390 Statistics of Medical Imaging
In the limiting case, because lim
J→∞
1 +
∂b(σ
2
k
)
∂σ
2
k
=
1, CRLB
ˆσ
2
k
=
2σ
4
k
/Jπ
k
. The Cramer-Rao Low Bound of variance of the sample variance
(with the sample size Jπ
k
) is (2σ
4
k
/Jπ
k
)(1 −
1
Jπ
k
)
[45], which is exactly the
same as E q. (12.46). This result indicates that both the EM and CM algo-
rithms finally lead to the sample variance.
Eqs. (12.37), (12.41), and (12.46) show that the Cramer–Rao Low Bounds
of varia nc es of the iFNM model pa rameter estimates are functions of σ
2
k
, J,
and π
k
. More specifically, these bounds will decrease when (a) image quality
becomes better (i.e., higher SNR or lower σ
2
k
), (b) the reso lution is higher
(i.e., larger J), and (c) the complexity is less (i.e., smaller K
0
or bigger π
k
).
These observations are similar to and consistent with the discussions under
detection performance (Sectio n 12.2.1.2).
12.2.2.3 Results of Estimation Performance
The relative error and the estimation interval of parameter estimates a re uti-
lized to illustrate the accuracy of estimates. The relative errors of the param-
eter estimates are defined by
ε
π
= max
1≤k≤4
{
ˆπ
k
−π
k
π
k
}
ε
µ
=
max
1≤k≤4
{
ˆµ
k
−µ
k
µ
k
}
ε
σ
=
max
1≤k≤4
{
ˆσ
2
k
−σ
2
k
σ
2
k
},
(
12.47)
the estimation intervals of the parameter estimates are defined by
̟
ˆπ
k
= (π
k
−
q
CRLB
ˆπ
k
,
π
k
+
q
CRLB
ˆπ
k
)
̟
ˆµ
k
=
(µ
k
−
q
CRLB
ˆµ
k
,
µ
k
+
q
CRLB
ˆµ
k
)
̟
ˆσ
2
k
=
(σ
2
k
−
q
CRLB
ˆσ
2
k
,
σ
2
k
+
q
CRLB
ˆσ
2
k
),
(
12.48)
where (π
k
, µ
k
, σ
2
k
) and (ˆπ
k
, ˆµ
k
, ˆσ
2
k
) a re the true , and the estimated values of
the parameters, (CRLB
ˆπ
k
, CRLB
ˆµ
k
, CRLB
ˆσ
2
k
) are given by Eqs. (12.37),
(12.41), and (12.46).
1) Results from simulated images
The simulated images shown in Figure 12.1 are also used here for evaluating
estimation performance. The parameter settings (i.e., the true values of the
parameters) are listed in Table 12.1. The parameter es tima tes and the Cr amer-
Rao Low Bounds of the variances of these estimates are summarized in Table
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