Adaptive multiwavelet initialization

The multiwavelet concept uses a set of scaling functions and a set of wavelets to generate an orthogonal multichannel multiresolution pyramid decomposition for finite energy signals. The multiple scaling functions extract the lowpass information of the input data set, and the multiwavelets extract the bandpass information. When the lowpass filters associated with the scaling functions are cascaded, a multiresolution pyramid decomposition/reconstruction system is formed with each pyramid convolution operator having several inputs and several outputs. However, there is only one data set available to initialize this process. This paper addresses the question of how to best modify the data set using prefilters so that its decomposition contains the most useful information for the chosen application. The “best” prefilters are determined by the minimization of the energy of preselected decomposition components. The resulting decomposition is, therefore, signal adaptive, and under appropriate conditions, perfect reconstruction of the input data set can be achieved with a proper postfiltering. The detailed discussions in the paper are focussed on the two-wavelet and two-scaling function case; the general multiwavelet case is addressed at the end of the paper. A compression example is provided to demonstrate the performance of the optimally initialized multiwavelet method and to compare it with a single wavelet method and another multiwavelet initialization method proposed by other authors