Image Processing Reference
In-Depth Information
for noise-free data. However, in practice, the data will be corrupted by noise, and
for particular realizations of the noise, the condition in Equation 4.106 may not
be met. Then A
0 will be a maximum. Moreover, if the condition in
Equation 4.106 is not met, b in Equation 4.111 is negative and thus ln L is
convex, which means that A
=
0 will be the only and, therefore, global maximum
of the ln L function. This implies that under the influence of noise, the two
maxima and one minimum will merge into one single maximum at A
=
0. This
maximum then corresponds to the ML estimate. Note that because the condition
in Equation 4.106 is identical to (and therefore can be replaced by) the condi-
tion
=
A c 2
>
0
, the probability that the ML estimate is found at A
=
0 is equal to
A c 2
the probability that
0
. This probability can be computed from the PDF given
in Equation 4.87.
It follows from these considerations that when the conventional estimator
becomes invalid, the ML estimator will still yield physically relevant results.
4.4.3.2
Region of Constant Amplitude and Unknown
Noise Variance
If the noise variance is unknown, the signal amplitude and the noise variance have
to be estimated simultaneously (i.e., the noise variance is a nuisance parameter).
4.4.3.2.1 CRLB
The Fisher information matrix of the data with respect to the parameters ( A ,
σ
2 )
is given by
2
ln
p
∂∂
2
ln
p
m
m
A
2
A
σ
2
I
=−
.
(4.112)
E
∂∂
2
ln
p
A
2
ln
p
m
m
σ
(
σ
)
2
22
After some calculations, it can be shown that
N
A
2
I
()
11
,=
Z
,
(4.113)
σ
σ
2
2
NA
A
2
I
() ()
12
, = , =
I
21
1
+ −
Z
,
(4.114)
σ
σ
4
2
NA Z
2
A
4
I
(
22
, =
)
1
+
(
−−
1
)
,
(4.115)
σ
σ
σ
4
2
4
where I ( i , j ) denotes the ( i , j )th element of the matrix I , and Z is given by
Equation 4.79. Finally, the CRLB for unbiased estimation of ( A ,
σ
2 ) is obtained
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