Design of Successively Refinable Unrestricted Polar Quantizer

This paper addresses the design of two-stage successively refinable unrestricted polar quantizers for bivariate circularly symmetric sources in the entropy-constrained and fixed-rate cases. The proposed solutions are globally optimal when the thresholds of the magnitude quantizers are confined to finite discretizations of the interval [0, ∞). The algorithm developed for the entropy-constrained case involves a series of stages, including solving the minimum-weight path problem for multiple node pairs in certain weighted directed acyclic graphs. The asymptotical time complexity is O(K
₁
K
₂
²
P
_max
), where K
₁
and K
₂
are the sizes of the sets of possible magnitude thresholds of the coarse and refined unrestricted polar quantizers (UPQs), respectively, while P
_max
is an upper bound on the number of phase levels in any phase quantizer of the coarse UPQ. The solution algorithm for the fixed-rate case is based on solving a succession of dynamic programming problems for multiple coarse quantizer bins. The time complexity in the fixed-rate case amounts to O(K
₁
K
₂
N
²
N
₁
), where N
₁
is the number of cells of the coarse UPQ and N is the ratio between the number of bins of the fine and coarse UPQs. The extensive experimental results on a bivariate circularly symmetric Gaussian source show the effectiveness of the proposed schemes.