Image Processing Reference
In-Depth Information
Since the PDF is symmetric, the optimum quantizer would be symmetric, that is,
d
0
¼
1
d
1
¼dd
2
¼
0
d
3
¼dd
4
¼
1
and
r
1
þ
r
2
2
r
2
þ r
3
2
r
3
þ
r
4
2
d
1
¼
¼d,
d
2
¼
¼
0 and d
3
¼
¼ d
Therefore the reconstruction levels are
r
1
¼
2d r
2
r
3
¼r
2
r
4
¼
2d þ r
2
where r
2
and r
1
are given by
Ð
0
d
xp(x)dx
Ð
0
i
Ð
0
d
x(1
þx)dx
2d
2
3d
r
2
¼
d
p(x)dx
¼
Ð
0
i
d
(1
þx)dx
¼
6
3d
and
Ð
d
Ð
d
1
xp(x)dx
Ð
d
i
1
x(1
þx)dx
3d
2
2d
3
1
r
1
¼
1
p(x)dx
¼
þx)dx
¼
Ð
d
i
3d
2
6d þ
3
(1
1
The above equations can be reduced to
(
2d
2
3d
r
2
¼
6
3d
3d
2
2d
3
2d r
2
¼
1
3d
2
6dþ
3
Eliminating r
2
, we have
d
4
5d
3
8d
2
þ
5d þ
1
¼
0
Factoring this polynomial yields
(d
1)(d
1)(d
0
:
382)(d
2
:
618)
¼
0
The only acceptable solution is d ¼
0
:
382, which results in r
2
¼
0
:
176. There-
fore, the decision boundaries and the reconstruction levels are
d
0
¼
1
d
1
¼
0
:
382
d
2
¼
0
d
3
¼
0
:
382
d
4
¼
1
r
1
¼
0
:
588 r
2
¼
0
:
176 r
3
¼
0
:
176 r
4
¼
0
:
588
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