The wide designed discrete wavelet transforms based on the perfect translation invariance theorem

The perfect translation invariance theorem has already been proven, which gives the condition of the perfect translation invariance for complex discrete wavelet transforms of an arbitrary square integrable function. In this paper, three types of discrete wavelet transforms based on this theorem are introduced. One of them is designed with variable density of complex wavelets in the time and frequency domains, and another of them is designed with customizable complex wavelets, and the other is designed as a real-valued discrete wavelet transform having perfect translation invariance.