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Using the equality log
p
(
θ
|
y
)=log
p
(
y
|
θ
)+log
p
(
θ
)
−
log
p
(
y
) it follows
log
p
(
θ
|
y
)=
F
(
q, θ
)+
KL
(
q, p
)+log
p
(
θ
)
−
log
p
(
y
) from which, we get a lower
bound
log
p
(
y
)
.
Maximizing this lower bound alternatively over
q
and
θ
leads to a sequence
{
L
(
q, θ
)onlog
p
(
θ
|
y
)givenby
L
(
q, θ
)=
F
(
q, θ
)+log
p
(
θ
)
−
(
q
(
r
)
,θ
(
r
)
). The maximization over
q
corresponds to the standard E-step and leads to
q
(
r
)
(
z
)=
p
(
z
q
(
r
)
,θ
(
r
)
}
r∈
N
satisfying
L
(
q
(
r
+1)
,θ
(
r
+1)
)
≥L
|
y
,θ
(
r
)
). It fol-
lows that
L
(
q
(
r
)
,θ
(
r
)
)=log
p
(
θ
(
r
)
|
y
) which means that the lower bound reaches
the objective function in
θ
(
r
)
y
)
at each step. It then appears that when considering our MAP problem, we re-
place (see
eg.
[12]) the function
F
(
q, θ
)by
F
(
q, θ
)+log
p
(
θ
). The corresponding
alternating procedure is: starting from a current value
θ
(
r
)
and that the sequence
{
θ
(
r
)
}
r∈
N
increases
p
(
θ
|
∈
Θ
, set alternatively
q
(
r
)
=argmax
q∈D
F
(
q, θ
(
r
)
) = arg max
q∈D
y
,θ
(
r
)
)
q
(
z
)+
I
[
q
]
,
log
p
(
z
|
(2)
z
∈Z
and
θ
(
r
+1)
= arg max
θ∈Θ
F
(
q
(
r
)
,θ
)+log
p
(
θ
) = arg max
θ∈Θ
log
p
(
y
,
z
,θ
)
q
(
r
)
(
z
)+log
p
(
θ
)
z
∈Z
z
∈Z
log
p
(
θ|
y
,
z
)
q
(
r
)
(
z
)
.
= arg max
θ∈Θ
The last equality in (2) comes from
p
(
y
,
z
|
θ
)=
p
(
z
|
y
,θ
)
p
(
y
|
θ
)andthefact
that
p
(
y
|
θ
) does not depend on
z
. The last equality in (3) comes from
p
(
y
,
z
|
θ
)=
p
(
θ
y
,
z
)
p
(
y
,
z
)
/p
(
θ
) and the fact that
p
(
y
,
z
) does not depend on
θ
.Theopti-
mization with respect to
q
gives rise to the same E-step as for the standard EM
algorithm, because
q
only appears in
F
(
q, θ
). It can be shown (eg. [12] p.319)
that EM converges to a local mode of the posterior density except in some very
special cases. This EM framework appears as a reasonable framework for in-
ference. In addition, it appears in (2) and (3) that inference can be described
in terms of the conditional models
p
(
z
|
y
,
z
). In the following sec-
tion, we show how to define our joint model so as to take advantage of these
considerations.
|
y
,θ
)and
p
(
θ
|
4 A Bayesian Model for Robust Joint Tissue and
Structure Segmentations
In this section, we describes the Bayesian framework that enables us to model
the relationships between the unknown linked tissue and structure labels, the
observed MR image data and the tissue intensity distributions parameters.
We consider a finite set
V
of
N
voxels on a regular 3D grid. We denote by
y
=
{
y
1
,...,y
N
}
the intensity values observed respectively at each voxel and by
t
=
e
1
,e
2
,e
3
}
where
e
k
is a 3-dimensional binary vector whose
k
th
component is 1, all other
components being 0. In addition, we consider
L
subcortical structures and denote
by
s
=
{
t
1
,...,t
N
}
the hidden tissue classes. The
t
i
's take their values in
{
{
s
1
,...,s
N
}
the hidden structure classes at each voxel. Similarly, the
s
i
's
e
1
,...,e
L
,e
L
+1
}
where
e
L
+1
corresponds to an additional
take their values in
{
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