Graphics Reference
In-Depth Information
3DWWs are capable to quantitatively measure the relative spatial arrangement of two
extended sets of 3D points by computing 27 weights and organizing them in a 3
3
matrix. As a particular case, the 3DWWs computed between an extended 3D entity and itself
also account for intrinsic shape information.
By developing on the properties of integrals, it can be easily proved that weights
w
i
,
j
,
k
(
A
×
3
×
,
B
)
are reflexive (i.e.,
w
i
,
j
,
k
(
A
A
)) and invariant with respect to shifting and
scaling. In addition, 3DWWs are compositional, in that the walkthroughs between
A
and the
union
B
1
∪
,
B
)
=
w
−
i
,
−
j
,
−
k
(
B
,
B
2
can be derived by the linear combination of the 3DWWs between
A
and
B
1
and
A
and
B
2
. The demonstrations for the 2D case can be found in Berretti et al. (2003). According
to this, the integral of Equation 3.21 can be reduced to the linear combination of subintegrals
computed on any partition of
A
and
B
.
Computer Implementation
=
n
A
n
,
Considering the uniform voxelization of two 3D entities
A
and
B
(
A
and
=
m
B
m
), the integral of Eq. 3.21 can be replaced by a linear combination of terms
w
i
,
j
,
k
(
A
n
,
B
B
m
) computed on the voxels pairs
A
n
,
B
m
:
w
i
,
j
,
k
(
n
B
)
n
1
K
i
,
j
,
k
(
A
A
n
,
B
m
)
=
K
i
,
j
,
k
(
A
n
,
B
m
)
·
w
i
,
j
,
k
(
A
n
,
B
m
)
(3.24)
,
m
m
Two 3D voxels in the 3D space can be posed, one with respect to the other, in a set of 27
different mutual arrangements (
basic
arrangements). Because the
w
i
,
j
,
k
coefficients for these
basic arrangements can be computed in closed form using Equation 3.21, the relationships
between two extended entities can be reduced to the combination through Equation 3.24 of
the coefficients computed in the basic cases, thus avoiding the numerical evaluation of the
integral measure.
As an example, Figure 3.32 shows a 3D face with the voxelization of two facial stripes
(for the clarity of the visualization, the cube shape of a 3D voxel is approximated with
a sphere).
Figure 3.32
The 3D voxelization of a pair of facial stripes
