A Zador-Like Formula for Quantizers Based on Periodic Tilings

N. J. A. Sloane and Vinay A. Vaishampayan  
Information Sciences Research Center 
AT&T Shannon Lab 
Florham Park, New Jersey 07932-0971, USA 


ABSTRACT

We consider Zador's asymptotic formula for the distortion-rate 
function for a variable-rate vector quantizer in the high-rate case.
This formula involves the differential entropy of the source,
the rate of the quantizer in bits per sample, and a
coefficient G which depends on the geometry of the quantizer
but is independent of the source.
We give an explicit formula for G in the case when the
quantizing regions form a periodic tiling of n-dimensional space,
in terms of the volumes and second moments of the
Voronoi cells.

As an application we show, extending earlier work of Kashyap and 
Neuhoff, that even a variable-rate three-dimensional quantizer
based on the ``A15'' structure is still inferior to a quantizer
based on the body-centered cubic lattice.

We also determine the smallest covering radius of such a structure.

Keywords: vector quantizer, Zador bound, distortion-rate function,
Voronoi cell, A15 structure, optimal quantizer, optimal covering, honeycomb.

2000 Mathematical Subject Classification: 11H31, 11H06, 52A99, 94A34.


For the full paper see
http://www.research.att.com/~njas/doc/zador.pdf or
http://www.research.att.com/~njas/doc/zador.ps