Image Processing Reference
In-Depth Information
A
c
for low SNR, cannot be a valid estimator of
A
unless a large number of data
points are used for the estimation. Therefore, in practice,
A
c
will only be a useful
estimator if the SNR is high.
However, even if the condition of high SNR is met, the use of as an
estimator of
A
should still not be recommended, because the results obtained are
biased because of the square root operation in Equation 4.65. For high SNR, the
expectation value of
A
c
A
c
is approximately given by (see
Appendix
)
.
σ
2
ˆ
AA
≈−
1
(4.91)
E
c
2
NA
2
Equation 4.91 is valid for high SNR. The bias appears in the second term of
Equation 4.72. Note that it decreases with increasing SNR and increasing number
of data points
N
. Furthermore, as
[
ˆ
]
A
A
c
A
c
E
2
=
2
, the variance of
can be easily
found from Equation 4.91. It is given by
()
=
2
ˆ
ˆ
ˆ
−
Var
A
A
2
A
(4.92)
E
E
c
c
c
σ
2
σ
4
≈−
(4.93)
NNA
22
Then, it follows that the MSE is given by
≈
σ
2
N
MSE
.
(4.94)
4.4.3.1.4 ML Estimation
In what follows, we will consider the ML estimator of
A
from a set
N
Rician-
distributed magnitude data points
m
=
(
1
mm
N
,...,
)
.
The joint PDF
p
m
is given by
N
m
22
2
A
m
m
+
A
∏
n
−
p
=
n
e
I
n
ε
()
m
,
(4.95)
2
σ
m
0
n
σ
σ
2
2
n
=
1
where are the magnitude variables corresponding to the magnitude obser-
vations . The ML estimate of
A
is constructed by substituting the available
observations in the expression for the joint PDF (Equation 4.75) and max-
imizing the resulting function
L
(
A
), or equivalently ln
L
(
A
), with respect to
A
.
Hence, it follows that
{}
m
n
{
m
n
{}
m
n
N
N
N
∑∑∑
1
m
−
m
2
+
A
2
Am
n
n
n
ln
L
=
ln
+
l
n
I
,
(4.96)
0
σ
2
σ
σ
2
2
2
n
=
n
=
1
n
=
1
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