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

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <formula formulatype="inline"><tex>$({\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}})$</tex></formula>, 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 <formula formulatype="inline"><tex>$({\hbox {DT}}{\hbox {-}}\BBC{\hbox {WPT}})$</tex></formula>, extending the <formula formulatype="inline"><tex>${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}}$</tex> </formula> 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 <formula formulatype="inline"><tex>${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WPT}}$</tex></formula>. Lastly, we show how to extend the two-band <formula formulatype="inline"><tex>${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}}$</tex> </formula> to an <formula formulatype="inline"><tex>$M$</tex></formula>-band <formula formulatype="inline"><tex>${\hbox {DT}}{\hbox {-}}\BBC{\hbox {WT}}$</tex> </formula> (provided that <formula formulatype="inline"><tex>$M=2^{b}$</tex> </formula>) using the same method. </para>