Image Processing Reference
In-Depth Information
Setting these equations to zero yields the ML estimators of
A
and
ϕ
n
:
N
1
∑
2
2
=
ww
+
,
(4.72)
A
rn
,
in
,
ML
N
n
=
1
w
w
ϕ
in
,
=
arctan
.
(4.73)
n
,
ML
rn
,
4.4.2.2.3 MSE
The ML estimator of the signal amplitude, given by Equation 4.73, is distributed
as the average of
N
independent, Rician-distributed variables. Therefore, its mean
value is simply given by the average of the mean values of the individual Rician-
distributed variables, whereas its variance is given by the sum of their variances
divided by
N
2
. Hence, we have for the MSE:
( )
=
( )
( )
2
+
MSE
b
A
Var
,
(4.75)
A
A
ML
ML
ML
(
)
++−
2
2
=−
A
A
2
2
σ
2
/
N
(4.76)
E
E
A
A
ML
ML
A
where
E
[
]
is now given by Equation 4.14 or equivalently:
ML
2
π
2
A
2
A
2
A
2
A
2
A
−
=
σ
e
1
+
I
+
I
.
E
2
(4.77)
A
4
σ
0
1
2
2
σ
4
σ
2
σ
4
σ
ML
2
2
2
Note that Equation 4.77 does not depend on
N
. Furthermore, Equation 4.77
is identical to Equation 4.65 if
N
=
1.
4.4.3
S
IGNAL
A
MPLITUDE
E
STIMATION
FROM
M
AGNITUDE
D
ATA
Though raw MR data are complex valued and Gaussian distributed, it is common
practice to transform them into magnitude MR data, because physiological and
anatomical information are more closely related to the magnitude of the magne-
tization vectors. However, as we have seen earlier, computing the magnitude
results in a change of the underlying data PDF, which has to be accounted for
when extracting quantitative information [38].
As earlier, we try to estimate the underlying, noiseless amplitude signal
A
from a region of interest (ROI), where
A
is assumed to be constant. The ROI now
consists of
N
independent, Rician-distributed data points
Unlike that from complex data, estimation of the signal amplitude from
magnitude data requires either prior knowledge of the noise variance or simul-
taneous estimation of signal amplitude and noise variance:
m
=, ,
(
1
m
…
).
N
•
The noise variance may be estimated separately if a background region
is available, i.e., a region in which the underlying signal is zero (see
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