Design of General Entropy-Constrained Successively Refinable Unrestricted Polar Quantizer

Abstract

This paper presents an algorithm for the optimal design of general entropy-constrained successively refinable unrestricted polar quantizer, i.e., with arbitrary number $L$ of refinement levels, for bivariate circularly symmetric sources. The optimization problem is formulated as the minimization of a weighted sum of distortions and entropies for the scenario where the magnitude quantizers’ thresholds are confined to a predefined finite set. The proposed solution algorithm is globally optimal. It involves $L$ stages, where each stage corresponds to an unrestricted polar quantizer (UPQ) level, and includes solving the minimum-weight path problem for multiple node pairs in a series of weighted directed acyclic graphs. Additionally, we derive an upper bound $P^{(l)}_{\max }, l\in [1:L]$ , on the possible number of phase levels in any phase quantizer of the $l$ -th level UPQ, which grows linearly with $l$ . The time complexity of the proposed approach is $O(L^{2}K^{3} P^{(1)}_{\max })$ , where $K$ is the cardinality of the predefined set of possible magnitude thresholds. Finally, the experimental results for $L=3$ demonstrate the effectiveness in practice of the proposed scheme.