374 Statistics of Medical Imaging
Substituting Eq. (12.3) into Eq. (12 .1) and then applying Eq. (12.1) to I
K
0
and I
K
1
in Eq. (12.4), we have
P
e
= P (ln L
K
0
(
ˆ
r) − ln L
K
1
(
ˆ
r) < (1.5 ln J)(K
0
− K
1
)). (12.6)
In Eq. (12.2), when ˆσ
2
k
>
1
2π
, (k = 1, ···, K) (this conditio n is always
satisfied in medical images), 0 < f(x
j
|
ˆ
r) < 1. [44] shows that ln w =
P
∞
k=1
(−1)
k+1
(w−1)
k
k
(0 < w ≤ 2). Using the linear term (w − 1) of this
series representation as an approximation o f ln w (Section A.5 of Appendix
12A gives a detailed discussion on this approximation) and applying this sim-
plification to Eq. (12.4), we have
ln L
K
(
ˆ
r) =
J
X
j=1
ln f (x
j
|
ˆ
r) ≃
J
X
j=1
f(x
j
|
ˆ
r) − J. (1 2.7)
Assuming that ˆπ
k
= 1/K and ˆσ
2
k
= ˆσ
2
(k = 1 , ···, K)
∗
, Eq. (12.7) becomes
ln L
K
(
ˆ
r) =
1
√
2πK ˆσ
J
X
j=1
K
X
k=1
exp(−
(x
j
− ˆµ
k
)
2
2ˆσ
2
) − J. (12.8 )
Pixel x
j
is a sample from the iFNM f(x|
ˆ
r). After classification, however, x
j
belongs to one and only one component of iFNM f (x|
ˆ
r), say g(x|θ
k
); that is,
to one and only one image reg ion, say R
k
. Let n
0
= 0, n
K
= J. Without loss
of generality, assume {x
n
k−1
+1
, ······, x
n
k
} ∈ R
k
(k = 1, ···, K). [44] also
shows that e
−
1
2
w
=
P
∞
k=
0
(−1)
k
(w/2)
k
k!
(−∞ < w < ∞). Using the linear term
(1 −
1
2
w) of this series representation as a n approximation of e
−
1
2
w
(
Section
A.5 of App endix 12A gives detailed disc ussion on this approximation) and
applying this simplification to Eq. (12.8), we have
ln L
K
(
ˆ
r) ≃
1
√
2πKˆσ
K
X
k=1
n
k
X
j=n
k−1
+1
exp(−
(x
j
− ˆµ
k
)
2
2ˆσ
2
) − J (
12.9)
∗
(1) When π
k
s are very different, the minor regions (i.e., regions with smaller π
k
) may be
ignored in the image analysis. The performance evaluation of this section is confined to
the case where all π
k
are si milar. As a result, the error-detection probability will truthfully
represent the detection ability of the image analysis technique itself and will not be affected
by the sizes of regions which are difficult to be assumed generally. (2) σ
2
k
s are determined
by two factors: the inhomogeneity in each object and the total noise. Inhomogeneity in each
object causes a slow variation of gray levels in each image region, which can be taken into
account in the region mean estimation. In this way, σ
2
k
will be solely affected by noise. When
σ
2
k
s are different, Eq. (12.21) is used to define ˆσ
2
. (3) The above two assumptions (with
respect to π
k
and σ
2
k
) can be satisfied by some simulation methods. For example, when the
Gibbs sampler [28, 33] is used to generate a Markov random field image and Gaussian noise
is superimposed on the image, then the sizes of the image regions can be controlled (to be
similar), the means of the image regions will be different, and the variances of the image
regions will be the same. Examples are given in the following sections.
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