Performance Evaluation of Image Analysis Methods 387
The expectation of ˆσ
2
k
of Eq. (12.27), using Eq. (12.30), is
E{ˆσ
2
k
} =
1
J ˆπ
k
J
X
j=
1
E{z
jk
(x
j
− ˆµ
k
)
2
}
=
1
J ˆπ
k
X
x
j
∈G
k
E{(x
j
− ˆµ
k
)
2
} = σ
2
k
(
1 −
1
J ˆπ
k
), (
12.33)
which finally leads to
lim
J→∞
E{ˆσ
2
k
} = σ
2
k
. (12.34)
Thus, (ˆπ
k
, ˆµ
k
, ˆσ
2
k
) are the asymptotica lly unbiased ML estimates of (π
k
, µ
k
, σ
2
k
)
in the cas e of no-overlap. The refore, (π
(m+1)
k
, µ
(m+1)
k
, σ
2
k
(m+1)
) of Eqs. (10.12)
and (10.16) are the asymptotically unbiased ML estimates of (π
k
, µ
k
, σ
2
k
)
under the same condition. The stopping criterion of the EM algorithm is
Eq. (10.13): |L
(m+1)
− L
(m)
| < ǫ. When this criterion is s atisfied, as shown
in Tables 12.7 and 12.8, the probability memberships z
(m)
jk
(j = 1, ···, J, k =
1, ···, K) do not change with further iterations. From Eq. (10 .12), π
(m)
k
, µ
(m)
k
,
σ
2(m)
k
(k = 1 , ···, K) will not change. T he stopping criterion of CM algorithm
is Eq. (10.17): µ
(m+1)
k
= µ
(m)
k
and σ
2
k
(m+1)
= σ
2
k
(m)
(k = 1, ···, K). When
this criterion is satisfied, there will be no pixel interchange among the imag e
regions in future iterations. Thus, when the stopping criteria (E qs. (10.13) and
(10.17)) are satisfied, the classification of pixels into image regions will not
change. Therefor e, in the case of no-overlap, the EM and CM algo rithms pro-
duce a symptotically unbiased ML estimates of the iFNM model para meters
when the stopping criteria are satisfied.
12.2.2.2 Cramer–Rao Low B ounds of Variances of the Parameter
Estimates
1) For Weight Estimation
Parameters θ
k
and π
k
of the iFNM model are linked by π
k
= P (θ = θ
k
),
which shows that π
k
is the probability of occurrence of the k-th c omponent
g(x
j
|θ
k
) of iFNM. The discrete distribution P (θ = θ
k
) can be expressed by
the matrix
θ = θ
1
, ···, θ
k
, ···, θ
K
p
θ
= π
1
, ···, π
k
, ···, π
K
or
θ = θ
k
, θ 6= θ
k
p
θ
= π
k
, p
θ
= 1 − π
k
. (12.35)
Thus
E
"
∂ ln p
θ
∂π
k
2
#
=
X
θ
∂ ln p
θ
∂π
k
2
p
θ
=
1
π
k
(
1 − π
k
)
. (12.36)
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