Staggered Quantizers for Perfect Perceptual Quality: A Connection between Quantizers with Common Randomness and Without

TL;DR

By analyzing an idealized setting, this work provides an interpretation of the advantage of dithered quantization in the RDP setting, which further allows to make a conceptual connection between randomized (dithered) quantizers and quantizers without common randomness.

Abstract

The rate-distortion-perception (RDP) framework has attracted significant recent attention due to its application in neural compression. It is important to understand the underlying mechanism connecting procedures with common randomness and those without. Different from previous efforts, we study this problem from a quantizer design perspective. By analyzing an idealized setting, we provide an interpretation of the advantage of dithered quantization in the RDP setting, which further allows us to make a conceptual connection between randomized (dithered) quantizers and quantizers without common randomness. This new understanding leads to a new procedure for RDP coding based on staggered quantizers.

Figures (4)

• Figure 1: 1-bit quantizers on the unit-circle with perfect perceptual quality: "$\times$" indicates a sample realization of $X$; "$\small \bullet$" indicate the distribution of reconstruction $\hat{X}$; red and blue regions indicate the partition region associated with indices $+1$ and $-1$, respectively. In (a), the deterministic encoder is used. The sample is encoded as $+1$ and its reconstruction is distributed uniformly over the red region. In (b), the dithered approach is used, and the reconstruction would be distributed uniformly over the arc centered at the sample. There are no clear partitions in this case, and thus purple is used as a mixture of red and blue regions. In (c), "$\circ$" indicates realizations of negative common randomness $-Z$, and the dithered quantization is viewed as a mixture of uncountably many deterministic quantizers, each associated with a realization of $Z$.
• Figure 2: Staggered quantizers with $1$ bit coding rate and $2$ bits common randomness.
• Figure 3: Staggered quantizers for general probability distributions.
• Figure 4: Quantization of a uniformly distributed source on an interval

Theorems & Definitions (7)

• Theorem 1
• proof : Proof of Theorem \ref{['thm:LN']}
• Theorem 2
• proof : Proof of Theorem \ref{['thm:RDPunitcircle']}
• Theorem 3
• proof : Proof of Theorem \ref{['thm:OptimalQuan']}
• proof : Optimality of Uniform Quantizers in the Proof of Theorem \ref{['thm:OptimalQuan']}