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Shift invariant properties of the dual-tree complex wavelet transform

We discuss the shift invariant properties of a new implementation of the discrete wavelet transform, which employs a dual-tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (2/sup m/:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses.