The dual-tree complex wavelet transform: a new technique for shift invariance and directional filters

A new implementation of the Discrete Wavelet Transform is presented, suitable for a range of signal and image processing applications. It employs a dual tree of wavelet lters to obtain the real and imaginary parts of complex wavelet coeecients. This introduces limited redundancy (4:1 for 2-dimensional signals) and allows the transform to provide approximate shift in-variance and directionally selective lters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational eeciency. An application to texture synthesis is presented.