A shift-invariant discrete wavelet transform

This article presents a unifying approach to the derivation and implementation of a shift-invariant wavelet transform of one- and two-dimensional (1-D and 2-D) discrete signals. Starting with Mallat's (1989) multiresolution wavelet representation (MRWAR), it presents an analytical process through which a shift-invariant, orthogonal, discrete wavelet transform called the multiscale wavelet representation (MSWAR) is obtained. The coefficients in the MSWAR are shown to be inclusive of those in the MRWAR with the implication that the derived representation is invertible. The computational complexity of the MSWAR is quantified in terms of the required convolutions, and its implementation is shown to be equivalent to the filter upsampling technique.