Gruber P. Convex and Discrete Geometry

33 Optimum Quantization 497
encoder now works as follows: If x is a signal, it finds out to which set of the dissection
x belongs. If it belongs to Ci, the encoder assigns the code-word ci, here also
called code-vector, tox. In case of ambiguity, which can occur only for x in a set of
Jordan measure 0, choose any code-vector ci with x ∈ Ci. Common measures for the
quality of the thus defined encoder or (vector-)quantizer on C can be described as
follows: Let α>0. The corresponding (average) distortion of the encoder is defined
to be
�
�
�x − ci� α dx.
i
Ci
How should the cells Ci and the code-vectors ci be chosen in order to minimize
the distortion? Given the code-book {c1,...,cn}, the distortion is minimized if
C1,...,Cn are the Dirichlet–Voronoĭ cells in C corresponding to the code-book
{c1,...,cn}. Then the distortion is easily seen to be
�
min
i∈{1,...,n} {�x − ci� α }dx.
Thus the minimum distortion is given by
�
min
c∈S {�x − c�α }dx.
C
inf
S⊆E d
#S=n
C
The literature on vector quantization began with fundamental papers of Shannon
[929] and Zador [1034, 1035]. Shannon considered the minimum distortion as
d →∞, while in Zador’s high resolution theory the case n →∞is investigated.
An immediate consequence of Zador’s theorem proved earlier, actually a reformulation
of it, is the following result of Zador [1034, 1035].
Corollary 33.3. Let α>0. Then there is a constant δ = δα,d > 0, depending only on
α and d, such that the following holds: let C ⊆ Ed be a compact, Jordan measurable
body. Then the minimum distortion of a vector-quantizer on C with n code-words is
asymptotically equal to
α+d
V (C) α
δ as n →∞.
Remark. See Gruber [443] for more precise results.
Minimum Error of Numerical Integration Formulae
n α d
Let J ⊆ Ed be a compact Jordan measurable set with V (J) >0 and F aclassof
Riemann integrable functions on J. For given sets of n nodes N ={p1,...,pn} ⊆J
and n weights W = {w1,...,wn} ⊆ R the error of the numerical integration
formula �
f (x) dx ≈ �
f (pi)wi for f ∈ F
J
i