Analytic Wavelet Packets—Combining the Dual-Tree Approach With Wavelet Packets for Signal Analysis and Filtering

Real wavelets (or real basis functions in general) suffer from two major disadvantages: shift-variance and oscillating coefficients. This paper provides a review of these problems as well as the undecimated wavelet transform which solves the problem of shift-variance. However, undecimated transforms have very high computational costs and still have oscillating coefficients. The work of Kingsbury is summarized afterwards. They have presented a dual-tree complex wavelet transform (DTCWT) which is nearly shift-invariant and has smooth coefficients. Subsequently, a filter swapping scheme is developed in order to create complex wavelet packets with analytic basis functions. Finally, the effectiveness of the proposed method is demonstrated by five examples.