Databases Reference
In-Depth Information
use an adaptive strategy. One of the more popular approaches to adaptive quantization is the
Jayant quantizer. We also looked at the issues involved with entropy-coded quantization.
Further Reading
With an area as broad as quantization, we had to keep some of the coverage rather cursory.
However, there is a wealth of information on quantization available in the published literature.
The following sources are especially useful for a general understanding of the area:
1.
A very thorough coverage of quantization can be found in
Digital Coding of Waveforms
,
by N.S. Jayant and P. Noll [
134
].
2.
The paper “Quantization,” by A. Gersho, in
IEEE Communication Magazine
, September
1977 [
127
], provides excellent tutorial coverage of many of the topics listed here.
3.
The original paper by J. Max, “Quantization for Minimum Distortion,”
IRE Transactions
on Information Theory
[
122
], contains a very accessible description of the design of
pdf
-optimized quantizers.
4.
A thorough study of the effects of mismatch is provided by W. Mauersberger in [
135
].
9.9 Projects and Problems
1.
Show that the derivative of the distortion expression in Equation (
18
) results in the
expression in Equation (
19
). You will have to use a result called Leibnitz's rule and the
idea of a telescoping series. Leibnitz's rule states that if
a
(
t
)
and
b
(
t
)
are monotonic,
then
δ
δ
b
(
t
)
b
(
t
)
δ
f
(
x
,
t
)
)
δ
b
(
t
)
)
δ
a
(
t
)
f
(
x
,
t
)
dx
=
dx
+
f
(
b
(
t
),
t
−
f
(
a
(
t
),
t
(74)
δ
δ
δ
t
t
t
t
a
(
t
)
a
(
t
)
2.
Use the program
falspos
to solve Equation (
19
) numerically for the Gaussian and
Laplacian distributions. You may have to modify the function
func
in order to do this.
3.
Design a 3-bit uniform quantizer (specify the decision boundaries and representation
levels) for a source with a Laplacian
pdf
, with a mean of 3 and a variance of 4.
4.
The pixel values in the Sena image are not really distributed uniformly. Obtain a histogram
of the image (you can use the
hist_image
routine), and using the fact that the quantized
image should be as good an approximation as possible for the original, design 1-, 2-, and
3-bit quantizers for this image. Compare these with the results displayed in Figure
9.7
.
(For better comparison, you can reproduce the results in the topic using the program
uquan_img
.)
5.
Use the program
misuquan
to study the effect of mismatch between the input and
assumed variances. How do these effects change with the quantizer alphabet size and
the distribution type?
6.
For the companding quantizer of Example 9.6.1, what are the outputs for the following
inputs:
0.3? Compare your results with the case when the input
is directly quantized with a uniform quantizer with the same number of levels. Comment
on your results.
−
0.8, 1.2, 0.5, 0.6, 3.2,
−
