Optimal pyramidal and subband decompositions for the hierarchical representation of vector valued signals

Abstract

Filters are determined which achieve optimal pyramidal decomposition of vector signals by minimising the variance of the errors between successive pyramid levels. Further, the filters are also determined which minimise the error produced when only one of the channels is retained from a multi-channel perfect reconstruction filter bank. The effect of either decomposition is to ensure that the lower-resolution image produced by the pyramid or the primary subband, bears maximum resemblance to the input image. This property is needed in applications requiring hierarchical (scalable, progressive) coding of vector fields. The noiseless case is examined first. Given arbitrary filters in one of the analysis or synthesis stages of the pyramid or the filter bank, the optimal corresponding synthesis, respectively analysis filters, are determined. Further, the globally optimal pairs of analysis and synthesis filters are determined. It is seen that under noiseless or lossless transmission conditions, the two above decomposition methods are optimised by identical families of analysis and synthesis filters. This is not the case if it is assumed that additive transmission noise corrupts the downsampled signal prior to the synthesis stage. This noise may represent transmission noise or the effect of quantisation between the analysis and synthesis stages. Given arbitrary analysis filters and arbitrary noise characteristics, the optimal synthesis filters are then determined for each decomposition method. The results are initially presented for the binary case and subsequently generalised to the case of M-factor pyramids and M-channel filter banks.