Image Processing Reference
In-Depth Information
by simple inversion of the 2
×
2 matrix given in Equation 4.90:
I
()
22
,
−,
I
()
21
1
CRLB
=
(4.116)
det
I
−,
I
() ()
12
I
11
,
4.4.3.2.2 Geometric Average
In subsection 4.4.3.1, it was shown that a simple modified root-mean-square
(RMS) estimator can be used to estimate the signal amplitude. However, this
estimator requires the knowledge of the noise variance. An estimator that does
not require this knowledge is given by the geometric average defined as [42]
N
∏
1
2
.
=
N
m
(4.117)
A
n
geom
n
=
4.4.3.2.3 Discussion
The bias of this estimator is given by
2
21
σ
b
A
(
)
=
,
(4.118)
geom
(
NA
−
)
and the variance is given by
σ
2
Var
(
)
=
.
(4.119)
A
geom
N
−
1
4.4.3.2.4 ML Estimation
If a background region is not available for noise variance estimation, the signal
A
and variance
2
have to be estimated simultaneously from the
N
available data
points by maximizing the log-likelihood function with respect to
A
and
σ
σ
2
:
{
}
{
}
=
,
arg max(ln
L
) ,
(4.120)
A
σ
2
ML
ML
A
,
σ
2
where ln
L
is given by Equation 4.96. Although optimization of a two-dimensional
function is more difficult, computational requirements were observed to be limited
because the likelihood function was observed to yield only one maximum.
4.4.4
D
ISCUSSION
In this subsection, the CRLB for unbiased estimation of the signal amplitude and
the performance of the ML signal amplitude estimators, elaborated in previous
subsections, are discussed.
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