Image Processing Reference
In-Depth Information
provides a choice of 256 reconstruction vectors. To show that VQ outperforms SQ,
consider two random variables X
1
and X
2
with a joint PDF p
X
1
, X
2
(
x
1
, x
2
)
given by
1
2
(
x
1
, x
2
) 2 dashed area
p
X
1
, X
2
(
x
1
, x
2
) ¼
(
2
:
52
)
0
otherwise
The marginal densities of random variables X
1
and X
2
are
1
1
2
1
<
x
1
<
1
p
X
1
(
x
1
) ¼
p
X
1
, X
2
(
x
1
, x
2
)d
x
2
¼
(
2
:
53
)
0
otherwise
1
1
1
2
1
<
x
2
<
1
p
X
2
(
x
2
) ¼
p
X
1
, X
2
(
x
1
, x
2
)d
x
1
¼
(
2
:
54
)
otherwise
0
1
Therefore, X
1
and X
2
are uniformly distributed between
1 and
þ
1. They are not
independent
since p
X
1
, X
2
(
x
1
, x
2
) 6¼
p
X
1
(
x
1
)
p
X
2
(
x
2
)
. The
correlation coef
cient
between x
1
and x
2
is
1
E
(
X
1
X
2
)
E
(
X
1
)
E
(
X
2
)
s
X
1
s
X
2
4
0
r ¼
¼
12
q
1
q ¼
0
:
75
(
2
:
55
)
If we quantize X
1
and X
2
separately, each to 1 bit, the optimum quantizer will be a
uniform quantizer with decision boundaries
{
d
1
d
2
d
3
} ¼ {
101
}
and
quantization levels of
0.5 and 0.5. The total average distortion is
ð
ð
0
1
(
x
1
þ
1
0
(
x
1
2
p
(
x
1
)d
x
1
þ
2
p
(
x
1
)d
x
1
D
¼
0
:
5
)
0
:
5
)
ð
ð
0
1
(
x
2
þ
1
0
(
x
2
1
6
2
p
(
x
2
)d
x
2
þ
2
p
(
x
2
)d
x
2
¼
þ
0
:
5
)
0
:
5
)
(
2
:
56
)
The quantization levels for X
1
and X
2
are shown in Figure 2.18. In this case, 2 bits
are used and the average distortion is
6
. The same average distortion can be achieved
using 1 bit VQ. The reconstruction vectors would be r
1
¼
T
and
½
0
:
50
:
5
½
T
as shown in Figure 2.18. As can be seen from Figure 2.18,
the two quantization vectors c and d are wasted if we use SQ.
The performance of the vector quantizer improves by increasing vector dimen-
sion. The design of the MMSE vector quantizer is conceptually similar to the scalar
quantizer but is computationally more complex. The generalized iterative Lloyd
algorithm (also known as the LBG (Linde
r
2
¼
0
:
5
0
:
5
Gray) or K-means algorithm) is
used to design the optimum decision regions and reconstruction vectors.
Buzo
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