Optimal Piecewise Uniform Vector Quantization of the Memoryless Two Dimensional Laplacian Source

In this paper we will present a simple and complete asymptotical analysis of an optimal piecewise uniform quantization (PUQ) of two-dimensional memoryless Laplacian source with the respect to distortion (D) i.e. the mean-square error (MSE). PUQ is based on uniform vector quantizers. PUQ consists of L different uniform vector quantizers. Uniform quantizer optimality conditions and all main equations for number of optimum number of output points and optimal number of levels for each partition are presented. These systems, although not optimal, may have asymptotic performance arbitrary close to the optimum. Furthermore, their analysis and implementation can be simpler than those of optimal systems. PUQ has complexity implementation between optimal nonuniform quantization (NQ) and uniform quantization (UQ). We derive the optimal granular distortion D_g ^opt (i) for each partition in a closed form. The two- dimensional space is partitioned using rectangular cells.