On the suboptimality of orthogonal transforms for single- and multi-stage lossless transform coding

Orthogonal transforms are compared with the causal transform in lossless transform coders. For single-stage lossless coding, it was shown in [1] that the integer-to-integer implementation of the best orthogonal decorrelating transform, the KLT, leads to lower compression performance than its causal counterpart. In this work, we pursue this analysis in the framework of a multi-stage lossless coding scheme, which yields a low resolution (lossy) signal, and an error signal. This scheme allows one to choose the respective bitrates of both complementary signals, depending for example on the bandwidth of the transmission link. We show that the causal approach allows one to code the data (almost) without causing any excess bitrate as compared with a single-stage coder, whereas for orthogonal transforms, the price paid for the multiresolution approach is a bitrate penalty of 0:25 bit per sample. This excess bitrate is due to a "gaussianization effect" of the transforms. Also, the approach based on the causal transform allows one to easily switch between a single- or a multi-stage compressor. Moreover, in the framework of interchannel redundancy removal, this approach allows one to easily fix the distortion and rate for both the low resolution and the error signal of each channel, by using different stepsizes in the quantization stage. Any of the channels may, as a particular case, be chosen to be directly losslessly coded. Finally, a side advantage of the causal approach is that entropy coding of the error signal is made very simple since for odd quantization stepsizes, the discrete error sources are uniformly distributed, so that the optimal codewords have the same length, and fixed rate coding is optimal.