Design of Threshold-Constrained Indirect Quantizers

TL;DR

This paper develops methods for designing optimal indirect quantizers under hardware constraints, deriving necessary conditions and algorithms for scalar and general cases to minimize mean-squared error.

Contribution

It introduces generalized Lloyd-Max conditions for constrained indirect quantization and proposes iterative and non-iterative algorithms for their design.

Findings

Derived necessary conditions for optimal constrained quantizers.

Proposed an iterative algorithm based on Lloyd-Max principles.

Developed a non-iterative dynamic programming algorithm for scalar cases.

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.