Graphics Reference
In-Depth Information
y
a
n
000
000
¼½¼
000
000
½1½
000
000
¼½¼
x
b
m
w
k
=1
(
a
n
,
b
m
) =
w
k
=0
(
a
n
,
b
m
) =
w
k
=-1
(
a
n
,
b
m
) =
z
Figure 3.33
Two 3D voxels in one of the
basic
spatial arrangements (
b
n
is lower positioned with
respect to
a
n
, and aligned with
a
n
along the
X
and
Z
axes) and the resulting values of the
w
i
,
j
,
k
coefficients
Suggestion:
As an example and suggestion for the derivation of the other cases, in the following
the closed form computation of the w
+
1
,
−
1
,
+
1
coefficient is detailed. To make easier the analytical
derivation, we assume the two voxels are cubes with side equal to T . The voxels have also the 3D
coordinates in the reference system shown in Figure
. 3.34.
According to Equation
3.21
, the relationship between voxels a
n
and b
m
can be written as
follows:
1
K
+
1
,
−
1
,
+
1
C
+
1
(
x
b
−
x
a
)
C
−
1
(
y
b
−
y
a
)
C
+
1
(
z
b
−
z
a
)d
b
d
a
.
w
+
1
,
−
1
,
+
1
(
a
n
,
b
m
)
=
a
n
b
m
(3.25)
According to the coordinates of Figure
. 3.34
for the two voxels, the integral of Equation
3.25
can
be split into
V
b
+
T
d
y
b
V
a
+
T
V
a
d
y
a
L
+
T
L
+
T
D
+
T
D
+
T
1
K
+
1
,
−
1
,
+
1
=
d
x
b
d
x
a
d
z
b
d
z
a
,
(3.26)
V
b
D
z
a
L
x
a
y
Va + T
a
n
Va
x
L
L+T
Vb + T
D+T
D
Vb
b
m
z
Figure 3.34
The two voxels are cubes with side equal to
T
and the coordinates given in the figure
