386 Statistics of Medical Imaging
TABLE 12.8
z
(m)
j,k
of EM Algorithm – Example #2: Pixels from Image in Figure 12.1h
z
(m)
2
048,k
z
(m)
2
049,k
k = 1 k = 2 k = 3 k = 4 k = 1 k = 2 k = 3 k = 4 ln L
m=1 0.11 2 0.594 0.28 2 0.012 0.005 0.180 0.623 0.192 −20485
11 0.000 0.714 0.28 6 0.000 0.000 0.182 0.813 0.005 −19603
21 0.000 0.943 0.05 7 0.000 0.000 0.059 0.941 0.000 −19502
31 0.000 0.996 0.00 4 0.000 0.000 0.004 0.995 0.000 −19413
41 0.000 0.996 0.00 4 0.000 0.000 0.000 0.999 0.001 −19391
51 0.000 0.994 0.00 5 0.000 0.000 0.000 0.999 0.001 −19389
53 0.000 0.994 0.00 6 0.000 0.000 0.000 0.999 0.001 −19389
55 0.000 0.994 0.00 6 0.000 0.000 0.000 0.999 0.001 −19389
shown in subsections 10.3.1, 10.3.2, and 12.2.2.1.1, respectively. This subsec-
tion first shows that EM and CM solutio ns provide ML estimates of iFNM
model parameters in the case of no-overlap. Eq. (1 2.27) is the ML solution,
but it cannot be really used to compute the parameter estimates because
z
jk
depe nds on the parameter estimates (ˆπ
k
, ˆµ
k
, ˆσ
2
k
) (k = 1, ···, K), which
are unknown and to be determined. Eq. (12.27), however, is an intuitively
appealing form for the solution of Eq. (12.26), and is also analogous to the
corresponding EM solution Eq. (10.12 ). Section 12.2.2.1.2 established that, in
the case of no-overlap, z
jk
= z
(m)
jk
. Thus, Eq. (10.12) (EM solution) will be
exactly the same as Eq. (12.27) (ML solution). Moreover, Eq. (10.16) (CM
solution) is a special ca se of Eq. (10.12) when z
(m)
jk
= 1. The refore, in the case
of no-overlap, Eqs . (10.12) and (10.16) produce ML estima tes of the iFNM
model parameter s.
Section 12.2.2.1.2 also showed that in the case of overlap, z
jk
≃ z
(m)
jk
. That
is, para meter estimates by EM algorithm may not be exactly the sa me as that
by the ML procedur e. The reason for this difference is that the EM solution
is obtained by maximizing the expectation of the iFNM likelihood function,
while the ML solution is obtained by maximizing the iFNM likelihood function
itself.
Next, this subsection shows that the EM and CM solutions pr ovide asymp-
totically unbiased ML estimates of iFNM model parameters in the case of
no-overlap. In the case of no-overlap, the expectation of ˆπ
k
of Eq. (12.27), by
using Eq. (12.29), is
E{ˆπ
k
} =
1
J
J
X
j=1
E{z
jk
} = π
k
. (12.31)
The expectation of ˆµ
k
of Eq. (12.27), using Eq. (12 .30), is
E{ˆµ
k
} =
1
J ˆπ
k
J
X
j=
1
E{z
jk
x
j
} =
1
J ˆπ
k
X
x
j
∈G
k
E{x
j
} = µ
k
. (
12.32)
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