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Optimal entropy-constrained scalar quantization of a uniform source

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2 Author(s)
A. Gyorgy ; Dept. of Math. & Stat., Queen's Univ., Kingston, Ont., Canada ; T. Linder

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=[eR] interval cells such that N-1 cells have equal length d and one cell has length c⩽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 Dh (R) as a function of the entropy constraint R is obtained for a uniform source. The Dh(R) curve turns out to be nonconvex in general. Moreover, for the squared-error distortion it is shown that D h(R) is a piecewise-concave function, and that a scalar quantizer achieving the lower convex hull of Dh(R) exists only at rates R=log N, where N is a positive integer

Published in:

IEEE Transactions on Information Theory  (Volume:46 ,  Issue: 7 )