Adaptive Quantization and Transform Coding

Adaptive Quantization and Transform Coding by Jun Zhuang Master of Science in Engineering-Electrical Engineering and Computer Sciences University of California at Berkeley Professor Martin Vetterli, Research Advisor Universal lossless codes have been proven to exist [22], and practical universal lossless coding schemes have been constructed [2][3][4]. Even though the existence of uni versal lossy compression algorithms are established in the early 1970's, the development of universal lossy codes remained most in the theoretical domain so far. The first part of this project is motivated by the work by C. Chan and M. Vetterli [12]. The approach here is to first collect some information about the source based on caus al past real samples and then decide whether it is worthwhile to update the coder and send the update as the side information. Unlike in [12], we confine the coder to be a tree struc tured vector quantizer, which has a low complexity. As a simplified case, we showed how to design such an adaptivecoder in the case of scalar quantization. Though similardesign can be extended to the vector quantization case, we concluded that though we achieve low complexity byconfining thecoderto be treestructured, theflexibility to adaptthequantizer suffers as an expense for lowcomplexity, andtheside information is increased drastically. Thus adaptive tree structured quantizerremains interesting in theory and after a brief dis cussion we focusour attention on themorepromising backward adaptive Karhunen-Loeve transform coding scheme. Most the previous work on universal lossy compression are addressed in the vector quantization framework, relativelylittle workhas beendone in the transformcodingframe work. Transform coding techniques are widely used and among various transforms the Karhunen-Loevetransform (KLT) is proven to be the optimal transformunder high resolution. real bitassignment and Gaussian assumptions. In the second part ofthis project, ourobjective is todevelop apractical backward adaptive Karhunen-Loeve transform coding scheme. Traditionally, despite its existence as the "optimal" transform, KLT is seldomused in practical transform coding schemes. Oneof the major difficulties in applying KLTin transform coding is that theKLT issignal dependent andneeds to besentthrough thecom munication channels which is usually too expensive. In this project, weuse a completely different approach. The essence of this approach is that we measure the autocorrelation of the source signal based on the causal past decoded samples and derive the approximated Karhunen-Loeve transform based on the measurement. Because both the encoder and the decoder have access to these samples, nosideinformation is required. Asa further step, we then proved when the source signal can be blocked into i.i. d. Gaussian random vectors, whenthe stepsize of thescalarquantizer used is small, the adapted transform converges to the true Karhunen-Loeve transform when both the number of samples used to measure the autocorrelation and the numberof adaptation steps go to infinity. In practice, our simula tions on AR(1) signal show that for finite number of samples and finite number of adapta tions, the adapted transform is very "close" to the true Karhunen-Loeve transform.