M. and Villasenor, J., Complex, linear-phase filters for efficient image coding, IEEE Trans. Signal Processing., vol. 43 (1995), pp. 24252427.CrossRefGoogle Scholar
[11]Lawton, W., Application of complex valued wavelet transforms to subband decomposition, IEEE Trans. Signal Processing., vol. 41 (1993), pp. 35663568.CrossRefGoogle Scholar
[12]Vo, A. and Oraintara, S., A Study of relative phase in complex wavelet domain: Property, statistics and applications in texture image retrieval and segmentation, Signal Processing: Image communication, vol. 25 (2010), pp. 2846.Google Scholar
[13]Ozmen, B. and Ozkaramanlia, H., Complex linear-phase biorthogonal filterbanks with approximately analytic wavelets, Signal Processing, vol. 89 (2009), pp. 599604.CrossRefGoogle Scholar
[14]Oppenheim, A.V. and Schafer, R. W., Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989.Google Scholar
[15]Yang, S. Z. and Xue, Y M., Construction of compactly supported conjugate symmetric complex tight wavelet frames, Int. J. Wavelets Multiresolut. Inf. Process, vol. 8 (2010), pp. 861874.CrossRefGoogle Scholar
[16]Chui, C. K., An introduction to wavelet, New York: Academic Press, 1992.CrossRefGoogle Scholar
[17]Ron, A. and Shen, Z., Affine system in L2(Rd): The analysis of analysis operator, J. Funct. Anal., vol. 148 (1997), pp. 408447.CrossRefGoogle Scholar
[18]Benedetto, J. J. and Li, S., The theory of multiresolution analysis frames and applications to filterbanks, Appl. Comput. Harmon. Anal., vol. 5 (1998), pp. 389427.CrossRefGoogle Scholar