Image Processing Reference
In-Depth Information
P
(
e
)
1
Δ
e
Δ
Δ
-
2
2
FIGURE 2.15
PDF of quantization noise.
Figure 2.15. In such a case, the quantization error for any signal sample would be
uniformly distributed between
2
and
2
.
The quantization noise power is the variance of this uniform distribution, that is,
2
1
ð
2
e
2
1
D
d
e
¼
D
2
e
¼
e
2
P
(
e
)d
e
¼
s
(
2
:
44
)
12
1
2
A B-bit quantizer for a signal in the range of
[
x
min
x
max
]
has a step size of
x
max
x
min
2
B
D ¼
(
2
:
45
)
The signal to quantization noise power ratio (SQNR) is given by
SQNR ¼
s
2
s
12
s
2
s
s
2
2B
(
x
max
x
min
)
12
s
2
n
¼
2
¼
(
2
:
46
)
2
s
2
D
The SQNR in decibels (dB) is
¼
2
s
s
2
s
(SQNR)
dB
¼
10
log
6B
þ
10
log
10
12
s
20
log
10
(
x
max
x
min
)(
2
:
47
)
s
2
n
Therefore, the SQNR increases by 6 dB for each additional quantization bit.
2.4.2.3 Optimum Minimum Mean-Square Error Quantizer
Assume that signal X has a PDF p
(
x
)
.Ifp
(
x
)
is uniformly distributed between x
min
and x
max
, then the optimum quantizer in the minimum mean-square error (MMSE)
sense is a uniform quantizer. However, if the PDF of X is nonuniform, the optimum
quantizer will also be nonuniformly spaced. When the PDF of the input signal is not
available, the smoothed data histogram can be used as an estimate of the PDF. In
general, one would expect that in order to minimize the average error the quantizer
structure should be
finer around the peaks of the signal distribution and coarser in
regions where the signal occurs infrequently. Mathematically, the reconstruction and
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