Multiple Description Vector Quantization with Lattice Codebooks:
                    Design and Analysis

by

Vinay A. Vaishampayan, N. J. A. Sloane
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
and
Sergio D. Servetto
Ecole Polytechnique F'ed'erale de Lausanne
CH-1015 Lausanne, Switzerland

Abstract

The problem of designing a multiple description vector quantizer
with lattice codebook Lambda is considered.
A general solution is given to a labeling problem which
plays a crucial role in the design of such quantizers.
Numerical performance results are obtained
for quantizers based on the lattices A_2 and Z^i, i=1,2,4,8,
that make use of this labeling algorithm.

The high-rate squared-error distortions for this family of
L-dimensional vector quantizers are then analyzed for a
memoryless source with probability density function p and
differential entropy h(p) < infty.

For any a in (0,1) and rate pair (R,R), it is shown that the
two-channel distortion d_0 and the channel 1 (or channel 2) 
distortions d_s satisfy 
 
lim_{R -> infty} d_0 2^(2R(1+a)) = (1/4) G(Lambda) 2^{2h(p)}
 
and 
 
lim_{R -> infty} d_s 2^(2R(1-a)) = G(S_L) 2^2h(p),
 
where G(Lambda) is the normalized second moment of a Voronoi
cell of the lattice Lambda and G(S_L) is the normalized second
moment of a sphere in L dimensions.


Index Terms: Source Coding, Quantization, Multiple
Descriptions, Lattice Quantization, Vector Quantization,
Hexagonal Lattice, Cubic Lattice.


This paper was published (in a somewhat different form) in
IEEE Trans. Information Theory, 47 (2001), 1718-1734.


For the full version see
http://www.research.att.com/~njas/doc/vinayIT.pdf (pdf) or
http://www.research.att.com/~njas/doc/vinayIT.ps (ps)