Lagrangian Vector Quantization With Combined Entropy and Codebook Size Constraints

In this paper, the Lagrangian formulation of variable-rate vector quantization is extended to quantization with simultaneous constraints on entropy and codebook size, including variable- and fixed-rate quantization as special cases. The formulation leads to a Lloyd quantizer design algorithm and generalizations of Gersho's approximations characterizing optimal performance for asymptotically large rate. A variation of Gersho's approach is shown to yield rigorous results partially characterizing the asymptotically optimal performance.