On the existence of optimal quantizers (Corresp.)

The existence of optimal
N
-level quantizers is demonstrated for several classes of quantizers including block quantizers, memoryless quantizers, and uniform step-size quantizers. No assumptions whatsoever are made regarding the input probability distribution. The distortion measure considered generalizes many of the commonly used error-based distortion measures, including mean squared-error distortion and
r
th law distortion.