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c
with energy
function denoted by
H
Θ
(
θ
) where by extension
θ
denotes the set of distinct val-
ues
θ
=
∈C
and all
i
∈
V
c
,
θ
i
=
θ
c
and consider a pairwise MRF on
C
. Outside the issue of estimating
θ
in the M-step, having
parameters
θ
i
's depending on
i
is not a problem. For the E-steps, we go back to
this general setting using an interpolation step specified in Section 5.2. It follows
that
p
(
θ
{
θ
c
,c
∈C}
|
y
,
t
,
s
) is defined as a MRF with the following energy function,
y
,
t
,
s
)=
H
Θ
(
θ
)+
c∈C
log
i∈V
c
H
Θ|Y,T,S
(
θ
|
g
S
(
y
i
|
t
i
,s
i
,θ
c
)
,
where
g
S
(
y
i
|
t
i
,s
i
,θ
c
) is the expression in (5). The specific form of the Markov
prior on
θ
is specified in Section 5.
5 Estimation by Generalized Alternating Minimization
The particularity of our segmentation task is to include two label sets of inter-
est,
t
and
s
which are linked and that we would like to estimate cooperatively
using one to gain information on the other. We denote respectively by
T
and
S
the spaces in which
t
and
s
take their values. Denoting
z
=(
t
,
s
), we apply
the EM framework introduced in Section 3 to find a MAP estimate
θ
of
θ
using
the procedure given by (2) and (3) and then generate
t
and
s
that maximize
the conditional distribution
p
(
t
,
s
y
, θ
). Note that this is however not equiva-
lent to maximizing over
t
,
s
and
θ
the posterior distribution
p
(
t
,
s
,θ
|
|
y
). Indeed
p
(
t
,
s
,θ
|
y
)=
p
(
t
,
s
|
y
,θ
)
p
(
θ
|
y
) and in the EM setting,
θ
is found by maximizing
the second factor only.
However, solving the optimization (2) over the set
D
of probability distri-
leads for the optimal
q
(
r
)
(
T,S
)
y
,θ
(
r
)
)whichis
intractable for the joint model defined in Section 4. We therefore propose an
EM variant appropriate to our cooperative context and in which the E-step is
not performed exactly. The optimization (2) is solved instead over a restricted
class of probability distributions
butions
q
(
T,S
)
on
T×S
to
p
(
t
,
s
|
˜
which is chosen as the set of distributions
that factorize as
q
(
T,S
)
(
t
,
s
)=
q
T
(
t
)
q
S
(
s
)where
q
T
D
(resp.
q
S
) belongs to the
set
). This variant
is usually referred to as Variational EM [15]. It follows that the E-step becomes
an approximate E-step,
D
T
(resp.
D
S
) of probability distributions on
T
(resp. on
S
(
q
(
r
)
T
,q
(
r
)
S
(
q
T
,q
S
)
F
(
q
T
q
S
,θ
(
r
)
)
.
) = arg max
This step can be further generalized by decomposing it into two stages. At it-
eration
r
, with current estimates denoted by
q
(
r−
1)
T
,q
(
r−
1)
S
and
θ
(
r
)
,weconsider
the following updating,
E-T-step:
q
(
r
)
T
q
(
r−
1)
S
,θ
(
r
)
)
=arg max
q
T
∈D
T
F
(
q
T
E-S-step:
q
(
r
)
S
F
(
q
(
r
)
T
q
S
,θ
(
r
)
)
.
=arg max
q
S
∈D
S
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