Design of Threshold-Constrained Indirect Quantizers

TL;DR

Necessary conditions an optimal quantizer within this class must satisfy are derived, in the form of generalized Lloyd-Max conditions, and an iterative algorithm for the design of such quantizers is proposed.

Abstract

We address the problem of indirect quantization of a source subject to a mean-squared error distortion constraint. A well-known result of Wolf and Ziv is that the problem can be reduced to a standard (direct) quantization problem via a two-step approach: first apply the conditional expectation estimator, obtaining a ``new'' source, then solve for the optimal quantizer for the latter source. When quantization is implemented in hardware, however, invariably constraints on the allowable class of quantizers are imposed, typically limiting the class to \emph{time-invariant} scalar quantizers with contiguous quantization cells. In the present work, optimal indirect quantization subject to these constraints is considered. Necessary conditions an optimal quantizer within this class must satisfy are derived, in the form of generalized Lloyd-Max conditions, and an iterative algorithm for the design of such quantizers is proposed. Furthermore, for the case of a scalar observation, we derive a non-iterative algorithm for finding the optimal indirect quantizer based on dynamic programming.

Figures (12)

• Figure 1: Indirect quantization scenario subject to MSE distortion: An observation $X$, statistically dependent on an unobserved source $S$ is quantized, producing an index $I$, from which the reconstruction $S$ is produced.
• Figure 2: Induced quantization cells when scalar quantization applied to a two-dimensional ($n=2$) observation vector, with $T=4$ thresholds $t_1, t_2, t_3, t_4$.
• Figure 3: Rate-constrained vs. threshold-constrained scalar quantizer.
• Figure 4: Two-step indirect quantization.
• Figure 5: Necessary conditions for optimality for threshold-constrained scalar quantization: the two shaded (purple) cells are the ones affected when adjusting a single threshold ($t_3$).

Theorems & Definitions (14)

• Lemma 1: Indirect SQ, Rate Constraint fine1965optimum
• Remark 1
• Lemma 2: Indirect SQ Subject to a Thresholds Constraint
• Remark 2
• Remark 3
• Claim 1: Indirect VQ, Rate Constraint
• Lemma 3: Indirect Two-Dimensional Quantization, Thresholds Constraint
• Remark 4
• Remark 5
• Lemma 4: Indirect VQ , Thresholds Constraint