Fixed-rate entropy-coded vector quantization using linear (zero-one) programming

Consider two sets of points X, A and their n-fold cartesian products {A}ⁿ, {X}n. A non-negative cost is associated with each element of A. A measure of distance is defined between an element of A and an element of X. It is assumed that the cost and also the distance in the n-fold space has an additive property. The shaped set is composed of a subset of elements of {A}ⁿ of the least cost. Decoding of an element x∈{X}ⁿ is the process of finding the element of the shaped set which has the minimum distance to x. Using the additivity property of cost and distance measures, the decoding problem is formulated as a linear program. Using the generalized upper bounding technique of linear programming in conjunction with some special features of the problem, we present methods to substantially reduce the complexity of the corresponding simplex search. The proposed method is used for the fixed-rate entropy-coded vector quantization of a Gaussian source. For n=128 (space dimensionality) using 8 points per dimension and for a rate of 2.5 bits/dimension, we need about 52 additions, 87 comparisons, 0.2 divisions, and 0.4 multiplications per dimension to achieve SNR=13.31 dB (the bound obtained from the rate-distortion curve is 13.52 dB). This is substantially less complex than the traditional methods based on the dynamic programming