Image Processing Reference
In-Depth Information
4.5.2.2.1 CRLB
It can easily be shown that the Fisher information matrix of the data with respect
to the parameters (
A
,
ϕ
1
,
…
,
ϕ
N
,
σ
2
) is given by
I
=−
E
∂
2
ln
p
∂
∂∂
2
ln
p
∂
∂∂
2
ln
p
∂
2
ln
p
c
c
c
c
...
∂
A
2
A
ϕ
A
ϕ
∂∂
A
σ
2
N
1
N
0
...
0
0
00
σ
2
∂
∂∂
2
ln
p
A
∂
2
ln
p
∂
∂∂
2
ln
p
∂
2
ln
p
c
c
c
c
...
A
2
0
...
0
0
ϕ
∂
ϕ
2
ϕ ϕ
ϕσ
∂
2
1
1
1
N
1
σ
2
.
...
...
...
...
...
×
...
...
...
...
...
=
A
2
∂
∂∂
2
ln
p
A
∂
2
ln
p
∂
2
ln
p
∂
∂∂
2
ln
p
...
0
c
c
c
c
...
σ
2
ϕ
∂
ϕϕ
∂
ϕ
2
ϕ σ
2
N
N
1
N
N
N
00
...
0
∂
∂∂
2
ln
p
∂
∂
2
ln
p
∂
∂∂
2
ln
p
∂
∂
2
ln
()
p
σ
4
c
c
c
c
...
σ
2
A
σϕ
2
σ ϕ
2
σ
22
1
N
(4.140)
and the CRLB for unbiased estimation of (
A
,
ϕ
1
,
…
,
ϕ
N
,
σ
2
) is given by
σ
2
0
...
0
0
N
σ
2
0
...
0
0
A
2
CRLB
=
,
(4.141)
...
... ... ...
...
σ
2
00
..
0
A
2
σ
4
00
...
0
N
4.5.2.2.2 ML Estimation
In case of different phase values, we have
N
1
2
∑
2
2
(
ˆ
)(
ˆ
ˆ
cos
ˆ
in
ˆ
σ
2
=
A
ϕ
−
+
A
s
ϕ
n
−
)
,
(4.142)
w
w
ML
N
ML
n
,
ML
rn
,
ML
,
ML
in
,
n
=
1
ϕ
A
with
and
given by Equation 4.73 and Equation 4.74, respectively.
ML
n
,
ML
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