CONSTRUCTION OF SHIFT INVARIANT M-BAND TIGHT FRAMELET PACKETS

Year 2016, Volume: 6 Issue: 1, 102 - 114, 01.06.2016

F. A. Shah

Abstract

Framelets and their promising features in applications have attracted a great deal of interest and eort in recent years. In this paper, we outline a method for constructing shift invariant M-band tight framelet packets by recursively decomposing the multiresolution space VJ for a xed scale J to level 0 with any combined mask m = [m0;m1; : : : ;mL] satisfying some mild conditions.

Keywords

M-band wavelets, tight wavelet frame, framelet packet, extension principle, shift invariant, Fourier transform.

References

• Bi,N., Dai,X. and Sun,Q., (1999), Construction of compactly supported M -band wavelets, Appl. Comput. Harmonic Anal., 6, pp. 113-131.
• Chai,A. and Shen,Z., (2007), Deconvolution: A wavelet frame approach, Numer. Math., 106, pp. 529-587.
• Chui,C.K. and Li,C., (1993), Non-orthogonal wavelet packets, SIAM J. Math. Anal., 24, pp. 712-738. [4] Coifman,R.R., Meyer,Y. and Wickerhauser,M.V., (1992), Wavelet analysis and signal processing, In: M. B. Ruskai et al., eds., Wavelets and Their Applications. Jones and Bartlett, Boston, pp. 153-178. [5] Daubechies,I., Han,B., Ron,A. and Shen,Z., (2003), Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmonic Anal., 14, pp. 1-46.
• Debnath,L. and Shah,F.A., (2003), Wavelet Transforms and Their Applications, Birkh¨auser, New York.
• Han,B., (2012), Wavelets and framelets within the framework of non-homogeneous wavelet systems, In Springer Proceedings in Mathematics 13, Eds. Neamtu, M., Schumaker, L., Eds.; Springer, pp. 121-161.
• Han,J.C. and Chen,Q.J., (2008), Characterizations of a class of orthogonal multiple vector-valued wavelet packets, Int. J. Wavelets, Multiresolut. Inf. Process., 6(4), pp. 631-641.
• Han,J.C. and Cheng,Z., (2004), The construction of M -band tight wavelet frames, Proceedings of the Third International Conference on Machine Learning and Cybemetics, Shanghai, pp. 26-29.
• Jiankang,Z., Zheng,B. and Licheng,J., (1998), Theory of orthonormal M -band wavelet packets, J. Elect., 15, pp. 193-198.
• Lin,T., Xu,S., Shi,Q. and Hao,P., (2006), An algebraic construction of orthonormal M -band wavelets with perfect reconstruction, Appl. Comput. Harmonic Anal., 17, pp. 717-730.
• Lu,D. and Fan,Q., (2011), A class of tight framelet packets, Czechoslovak Math. J., 61, pp. 623-639. [13] Nielsen,M., (2002), Highly nonstationary wavelet packets, Appl. Comput. Harmonic Anal., 12, pp. 209-229.
• Ron,A. and Shen,Z., (1997), Affine systems in L2(Rd): the analysis of the analysis operator, J. Funct. Anal., 148, pp. 408-447.
• Shah,F.A., (2009), Construction of wavelet packets on p-adic field, Int. J. Wavelets, Multiresolut. Inf. Process., 7(5), pp. 553-565.
• Shah,F.A. and Debnath,L., (2011), p-Wavelet frame packets on a half-line using the Walsh-Fourier transform, Integ. Transf. Spec. Funct. 22(12), pp. 907-917.
• Shah,F.A. and Debnath,L., (2012), Explicit construction of M -band tight framelet packets, Analysis., 32, pp. 281-294.
• Shen,Z., (1995), Non-tensor product wavelet packets in L2(Rs), SIAM J. Math. Anal., 26(4) , pp. 1061-1074.
• Steffen,P., Heller,N., Gopinath,R.A. and Burrus,C.S., (1993), Theory of regular M -band wavelet bases, IEEE Trans.Sig. Proces., 41, pp. 3497-3510.