Codecell convexity in optimal entropy-constrained vector quantization

Properties of optimal entropy-constrained vector quantizers (ECVQs) are studied for the squared-error distortion measure. It is known that restricting an ECVQ to have convex codecells may preclude its optimality for some sources with discrete distribution. We show that for sources with continuous distribution, any finite-level ECVQ can be replaced by another finite-level ECVQ with convex codecells that has equal or better performance. We generalize this result to infinite-level quantizers, and also consider the problem of existence of optimal ECVQs for continuous source distributions. In particular, we show that given any entropy constraint, there exists an ECVQ with (possibly infinitely many) convex codecells that has minimum distortion among all ECVQs satisfying the constraint. These results extend analogous statements in entropy-constrained scalar quantization. They also generalize results in entropy-constrained vector quantization that were obtained via the Lagrangian formulation and, therefore, are valid only for certain values of the entropy constraint.