Biomedical Engineering Reference
In-Depth Information
∀
n
,
m
∈{
1
···
N
k
,
}
,
m
∈
V
k
,
(
n
)
⇔∃
s
∈
n
,
∃
r
∈
m
\
r
∈
V
(
s
)
. C
being the
set of neighboring pairs on
S
k
, we must now distinguish between two types
of such pairs: the pairs inside one cube and the pairs between two cubes:
∀
n
∈{
1
...
N
k
,
}
,<
s
,
r
>
∈
C
n
⇔
s
∈
n
,
r
∈
n
and
r
∈
V
(
s
)
.
∀
n
∈{
1
...
N
k
,
}
,
∀
m
∈
V
(
n
)
,<
s
,
r
>
∈
C
nm
⇔
m
∈
V
l
(
n
)
,
s
∈
n
,
r
∈
m
and
r
∈
V
(
s
)
.
For the sake of concision, we will now drop the resolution index
k
. With
these notations, the cost function (8.3) becomes
U
(
,δ
,β
;
w
,
f
)
=
N
s
∈
n
δ
s
∇
f
t
(
s
,
t
)
2
+
ψ
1
δ
s
f
s
P
s
n
+
n
=
1
⎡
⎣
m
∈
V
(
n
)
⎤
N
<
s
,
r
>
∈
C
nm
β
sr
||
w
s
+
P
s
n
−
w
r
+
P
r
m
||
+
ψ
2
β
sr
2
⎦
+
α
n
=
1
⎡
⎣
⎤
⎦
.
N
<
s
,
r
>
∈
C
n
β
sr
||
w
s
+
P
s
n
−
w
r
+
P
r
n
||
+
ψ
2
β
sr
2
+
α
(8.4)
n
=
1
Considering the auxiliary variables of the robust estimators as fixed, one
can easily differentiate the cost function (8.4) with respect to any
n
and get
a linear system to be solved. We use a Gauss-Seidel method to solve it for its
implementation simplicity. However, any iterative solver could be used (solvers
such as conjugate gradient with an adapted preconditioning would also be effi-
cient). In turn, when the deformation field is “frozen”, the weights are obtained
in a closed form [14, 24]. The minimization may therefore be naturally han-
dled as an alternated minimization (estimation of
n
and computation of the
auxiliary variables). Contrary to other methods (minmax problem like the de-
mon's algorithm for instance), that kind of minimization strategy is guaranteed
to converge [24, 42, 99] (i.e., to converge toward a local minimum from any
initialization).
Moreover, the multigrid minimization makes the method invariant to inten-
sity inhomogeneities that are piecewise constant. As a matter of fact, if the
intensity inhomogeneity is constant on a cube, the restriction of the DFD on
that cube is modified by adding a constant. As a consequence, minimizing the
cost function 8.3.2 gives the same estimate, whenever the cost at the optimum
is zero or a constant (see section 8.3.3 for an illustration on that issue).
