When the mean square distortion measure is used, asymptotically optimal quantizers of uniform bivariate random vectors correspond to the centers of regular hexagons (Newman, 1982), and if the random vector is non-uniform, asymptotically optimal quantizers are the centers of piecewise regular hexagons where the sizes of the hexagons are determined by a properly chosen density function (Su and Cambanis, 1996). This paper considers bivariate random vectors with finite # th (#>0) moment. If the # th mean distortion measure is used, a complete characterization of the asymptotically optimal quantizers is given. Furthermore, it is shown that the procedure introduced by Su and Cambanis ( 1996) is also asymptotically optimal for every #>0. Examples with a normal distribution and a Pearson type VII distribution are considered.
1997 Academic Press
1. Introduction
Assume that a random vector is distributed over a region D in the d dimensional Euclidean space R d . Optimal quantizers of the random vector is a finite set of points in D such that the # th mean distance of the random vector from this set is minimized, assuming that the # th moment of the random vector exists. Optimal quantizers are also called representative points [1] or principal points of the random vector [2,9]. If the number of points in this set is N, then they are termed as quantizers of level N or N principal points. Flury [2] and Tarpey [9] studied the problem of finding the N principal points for a fixed N and mainly elliptical type distributions. Linde et al. [5] gave an algorithm of simulation to search for the optimal quantizers which is the first method for general distributions. An asymptotic approach has also been extensively investigated in order to circumvent the difficulty for fixed N, which is to find a sequence of points so that their distance from the random vector is minimized as N goes to infinity.
Zador [8] characterized the performance of asymptotically optimal quantizers for every positive # and every dimension d. A more precise description of the performance of asymptotically optimal quantizers, for the mean square error (#=2) and in the two dimension (d=2), was given by Fejes Toth [10] earlier. However, their proofs are not constructive. A article no. MV971663 67 0047-259Xร97 25.00