Akfmct-An iterative descent algorithm based on a Lagrangian formulation is introduced for designing vector quantizers having minimum distortion subject to an entropy constraint. These entropy-constrained vector quantizers (ECVQ's) can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion. Experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates (about 1 bit/sample) under the squared error distortion measure, about 1.6 dB improvement in the signal-to-noise ratio can be expected over the best scalar and lattice quantizers when block entropy-coded with blocklength 4. Even greater gains are made over other forms of entropy-coded vector quantizers. For pattern recognition, it is shown that the ECVQ algorithm is a generalization of the k-means and related algorithms for estimating cluster means, in that the ECVQ algorithm estimates the prior cluster probabilities as well. Experiments on multivariate Gaussian distributions show that for clustering problems involving classes with widely different priors, the ECVQ outperforms the k-means algorithm in both likelihood and probability of error.

An algorithm recently introduced by Breiman, Friedman, Olshen, and Stone in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved. These include variable-rate and minimum-entropy tree-structured vector quantization, minimum expected cost decision trees, variable-order Markov modeling, optimum bit allocation, and computer graphics and image processing using quadtrees. A concentration on the first of these and a detailed analysis of variable-rate tree-structured vector quantization are provided. We find that variable-rate tree-structured vector quantization outperforms not only the fixed-rate variety but also full-search vector quantization as well. Furthermore, the "successive approximation" character of variable-rate tree-structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder. This has applications to the problem of buffer overflow.

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Three techniques for variable-rate vector quantizer design are applied to medical images. The first two are extensions of an algorithm for optimal pruning in tree-structured classification and regression due to Breiman et al. The code design algorithms find subtrees of a given tree-structured vector quantizer (TSVQ), each one optimal in that it has the lowest average distortion of all subtrees of the TSVQ with the same or lesser average rate. Since the resulting subtrees have variable depth, natural variable-rate coders result. The third technique is a joint optimization of a vector quantizer and a noiseless variable-rate code. This technique is relatively complex but it has the potential to yield the highest performance of all three techniques.