Image Processing Reference
In-Depth Information
Setting Equation 4.48 and Equation 4.59 to zero yields the ML estimators of
A
and
ϕ
:
2
2
N
+
N
1
∑∑
=
w
w
,
(4.60)
A
rn
,
in
,
ML
N
n
=
1
n
=
1
∑
∑
N
w
in
,
ϕ
=
arctan
n
=
1
.
(4.61)
N
ML
w
rn
,
n
=
1
A
Notice that the estimator is obtained by taking the square root of the
quadratic sum of two Gaussian-distributed variables. Hence,
ML
A
is Rician dis-
ML
tributed [18].
4.4.2.1.3 MSE
As is Rician distributed, we find for its MSE, which is the sum of the bias
(
b
) squared and the estimator's variance (cf. Equation 4.44):
A
ML
( )
( )
2
+
MSE
=
b
A
Var
(4.62)
A
ML
ML
=−
+
2
AA
A
2
2
2
σ
2
/
N
(4.63)
E
ML
(
)
+
−
=
2
AA
2
σ
2
/
N
,
(4.64)
E
A
ML
where the first moment of its PDF can be deduced from Equation 4.17:
+
π
2
NA
2
NA
2
NA
2
NA
2
NA
−
=
σ
N
e
1
+
I
I
.
(4.65)
E
2
A
4
σ
0
1
2
2
σ
4
σ
2
σ
4
σ
ML
2
2
2
2
4.4.2.2
Region of Constant Amplitude and Different Phases
Now assume that the complex data
c
=
{(
ww
,
)}
have an underlying signal
rn
,
in
,
amplitude
A
and arbitrary phase values
ϕ
1
,
…
,
ϕ
N
. Then, the joint PDF of the
p
c
,
complex data,
is given by
N
N
2
2
=
1
(
ω
rn
A
,
−
cos
ϕ
)
(
ω
,
−
A
sin
ϕ
)
∏
n
i n
n
−
−
p
({(
ww
,
)}
| ,
A
{
ϕ
})
e
e
.
(4.66)
2
2
2
σ
2
σ
c
rn
,
in
,
n
2
πσ
2
n
=
1
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