Biomedical Engineering Reference
In-Depth Information
point in the experimental data set. This time can be significantly reduced by
applying the following grid closest point (GCP) transform.
The GCP transform
GCP
:
3
that
encloses the two surfaces
S
and
S
to a displacement vector,
r
, which represents
the displacement from the closest point in the model set
S
. Thus for all
z
∈
G
3
3
maps each point in the 3D space
G
⊂
→
GCP
(
z
)
=
r
=
x
m
−
z
(1.5)
such that
d
(
z
,
x
m
)
=
min
x
i
∈
S
{
d
(
z
,
x
i
)
(1.6)
where
d
(
·
) is the Euclidean distance. For each point in
G
, the transform calculates
a displacement vector to the closest point in the model data set which can be used
subsequently to find matching points between
S
and
S
during the minimization
process.
In the discrete case, assume that
G
consists of a rectangular box that encloses
the two surfaces. Furthermore, assume that
G
is quantized into a set of
L
×
W
×
H
cells of size
δ
3
{
C
ijk
|
0
≤
i
≤
L
,
0
≤
j
≤
W
,
0
≤
k
≤
H
}
(1.7)
such that
W
=
(
W
max
−
W
min
)
/δ
(1.8)
L
=
(
L
max
−
L
min
)
/δ
(1.9)
and
H
=
(
H
max
−
H
min
)
/δ
(1.10)
Figure 1.2 shows a 2D illustration of such a grid.
Each cell
C
ijk
will hold a displacement vector
r
ijk
which is a vector from its
centroid, denoted by
c
0
ijk
, to its closest point in the model set.
The GCP transform is applied only once at the beginning of the registration
process. After its application, each cell in
G
has a displacement vector to its
closet point in the model set. During the minimization, to calculate the closest
point
x
=
C
(
y
v
,
S
) to a point
y
v
∈
S
, you first have to find the intersection of
y
v
and
G
. Assuming uniform quantization, the indices of the cell
C
ijk
in which the
