On the Dual-Tree Complex Wavelet Packet and $M$-Band Transforms

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 $({\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}})$, 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 $({\hbox {DT}}{\hbox {-}}\BBC{\hbox {WPT}})$, extending the ${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}}$ 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 ${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WPT}}$. Lastly, we show how to extend the two-band ${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}}$ to an $M$-band ${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}}$ (provided that $M=2^{b}$ ) using the same method.