On the design of an optimal quantizer

The problem of designing an optimal quantizer with a fixed number of levels for a wide class of error weighting functions and an arbitrary distribution function is discussed. The existence of an optimal quantizer is proved, and a two-stage algorithm for its design is suggested. In this algorithm, at the first stage, Lloyd´s iterative Method I is applied for reducing the region where, at the second stage, the search for an optimal quantizer is performed using a hybrid of the dynamic programming algorithm and the Lloyd-Max algorithm, which achieves the absolute optimality of dynamic programming with much less computational effort. For a continuous distribution with log-concave density and an increasing convex weighting function of the absolute quantization error, a reliable method is presented to compute the parameters of the optimal quantizer with a known precision using a generalization either of Lloyd´s Method I or of the Lloyd-Max algorithm