Alphabet-constrained rate-distortion theory is extended to coding of sources with memory. Two different cases are considered: when only the size of the codebook is constrained and when the codevector values are also held fixed. For both cases, nth-order constrained-alphabet rate-distortion functions are defined and a convergent algorithm for their evaluation is presented. Specific simulations using AR(1) sources show that performance near the rate-distortion bound is possible using a reproduction alphabet consisting of a small number of codevectors. It is also shown that the additional constraint of holding the codevector values fixed does not degrade performance of the coder in relation to the size-only constrained case. This observation motivates the development of a fixed-codebook vector quantizer, called the alphabet- and entropy-constrained vector quantizer, the performance of which is comparable to the entropy-constrained vector quantizer. A number of examples using an AR(1) and a speech source are presented to corroborate the theory.>

An algorithm given by R.P. Rao (see Proc. IEEE Inter. Telecomm. Symp., Sept. 1990) for multirate vector quantization is used for coding image pyramids. The algorithm, called alphabet- and entropy-constrained vector quantization (AECVQ), operates by optimally choosing sub-codebooks from a large generic codebook. In terms of performance and complexity, AECVQ equals the best single-rate VQ. Excellent results at 1 b/p and below, judged both visually and using a peak-to-peak SNR criterion, have been obtained by coding image pyramids using the AECVQ algorithm. these results demonstrate significant improvements over existing schemes. Although an AECVQ based image coding scheme is considerably complex, it can be implemented in real time using current VLSI technology.>

An information-theoretic analysis of multiresolution pyramid structures is presented. The analysis is carried out using the concept of spectral entropy which, for Gaussian sources, is linearly related to the differential entropy. The spectral entropy is used to define the spectral roughness measure that, in turn, is an indicator of the amount of memory in a source. The more the memory in a source, the greater is the value of its spectral roughness measure. The spectral roughness measure also plays an important role in lower bounding the rate-distortion function. The main result is that the spectral roughness measure of a pyramid structure is less than that of the original fullband process. This means that substantial reduction in bits can be obtained by merely representing a source as a pyramid. Simulations using one-dimensional and two-dimensional sources verifying these claims are presented.>