384 Statistics of Medical Imaging
The solutions of Eq. (12.26) are
ˆ
λ = −J
ˆπ
k
=
1
J
P
J
j=
1
z
jk
ˆµ
k
=
1
J ˆπ
k
P
J
j=
1
z
jk
x
j
ˆσ
2
k
=
1
J ˆπ
k
P
J
j=
1
z
jk
(x
j
− ˆµ
k
)
2
,
(12.27)
where k = 1, ···, K and the quantities
z
jk
=
ˆπ
k
g(x
j
|
ˆ
θ
k
)
f(x
j
|
ˆ
r)
(
12.28)
are known as Bayesian probabilities of the j-th pix el to be classified into the
k-th image r egion for every j = 1, ···, J and k = 1, ···, K.
2) Bayesian Probability z
jk
and Probability Membership z
(m)
jk
Theoretically, any Gaussian random variable X takes the values from the
interval (−∞, ∞). In medical imaging, due to the physical limitations, any
X that r epresents a measured quantity (such as pixe l intensity) takes values
from the interval (−M, M), where M > 0. That is, X has a trunca ted
Gaussian distribution. In the iFNM model, each Gaussian random variable
X
k
(k = 1, ···, K) that represents pixel intensity x
k
takes values from the
interval (x
′
k
, x
′′
k
).
§
If all these intervals are mutually exclusive, then it is said
that the r andom variables X
k
(k = 1 , ···, K) are no-overlapping.
The B ayesian probability z
jk
of Eq. (12.28) and the probability membership
z
(m)
jk
of Eq. (10.11) are identical in their functional forms. However, they are
different. z
jk
cannot be really computed, because it requires the parameter
estimates (ˆπ
k
, ˆµ
k
, ˆσ
2
k
) that are unknown and to be determined. While z
(m)
jk
can be computed us ing the incomplete data ({x
j
, j = 1, ···, J}) and the
current parameter estimates (π
(m)
k
, µ
(m)
k
, σ
2
k
(m)
), and w ill be used to compute
the updated parameter estimates (π
(m+1)
k
, µ
(m+1)
k
, σ
2
k
(m+1)
).
It is easy to prove that Bayesian probabilities z
jk
of Eq. (12.28) have two
properties
K
X
k=1
z
jk
= 1 (j = 1, ···, J) and E{z
jk
} = π
k
(k = 1 , ···, K). (12.29)
§
This notation and {x
n
k−1
+1
, · · · , x
n
k
} ∈ R
k
of Section 12.2.1.1 are equival ent, that is,
x
′
k
< {x
n
k−1
+1
, · · · , x
n
k
} ≤ x
′′
k
.
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