Image Processing Reference
In-Depth Information
4.5.2.2.3 MSE
It can be shown that, for large
N ˆ
N
, the quantity
2
σσ
ML
2
/
2
is approximately dis-
tributed as
χ
2
=
χ
2
[52]. This means that
2
NN
−+
(
1
)
N
1
σ
2
σ
2
1
( )
ˆ
b
σ
2
(
N
−− =−
1
)
σ
2
1
+
(4.143)
ML
2
N
2
N
and
σ
4
1
( )
ˆ
Var
σ
2
NN
1
.
(4.144)
ML
2
ˆ
Hence, the MSE of
σ ML
2
is given by
.
σ
4
41
( )
ˆ
MSE
σ
2
1
+−
(4.145)
ML
4
NN
2
4.5.2.3
Background Region
Next, consider the case in which the noise variance is estimated from a back-
ground region (i.e., a region where A is known to be zero).
4.5.2.3.1 CRLB
It can easily be shown that the CRLB for unbiased estimation of
σ
2 is given by
= σ
4
N
CRLB
,
(4.146)
independent of the underlying phase values.
4.5.2.3.2 ML Estimation
The ML estimator is given by
N
(
)
1
2
2
ˆ
σ ML
2
=
+
,
(4.147)
ww
2
N
rn
,
in
,
n
=
1
independent of the underlying phase values.
4.5.2.3.3 MSE
It can easily be shown that the ML estimator Equation 4.147 is unbiased and that
its variance equals the CRLB. Therefore, its MSE is simply given by
4
( ) = N
σ
ˆ
MSE
σ
2
.
(4.148)
ML
 
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