On the Minimum Distortion of Quantizers with Heterogeneous Reproduction Points

In quantization theory, one typically works with a unique distortion function (e.g. the squared-error distortion function) that quantifies the cost of quantizing a given source sample to any given reproduction point of the quantizer. Many applications, however, induce quantization problems where different distortion functions should be associated with different reproduction points. In this paper, we consider the case where the distortion of a given reproduction point is the squared distance to the source sample weighted by a factor that varies from one reproduction point to another. For a uniform distribution of source samples, we determine the corresponding optimal scalar quantizers and their distortions. We also find upper and lower bounds on the distortion of optimal vector quantizers. For non-uniform distributions, we provide a high resolution analysis of the minimum possible distortion. As a byproduct of our analysis, we show that for certain distributions of weights, a tessellation of non-congruent quantization cells can outperform tessellations of congruent polytopes. This suggests that Gersho's conjecture cannot be extended to the case of squared-error distortion functions with weighted reproduction points.