Biomedical Engineering Reference
In-Depth Information
8.3.2.8
Multimodal Non-Rigid Registration
We have proposed a multimodal version of Romeo [72] where the optical flow
is replaced by a more adapted and general similarity measure: mutual informa-
tion. Mutual information has been presented in the section dedicated to rigid
registration (8.3.2).
Let us note
T
w
as the transformation associated with the deformation field
w
.
The total transformation
T
w
◦
T
0
maps the floating volume
B
onto the reference
volume
A
. The field
w
is defined on
S
B
, where
S
B
denotes the lattice of volume
B
(pixel lattice or voxel lattice). The cost function to be minimized then becomes:
<
s
,
r
>
∈
C
B
||
w
s
−
w
r
||
2
U
(
w
;
A
,
B
,
T
0
)
=−
I
(
A
,
(
T
w
◦
T
0
)(
B
))
+
α
,
where
C
B
is the set of neighboring pairs of volume
B
(if we note
V
a neighborhood
system on
S
B
, we have:
<
s
,
r
>
∈
C
B
⇔
s
∈
V
(
r
)).
A multiresolution and multigrid minimization are also used in this con-
text. At grid level
and on each cube
n
, we estimate an affine displace-
ment increment defined by the parametric vector
n
:
∀
s
∈
n
,
d
w
s
=
P
s
n
,
with
P
s
= I
2
⊗
[1
x
s
y
s
] for 2D images, and
P
s
= I
3
⊗
[1
x
s
y
s
z
s
] for 3D images (operator
⊗
denotes the Kronecker product).
To be more explicit, in 3D we have:
⎛
⎝
⎞
⎠
.
1
x
s
y
s
z
s
00 0 000 0 0
00 0 01
x
s
y
s
z
s
00 0 0
00 0 000 0 01
x
s
P
s
=
y
s
z
s
Let us note
T
n
, as the transformation associated with the parametric field
n
. We have
T
T
d
w
and
T
n
=
T
d
w
|
n
, where
T
d
w
|
n
denotes the restriction
=
of
T
n
to the cube
n
.
A neighborhood system
V
on the partition
derives naturally from
V
:
∀
n
,
m
∈{
1
···
N
}
,
m
∈
V
(
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
}
,<
s
,
r
>
∈
C
n
⇔
s
∈
n
,
r
∈
n
and
r
∈
V
(
s
)
.
∀
n
∈{
1
...
N
}
,
∀
m
∈
V
(
n
)
,<
s
,
r
>
∈
C
nm
⇔
m
∈
V
l
(
n
)
,
s
∈
n
,
r
∈
m
and
r
∈
V
(
s
)
.
