A Simple Construction for the M-Band Dual-Tree Complexwavelet Transform

The dual-tree complex wavelet transform (DT-CWT) which utilizes two 2-band discrete wavelet transforms (DWT) was recently extended to M-band by Chaux et al. In this paper, we provide a simple construction method for an M-band DT-CWT, with M = rd where r, d isin Z. In particular, we show how to extend a given r-band DT-CWT to an rd -band one. For convenience, the case where r = 2, d = 2 is considered. However, the scheme can be extended to general {r, d} pairs straightforwardly. The extension to 2-D which achieves a directional analysis is also provided

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

[2]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[3]  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).

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

[5]  Peter N. Heller,et al.  Theory of regular M-band wavelet bases , 1993, IEEE Trans. Signal Process..

[6]  Josiane Zerubia,et al.  Satellite image deconvolution using complex wavelet packets , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[7]  Caroline Chaux,et al.  Hilbert pairs of M-band orthonormal wavelet bases , 2004, 2004 12th European Signal Processing Conference.

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

[9]  N. Kingsbury.  Image processing with complex wavelets , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  Richard Baraniuk,et al.  The dual-tree complex wavelet transform , 2005, IEEE Signal Processing Magazine.

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

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

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

[14]  Grant V. Welland,et al.  Construction of compactp-wavelets , 1993 .