Multiresolution analysis, its link to the discrete parameter wavelet transform, and its initialization

By establishing an equivalence between the discrete parameter (DP) wavelet transform and the multiresolution analysis (MRA) detail coefficients, a simple derivation of the two-scale wavelet equations is given. MRA computes the DP of a signal s(t) only if the MRA inputs s(n) are samples of the inner product of s(t) and the scaling function. This is seldom true in practice. A prefiltering of s(n), as an approximate initialization, is necessary to make the MRA detail coefficients closer to the DP coefficients. A new prefiltering scheme is proposed and shown to be effective in an example.