Image Processing Reference
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which equals the CRLB (cf. Equation 4.153). The estimator (Equation 4.156) is,
however, biased because of the square root operation. Its expectation value is
approximately equal to
.
1
[
]
σ
1
−
(4.158)
E
σ
4
NK
ML
This means that it is possible to apply a bias correction. This, however, would
increase the variance of the estimator.
4.5.3.1.4 Conventional Estimation
Another commonly used estimator of
can be found from the first moment of
the Rayleigh PDF. Because the mean value of the generalized Rayleigh PDF is
given by
σ
σ
Γ
((
K
+
12
2
) /
)
[]
=
2
,
(4.159)
E
m
n
Γ
(/)
K
an unbiased estimator of
σ
is easily seen to be
N
Γ
(/)
K
2
12
1
2
∑
σ
=
.
(4.160)
m
c
Γ
((
K
+
) /
)
n
N
n
=
1
The variance of this estimator is given by
2
σ
2
K
Γ
(/)
K
2
12
=
Var(
σ
)
−
1
,
(4.161)
c
N
2
Γ
((
K
+
) /
)
,
described in the preceding text by Equation 4.160 and Equation 4.156, in terms
of the MSE. The MSE ratio, defined as
which is always larger than the CRLB. Next, we can compare both estimators of
σ
MSE
MSE
σ
σ
()
c
MSE ratio
=
)
,
(4.162)
(
ML
is shown in
Figure 4.11
as a function of the number of data points for
K
=
2, 4,
and 6. For large
N
, the MSE of the common estimator (Equation 4.160) is
significantly larger than that of the ML estimator (Equation 4.156). The perfor-
mance of the conventional estimator, compared with the ML estimator, is not
good for conventional magnitude MR images where
K
=
2.
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