Hilbert pair of almost symmetric orthogonal wavelets with arbitrary center of symmetry

This paper proposes a new method for designing a class of Hilbert pairs of almost symmetric orthogonal wavelets with arbitrary center of symmetry. Two scaling low-pass filters are designed simultaneously to satisfy the specified degree of flatness of group delays, vanishing moments and orthogonality condition of wavelets, along with improved analyticity. Therefore, the resulting scaling low-pass filters have flat group delay responses and the specified number of vanishing moments. Moreover, the difference of the frequency responses between two scaling low-pass filters can be effectively minimized to improve the analyticity of complex wavelets. The condition of orthogonality is linearized, and then an iterative procedure is used to obtain the filter coefficients. Finally, several examples are presented to demonstrate the effectiveness of the proposed design procedure.

[1]  Hüseyin Özkaramanli,et al.  On the phase condition and its solution for Hilbert transform pairs of wavelet bases , 2003, IEEE Trans. Signal Process..

[2]  Hüseyin Özkaramanli,et al.  On the phase condition and its solution for Hilbert transform pairs of wavelets bases , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[3]  Ramesh A. Gopinath.  Lowpass delay filters with flat magnitude and group delay constraints , 2003, IEEE Trans. Signal Process..

[4]  N. Kingsbury.  Complex Wavelets for Shift Invariant Analysis and Filtering of Signals , 2001 .

[5]  Xi Zhang,et al.  Design of Q-shift filters with flat group delay , 2012, 2012 IEEE International Symposium on Circuits and Systems.

[6]  Unto K. Laine,et al.  Splitting the unit delay [FIR/all pass filters design] , 1996, IEEE Signal Process. Mag..

[7]  Xi Zhang,et al.  IIR-Based ${\rm DT}{\cal C}{\rm WTs}$ With Improved Analyticity and Frequency Selectivity , 2012, IEEE Transactions on Signal Processing.

[8]  Caroline Chaux,et al.  Image analysis using a dual-tree M-band wavelet transform , 2006, IEEE Transactions on Image Processing.

[9]  Xi Zhang.  Design of Q-shift filters with improved vanishing moments for DTCWT , 2011, 2011 18th IEEE International Conference on Image Processing.

[10]  Richard Baraniuk,et al.  The Dual-tree Complex Wavelet Transform , 2007 .

[11]  Nick G. Kingsbury,et al.  A dual-tree complex wavelet transform with improved orthogonality and symmetry properties , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[12]  Unto K. Laine,et al.  Splitting the Unit Delay - Tools for fractional delay filter design , 1996 .

[13]  Nick Kingsbury,et al.  The dual-tree complex wavelet transform: a new technique for shift invariance and directional filters , 1998 .

[14]  Ivan W. Selesnick,et al.  The design of approximate Hilbert transform pairs of wavelet bases , 2002, IEEE Trans. Signal Process..

[15]  I. Selesnick.  Hilbert transform pairs of wavelet bases , 2001, IEEE Signal Processing Letters.

[16]  Nick G. Kingsbury,et al.  Design of Q-shift complex wavelets for image processing using frequency domain energy minimization , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[17]  H. Ozkaramanli,et al.  Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions , 2005, IEEE Transactions on Signal Processing.