New Results on Robust Quantization

In this paper we consider the design of robust block quantizers when the number of quantization levels is large. The
r
th power distortion measure is utilized through the convenient expression developed by Bennett and Gersho. The robust design is formulated as a two-person game, and it is shown that for convex families of signal probability density functions there is a saddle point solution. The evaluation of the robust solution amounts to determining the maximum
s
-norm element in the class of signal densities. We then develop specific solutions for three classes of pdf: a) the class specified by generalized moment contraints, b) the class of εcontaminated densities, which has been a popular model in robust signal detection, and c) the class specified by upper and lower bounds to the probability density function of the signal. For high-quality quantization under fixed output entropy, the quantizer is uniform and the resulting distortion is an increasing function of the source entropy. The least favorable distribution is then the one having maximum entropy. For the ε-contaminated family and the "banded" family
(c)
, we derive the maxentropic distributions.