A theoretical high-rate analysis of causal versus unitary online transform coding

Backward adaptive or "online" transform coding (TC) of Gaussian sources is investigated. The Karhunen-Loe`ve transform (KLT, unitary approach) and the causal transform (CT, causal approach) are compared in this context. When the covariance matrix R__x of the source is used in the TC scheme, KLT and CT present similar coding gains at high rates. The aim of this study is to model analytically the behavior of these two coding structures when the ideal TC scheme gets perturbed, that is, when only a perturbed value R__x+ΔR is known at the encoder. In the online TC schemes considered here, this estimate is used to compute both the transform and the bit assignment. ΔR is caused by two noise sources: estimation noise (finite set of available data at the encoder) and quantization noise (quantized data at the decoder). Furthermore, not only does the transformation itself get perturbed, but also the bit assignment does as well. In this framework, theoretical expressions for the coding gains in both the unitary and the causal cases are derived under the high-rate assumption.