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y the field value and then summed. The final result is divided by the sum of the
field values, sum. The case where q is outside D (Figure 5.10) must also be handled.
There D(q) cannot be calculated as sum = 0, so 0 is returned as the field value.
Hence, the approximated scalar field fA ′ at q is defined as:
fA ′(q) =
D(q) = −
sum =
fA(q + D(q)) sum > 0
0 sum = 0
m
i=1
i=1
q−vi
g( ) · Tvi
R
sum
44
(5.6)
(5.7)
m q − vi
g( ) (5.8)
R
where fA is the scalar field before deformation and m is the number of grid vertices
which influence q. Here again g(x) is used as a smooth blending function. Since
fA ′(q) is 0 outside the voxel grid, the field has discontinuity (Figure 5.11b), however
as the iso-surface lies inside the voxels, this is acceptable. This technique produces
a real-time interactive approximation to the final deformation, which is described in
the next section.
Figure 5.10: The field value of q1 is calculated with the Wyvill function. Nevertheless,
the field value of q2 cannot be calculated because no vertex influences q2. In this case,
the system simply returns 0 as the field value of q2.