Design of filters for optimal pyramidal and subband decomposition

Abstract

Properties are determined of the filters which achieve optimal construction of multiresolution sequences by minimising the variance of the error signals between successive pyramid levels. A measure of the entropy reduction achieved by the pyramid is in this way maximised. Further, filters are also determined which minimise the error committed when only one of the channels is retained from a multi-channel perfect reconstruction filter bank. The effect of this is to ensure that the lower-resolution image produced by the primary subband bears maximum resemblance to the input image. In both cases, it is assumed that additive transmission noise corrupts the downsampled signal prior to the synthesis stage. It is seen that under noiseless transmission conditions, the above two families of optimal analysis and synthesis filters coincide. Given arbitrary filters in one of the analysis or synthesis stages of the filter bank, the optimal corresponding synthesis and analysis filters are determined, respectively. Also the globally optimal pairs of analysis and synthesis filters are found to be ideal filters with passbands depending on the input power spectrum. The construction of realisable suboptimal approximations to the ideal filters is investigated with and without constraints on the linearity of the phase of the filters.