A Lagrangian formulation of fixed-rate quantization

A Lagrangian formulation of fixed-rate vector quantization is presented. The formulation provides an alternative version of the classic high-rate quantization approximations for fixed-rate codes of Zador (1966), and Bucklew and Wise (1982) which parallels the Lagrangian results for variable-rate codes and it leads to a variation of the classic Lloyd (1982) algorithm for quantizer design. The approach also leads to a natural Lagrangian formulation combining both common rate constraints of alphabet size and entropy, effectively providing a Lagrangian formulation of memory and entropy constrained vector quantization.