Optimal biorthogonal wavelet bases for signal decomposition

The selection of scaling functions for optimal signal representation by general multidimensional biorthogonal wavelet bases is investigated. Criterion for optimality is the minimization of the mean-square approximation error at each level of the decomposition. Conditions are given under which the approximation error of the decomposition approaches zero as the level increases. Given arbitrary synthesis filters, the optimal corresponding analysis filters are determined. Globally optimal families of filters are also found, and suboptimal linear and nonlinear-phase filters for the realization of the optimal scaling functions are explicitly determined.