On the sizes of Voronoi cells in entropy-constrained vector quantization

Voronoi cells for vector quantization subject to an entropy constraint are considered. It is shown that the constraint on the output entropy leads to a weaker and even vanishing dependency of the Voronoi cell's volume on the probability density function. Using some simplifying assumptions like linearization of a small part of the r-dimensional input space and modeling of the cell shapes as hyperspheres leads to an analytic expression of the quotient of the volumes of two neighboring Voronoi cells. The results confirm the use of entropy coded lattice vector quantizers with optimized reproduction vectors in cases of vanishing dependency and may in other cases be exploited for the design of vector companders to be used in conjunction with lattice vector quantization.