Comparison between unitary and causal approaches to backward adaptive transform coding of vectorial signals

In a transform coding framework, we compare the optimal causal approach (LDU, Lower-Diagonal-Upper) to the optimal unitary approach (Karhunen-Loeve Transform, KLT). The criterion of merit used for this comparison is the coding gain, defined for a transformation T as the ratio of the average distortion obtained with the identity transformation over the average distortion obtained with T. Both transforms are known to yield the same gain when they are computed on the signal covariance matrix R. The purpose of this paper is to compare the behavior of these two transformations when the ideal transform coding scheme gets perturbed, that is, when only an estimate R + ΔR of R is known. In this case, not only the transformation itself will be perturbated, but also the bit allocation mechanism. We compare the two approaches in two cases. Firstly, ΔR is caused by a quantization noise: the coding scheme is based on the statistics of the quantized data. We find that the coding gain in the unitary case is higher than in the causal case. In a second case, ΔR corresponds to an estimation noise: the coding scheme is based on an estimate of R based on a finite amount of available data. In this case, both causal and unitary approaches are strictly equivalent, because of the unimodularity and decorrelating properties of the transformations. Simulations results confirming the predicted behavior of the coding gains with perturbations are reported.