Optimal construction of subband coders using Lloyd-Max quantizers

A new method is presented for the analysis of the effects of Lloyd–Max quantization in subband filterbanks and for the optimal design of such filterbanks. A rigorous statistical model of a vector Lloyd–Max quantizer is established first, consisting of a linear time-invariant filter followed by additive noise uncorrelated /with the input. On the basis of this model, an expression for this variance of the error of a subband coder using Lloyd–Max quantizers is explicitly determined. Given analysis filters that statistically separate the subbands, it is shown that this variance is minimized if the synthesis filters are chosen, which would achieve perfect reconstruction in lossless coding. The globally optimum such filterbank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. An alternative design method is also evaluated and optimized. In this, the errors correlated with the signal are set to zero, leaving a random error residue uncorrelated with the signal. This design method is optimized by choosing the analysis filters so as to minimize the random error variance. The results are evaluated experimentally in the realistic setting of a logarithmically split subband image coding scheme.