Complex wavelets for shift invariant analysis and filtering of signals

This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain thei r real and imaginary parts. This introduces limited redundancy (2(m) : 1 fo r m-dimensional signals) and allows the transform to provide approximate sh ift invariance and directionally selective filters (properties lacking in t he traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this ap proximation and design suitable filters to achieve this. We discuss two dif ferent variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dua l tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advant age of its unique properties. (C) 2001 Academic Press.