Optimal entropy-constrained scalar quantization of a uniform source

Optimal scalar quantization subject to an entropy constraint is studied for a wide class of difference distortion measures including rth-power distortions with r>0. It is proved that if the source is uniformly distributed over an interval, then for any entropy constraint R (in nats), an optimal quantizer has N=[e/sup R/] interval cells such that N-1 cells have equal length d and one cell has length c/spl les/d. The cell lengths are uniquely determined by the requirement that the entropy constraint is satisfied with equality. Based on this result, a parametric representation of the minimum achievable distortion D/sub h/(R) as a function of the entropy constraint R is obtained for a uniform source. The D/sub h/(R) curve turns out to be nonconvex in general. Moreover, for the squared-error distortion it is shown that D/sub h/(R) is a piecewise-concave function, and that a scalar quantizer achieving the lower convex hull of D/sub h/(R) exists only at rates R=log N, where N is a positive integer.