On the dual-tree complex wavelet packet and M-band transforms

Ilker Bayram, Ivan W. Selesnick

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2298-2310
Number of pages13
JournalIEEE Transactions on Signal Processing
Issue number6
StatePublished - Jun 2008


  • Dual-tree complex wavelet transform
  • Wavelet packet

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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