On the Asymptotics of Quantizers in Two Dimensions

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 finiteγth (γ>0) moment. If theγth 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 everyγ>0. Examples with a normal distribution and a Pearson type VII distribution are considered.