A shift-invariant discrete wavelet transform

This correspondence 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 multiresolution wavelet representation (MRWAR), the correspondence presents an analytical process through which a shift-invariant, orthogonal, discrete wavelet transform called the multiscale wavelet representation (MSWAR) is obtained, The coefficients in MSWAR are shown to be inclusive of those in MRWAR with the implication that the derived representation is invertible. The computational complexity of MSWAR is quantified in terms of the required convolutions, and its implementation is shown to be equivalent to the filter upsampling technique.