Image Processing Reference
In-Depth Information
where is given by Equation 4.128. The expectation value in Equation 4.121
can be evaluated numerically. The CRLB can easily be obtained by applying the
inverse operator.
p
m
n
4.4.5.1.2 Mean Estimator
The most intuitive way of estimating the unknown signal component is through
a simple averaging of pixel values in the region
. Without
a priori
knowledge
of the proper data PDF, this action would be justified as it is the optimal (i.e.,
ML) estimation procedure if the data is corrupted by Gaussian-distributed noise.
This average or “mean estimator” is given by
Ω
N
1
∑
ˆ
=
m
.
(4.122)
A
n
m
N
n
=
1
The variance of this mean estimator is given by
(
)
.
1
(
ˆ
2
2
−
Var
A
)
=
E
m
[]
(4.123)
E
m
m
N
n
n
However, as PCMR data are not Gaussian distributed, it is clear that a huge
bias would be introduced if the signal is estimated by averaging pixel values. The
bias, relative to the true signal component
A
, is in general defined by
[
ˆ
]
AA
A
−
E
Relative bias
=
×
100
%
.
(4.124)
In the definition, the absolute value was taken as to make the relative bias
logarithmically plottable. For the mean estimator Equation 4.22, the expectation
value is given by , because the average operator is an unbiased esti-
mator of the expectation value. Hence, the relative bias of can be computed
from the expression for the moments of the generalized Rician PDF as given in
Equation 4.30 with
v
[
ˆ
E
]
E m
[]
A
m
A
m
1. Note that it follows from Equation 4.30 and
Equation 4.124 that the relative bias can be written solely in terms of the SNR and
=
is independent of the number of averaged pixel values
N
.
Figure 4.8
shows the
A
relative bias of as a function of the SNR for various values of
K
. From the
figure, it is clear that the bias increases rapidly with decreasing SNR. Also, the bias
increases with increasing number of flow-encoding directions.
m
4.4.5.1.3 Modified RMS Estimator
An easy way to reduce the bias is by exploiting the second moment of the
generalized Rician distribution, as was given in Equation 4.31. Indeed, an
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