TY - JOUR
T1 - On the dual-tree complex wavelet packet and M-band transforms
AU - Bayram, Ilker
AU - Selesnick, Ivan W.
N1 - Funding Information:
Manuscript received January 7, 2007; revised October 15, 2007. This work was supported by the Office of Naval Research (ONR) under Grant N00014-03-1-0217. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Cedric Richard.
PY - 2008/6
Y1 - 2008/6
N2 - The two-band discrete wavelet transform (DWT) provides an octave-band analysis in the frequency domain, but this might not be "optimal" for a given signal. The discrete wavelet packet transform (DWPT) provides a dictionary of bases over which one can search for an optimal representation (without constraining the analysis to an octave-band one) for the signal at hand. However, it is well known that both the DWT and the DWPT are shift-varying. Also, when these transforms are extended to 2-D and higher dimensions using tensor products, they do not provide a geometrically oriented analysis. The dual-tree complex wavelet transform DT-CWT, introduced by Kingsbury, is approximately shift-invariant and provides directional analysis in 2-D and higher dimensions. In this paper, we propose a method to implement a dual-tree complex wavelet packet transform (DT-CWPT), extending the DT-CWT as the DWPT extends the DWT. To find the best complex wavelet packet frame for a given signal, we adapt the basis selection algorithm by Coifman and Wickerhauser, providing a solution to the basis selection problem for the DT-CWPT. Lastly, we show how to extend the two-band DT-CWT to an M-band DT-CWT (provided that M=2b) using the same method.
AB - The two-band discrete wavelet transform (DWT) provides an octave-band analysis in the frequency domain, but this might not be "optimal" for a given signal. The discrete wavelet packet transform (DWPT) provides a dictionary of bases over which one can search for an optimal representation (without constraining the analysis to an octave-band one) for the signal at hand. However, it is well known that both the DWT and the DWPT are shift-varying. Also, when these transforms are extended to 2-D and higher dimensions using tensor products, they do not provide a geometrically oriented analysis. The dual-tree complex wavelet transform DT-CWT, introduced by Kingsbury, is approximately shift-invariant and provides directional analysis in 2-D and higher dimensions. In this paper, we propose a method to implement a dual-tree complex wavelet packet transform (DT-CWPT), extending the DT-CWT as the DWPT extends the DWT. To find the best complex wavelet packet frame for a given signal, we adapt the basis selection algorithm by Coifman and Wickerhauser, providing a solution to the basis selection problem for the DT-CWPT. Lastly, we show how to extend the two-band DT-CWT to an M-band DT-CWT (provided that M=2b) using the same method.
KW - Dual-tree complex wavelet transform
KW - Wavelet packet
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U2 - 10.1109/TSP.2007.916129
DO - 10.1109/TSP.2007.916129
M3 - Article
AN - SCOPUS:44849125344
SN - 1053-587X
VL - 56
SP - 2298
EP - 2310
JO - IEEE Transactions on Signal Processing
JF - IEEE Transactions on Signal Processing
IS - 6
ER -