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Structure conditional Tissue model.
We define
p
(
t
|
s
,
y
,θ
)asaMarkov
Random Field in
t
with the following energy function,
⎛
⎝
t
t
i
γ
i
(
s
i
)+
j∈N
(
i
)
⎞
H
T |S,Y,Θ
(
t
|
s
,
y
,θ
)=
i∈V
U
ij
(
t
i
,t
j
;
η
T
)+log
g
T
(
y
i
;
t
θ
i
t
i
)
⎠
,
(3)
(
i
) denotes the voxels neighboring
i
,
g
T
(
y
i
;
t
θ
i
t
i
) is the Gaussian distri-
bution with parameters
θ
i
where
N
if
t
i
=
e
k
and the external field
γ
i
depends on
s
i
and
e
1
,...,e
L
}
is defined by
γ
i
(
s
i
)=
e
T
s
i
and
γ
i
(
s
i
)=
0
otherwise, with
T
s
i
denoting the tissue of structure
s
i
and
0
the 3-dimensional null vector. The
rationale for choosing such an external field, is that depending on the structure
present at voxel
i
and given by the value of
s
i
, the tissue corresponding to this
structure is more likely at voxel
i
while the two others tissues are penalized
by a smaller contribution to the energy through a smaller external field value.
When
i
is a background voxel, the external field does not favor a particular
tissue. The Gaussian parameters
θ
i
if
s
i
∈{
μ
i
,λ
i
}
are respectively the mean and
precision which is the inverse of the variance. We use similar notation such as
μ
=
=
{
{
μ
i
,i
∈
V, k
=1
...K
}
and
μ
k
=
{
μ
i
,i
∈
V
}
,etc.
Tissue conditional structure model.
Apriori
knowledge on structures is
incorporated through a field
f
=
where
f
i
=
t
(
f
i
(
e
l
)
,l
=1
...L
+1)
and
f
i
(
e
l
) represents some prior probability that voxel
i
belongs to structure
l
,
as provided by a registered probabilistic atlas. We then define
p
(
s
{
f
i
,i
∈
V
}
|
t
,
y
,θ
)asa
Markov Random Field in
s
with the following energy function,
H
S|T,Y,Θ
(
s
|
t
,
y
,θ
)=
i∈V
⎛
⎝
t
s
i
log
f
i
+
j∈N
(
i
)
⎞
⎠
(4)
U
ij
(
s
i
,s
j
;
η
S
)+log
g
S
(
y
i
|t
i
,s
i
,θ
i
)
where
g
S
(
y
i
|
t
i
,s
i
,θ
i
)isdefinedasfollows,
t
i
,s
i
,θ
)=[
g
T
(
y
i
;
t
θ
i
e
T
s
i
)
f
i
(
s
i
)]
w
(
s
i
)
[
g
T
(
y
i
;
t
θ
i
t
i
)
f
i
(
e
L
+1
)]
(1
−w
(
s
i
))
(5)
g
S
(
y
i
|
where
w
(
s
i
) is a weight dealing with the possible conflict between values of
t
i
and
s
i
. For simplicity we set
w
(
s
i
)=0if
s
i
=
e
L
+1
and
w
(
s
i
)=1otherwise
but considering more general weights could be an interesting refinement. Other
parameters in (3) and (4) include interaction parameters
η
T
and
η
S
which are
considered here as hyperparameters to be specified (see Section 6).
4.2 A Conditional Model for Intensity Distribution Parameters
To ensure spatial consistency between the parameter values, we define also
p
(
θ
y
,
t
,
s
) as a MRF. In practice however, in our general setting which allows
different values
θ
i
at each
i
, there are too many parameters and estimating them
accurately is not possible. As regards estimation then, we adopt a local approach
as in [4]. The idea is to consider the parameters as constant over subvolumes
oftheentirevolume.Let
|
be a regular cubic partitioning of the volume
V
in
a number of non-overlapping subvolumes
C
{
V
c
,c
∈C}
. We assume that for all
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