Image Processing Reference
In-Depth Information
where
τ
≡
τ
(
σ
2
)
=
σ
, and
I
−1
is given by Equation 4.151
=
1
2
2
σ
4
1
2
CRLB
(4.152)
σ
NK
σ
2
2
NK
.
σ
=
(4.153)
4.5.3.1.3 ML Estimation
We will now describe the ML method for the estimation of the noise standard
deviation or the noise variance from background magnitude MR data. Thereby,
it will be assumed that the available data is governed by a generalized Rician
distribution. The methods described in the following text can also be applied to
conventional Rician-distributed data, which would be the case when
K
2.
The likelihood function is obtained by substituting the available background
data points {
m
n
} for the variables
=
{}
m
n
in Equation 4.125. Then the log-likelihood
function, only as a function of
σ
2
, is given by:
N
1
∑
ln
LN
∼ −
ln
σ
2
−
m
2
.
(4.154)
n
K
σ
2
n
=
1
Maximizing with respect to
σ
2
yields the ML estimator of
σ
2
as
N
1
∑
2
=
n
.
(4.155)
m
σ
2
KN
ML
n
=
1
It can be shown that Equation 4.155 is an unbiased estimator, that is, its mean
is equal to
σ
2
. Furthermore, the variance of the ML estimator Equation 4.155 is
equal to 2
σ
4
/
NK
, which equals the CRLB given by Equation 4.151 for all values
of
N
.
One might be interested in the value of the standard deviation
σ
, e.g., to
estimate the SNR
A
/
σ
. Simply taking the square root of the ML estimator of
σ
2
in Equation 4.155 yields an estimator of
σ
as
N
1
∑
σ
2
=
m
n
,
(4.156)
KN
ML
n
=
1
because the square root
operation has a single-valued inverse (cf. Invariance property of ML estimators
This estimator is identical to the ML estimator of
σ
[
53
]). Its variance is approximately equal to
σ
2
Var
(
)
,
(4.157)
σ
ML
2
NK
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