Biomedical Engineering Reference
In-Depth Information
Experimental Data Set S
,
r
Wmax,0
r
0,0
r
2,Rmax
Model Data Set S
δ
δ
δ
Model Data Set S
δ
Figure 1.2:
(Left) superimposing a uniform grid
G
of size
δ
onto the space that
encloses the model and experimental data sets. (Right) for each cell
C
ijk
∈
G
,
calculate a displacement
r
ijk
, from the cell centroid,
c
0
ijk
, to its closet point
on
S
.
point
y
v
={
u
,v,w
}
lies can be found by
u
−
L
min
δ
j
=
v
−
W
min
δ
k
=
w
−
H
min
δ
i
=
(1.11)
,
,
If the content of the cell
C
ijk
is
r
ijk
then
x
=
C
(
y
v
,
S
)
=
r
ijk
+
c
0
ijk
(1.12)
An approximation of the closest point can be obtained by using the point itself
instead of the centroid of the cell in which it lies
x
=
C
(
y
v
,
S
)
≈
r
ijk
+
y
v
(1.13)
Equation (1.13) introduces an error which is a function of
δ
, the quantization step.
This error can be reduced to some extent by using a non-uniform quantization.
It should be noted that the GCP transform is spatially quantized and its accuracy
depends largely on the selection of
δ
. The error in the displacement vector is
≤
2
. Therefore, smaller values for
δ
will give higher accuracy but on the extent
of larger memory requirements and larger number of computations of the GCP
for each cell. To solve this problem, you can select a small value for
δ
in the
region that directly surrounds the model set, and a slightly larger value (in this
work 2
δ
) for the rest of the space
G
. This enables a coarse matching process
3
4
δ
