Perfectly translation-invariant complex wavelet packet transforms

The complex discrete wavelet transform having perfect translation invariance has already been proposed. However, due to complication of frequency divisions with wavelet packets, it is difficult to design a complex wavelet packet transform having perfect translation invariance. In this paper, a useful theorem for achieving perfect translation invariance is proved, and a novel complex wavelet packet transform is designed to create this perfect translate invariance. This complex wavelet packet transform is based on a Meyer wavelet, which has the important characteristic of having a wide range of shapes. Therefore, the complex wavelet packet transform having perfect translation invariance can be designed with the optimized shapes of the Meyer wavelet. One of them is based on a single Meyer wavelet and the other is based on a number of different shapes of the Meyer wavelets to create good localization of complex wavelet packets.