Image Processing Reference
In-Depth Information
The MMSE is
d
i
ð
ð
ð
0
:
382
0
D ¼
X
L
(x r
i
)
2
p(x)dx ¼
588)
2
p(x)dxþ
176)
2
p(x)dx
(xþ
0
:
(x þ
0
:
i¼
1
d
i1
1
0
:
382
ð
ð
0
:
382
1
176)
2
p(x)dx þ
588)
2
p(x)dx
þ
(x
0
:
(x
0
:
0
0
:
382
This can be simpli
ed as
2
ð
0
:
382
2
ð
1
176)
2
(1
588)
2
(1
D ¼
(x
0
:
x)dx þ
(x
0
:
x)dx ¼
0
:
01548
0
0
:
382
Notice that the resulting MSE from Lloyd
Max quantizer is smaller than the MSE
obtained in part (a) from uniform quantizer.
-
Example 2.6
Design a 3 bit optimum quantizer for a signal with unit variance Laplacian distri-
bution. The unit variance Laplacian distribution is given by the following PDF.
p
2
e
p
jxj
P(x)
¼
1< x < 1
S
OLUTION
Using the Lloyd
Max iterative algorithm, the following values are obtained for a 3
bit optimum quantizer
-
d
0
¼1 d
1
¼
2
:
3654
d
2
¼
1
:
2474
d
3
¼
0
:
5313
d
4
¼
0
d
5
¼
0
:
5313 d
6
¼
1
:
2474 d
7
¼
2
:
3654 d
8
¼1
r
1
¼
3
:
0658
r
2
¼
1
:
664
r
3
¼
0
:
83
r
4
¼
0
:
2327
r
5
¼
0
:
2327 r
6
¼
0
:
83 r
7
¼
1
:
664 r
8
¼
3
:
0658
The PDF of the Laplacian distribution and its 3 bit quantization levels are shown in
Figure 2.17.
2.4.2.4 Perceptual Quantization
Digital images are obtained either through the process of scanning a raw continuous
tone image (e.g., facsimile) or direct acquisition (e.g., digital cameras). In digital
cameras, the sampling is accomplished by a CCD sensor array. At each pixel, a
photodiode converts the incident photons into electrons that are subsequently counted
as the pixel intensity. In many imaging systems, it is customary to quantize the pixel
value to 8 bits (256 levels), since it is argued that the human eye cannot distinguish
more than 100
200 brightness levels. Quantization to 12 or even 14 bits is not
-
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