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

Orthogonal transforms are compared with the casual transform in lossless transform coders. For single-stage lossless coding, it was previously shown that the integer-to-integer implementation of the best orthogonal decorrelating transform, the KLT, leads to lower compression performance than its casual counterpart. The analysis in the framework of a multi-stage lossless coding scheme, which yields a lossy coded signal, and an error signal was pursued. This scheme allows one to choose the respective bit rates of both complementary signals, depending for example on the bandwidth of the transmission link. It was shown that the casual approach presents several advantages w.r.t. its orthogonal counterparts. For orthogonal transforms, the price paid for the multiresolution approach is a bit rate penalty of 0.25 bit per sample. This excess bit rate is due to a "gaussianization effect" of the transforms. Firstly, it was shown under the assumptions of smooth p.d.f.s for the sources, and of high resolution for the lossy coded signal, that the casual approach allows one to code the data without causing any excess bit rate as compared with a single-stage coder. Secondly, the approach based on the casual transform allows one to easily switch between a single- or a multi-stage compressor. Thirdly, in the framework of interchannel redundancy removal, this approach allows one to easily fixed 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 casual 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.