Biomedical Engineering Reference
In-Depth Information
can be 16, 32, 48 or 64 voxels on a side. As reported later, the selection of the
VOI size depends on the amount of warping required. A simplex optimization
algorithm varies the
x
,
y
, and
z
transformation parameters of the floating VOI
until the mutual information with the reference VOI is optimized. Each control
point is optimized independently and the 3D coordinates of the optimal CPs are
recorded.
The final major step is to warp the floating volume using the corresponding
optimal CPs coordinates to establish a three-dimensional thin-plate spline (TPS)
transformation [48, 49]. We now briefly go through the three computing steps
for the TPS transformation.
First, let
P
1
=
(
x
1
,
y
1
,
z
1
),
P
2
=
(
x
2
,
y
2
,
z
2
),...,
P
n
=
(
x
n
,
y
n
,
z
n
)be
n
control
points in the image coordinate of the
reference
volume. Write
r
ij
=
P
i
−
P
j
for
the distance between point
i
and
j
. We define matrices:
⎡
⎣
⎤
⎦
1
x
1
y
1
z
1
1
x
2
y
2
z
2
··· ··· ··· ···
1
P
=
,
n
×
4;
x
n
y
n
z
n
⎡
⎤
0
r
12
r
13
···
r
1
n
r
21
0
r
23
···
r
2
n
··· ··· ··· ··· ···
r
n
1
⎣
⎦
K
=
n
×
n
;
,
r
n
2
r
n
3
0
···
and
KP
P
T
L
=
(
n
+
4)
×
(
n
+
4);
,
O
where
T
is the matrix transpose operator and
O
isa4
×
4 matrix of zero.
Second, let
Q
1
=
(
u
1
,v
1
,w
1
),
Q
2
=
(
u
2
,v
2
,w
2
), ...,
Q
n
=
(
u
n
,v
n
,w
n
)be
n
corresponding control points in the image coordinate of the
floating
volume.
We get matrices:
⎡
⎣
⎤
⎦
,
u
1
u
2
···
u
n
V
=
3
×
n
,
v
1
v
2
···
v
n
w
1
w
2
···
w
n
V
|
0000
T
Y
=
3
×
(
n
+
4)
,
,
