Asymptotic optimality of scalar Gersho quantizers

TL;DR

This paper proves that scalar Gersho quantizers, which allocate equal distortion contribution per codecell, are asymptotically optimal for non-atomic scalar distributions, confirming a long-standing hypothesis.

Contribution

It establishes the existence of Gersho quantizers in the scalar case and proves their asymptotic optimality, advancing the theoretical understanding of quantizer design.

Findings

Gersho quantizers exist for non-atomic scalar distributions.

Gersho quantizers are asymptotically optimal.

Equal contribution to distortion per codecell is asymptotically achieved.

Abstract

In his famous paper [7] Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We will show that such quantizers of Gersho type - or Gersho quantizers for short - exist for non-atomic scalar distributions. As a main result we will prove that Gersho quantizers are asymptotically optimal.