Perfect translation invariance with a wide range of shapes of Hilbert transform pairs of complex wavelets

It is well-known that a Hilbert transform pair of a complex wavelet improves the lack of translation invariance of the discrete wavelet transform. However, its shapes and improvement are limited by the difficulty in applying the Hilbert transform pair to a discrete signal. In this paper, a novel Hilbert transform pair, which is based on a Meyer wavelet and has a wide range of shapes, is designed to create perfect translate invariance. A new frame structure is designed to apply an arbitrary Hilbert transform pair of the complex wavelet to any discrete signal. Therefore, perfect translation invariance is achieved with a wide range of shapes of the Hilbert transform pairs of complex wavelets.