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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