The name is absent

15

Figure 2.4 : The perspective view of a multi-material volume of size 33³ rendered
using GPU tri-linear contouring (a) and as polygonal contours generated by Dual
Contouring (b), showing the grid structure (c). (d) depicts the mesh generated from
Dual Contouring without the letters.

over the cell. For the three material case, t^k(x) and t^j(x) denote the largest and second
largest tri-linear interpolant at x. Using the difference of the top two interpolated
values will produce the exact gradient field, since we can view the local neighborhood
of a point on the two-material boundary as defined by the two dominant tri-linear
functions. Note that points where more than two materials meet are degenerate with
unknown gradients. Figure 2.1(a) show the gradient field for a two material pixel,
while Figure 2.3(a) shows the gradient field for a three material pixel.

2.2 Set Operations on Multi-material Contours

Given that the goal of this thesis is to apply multi-material contours to the problem
of representing segmentations, we next develop analogs of the set operations Union
and Intersection for multi-material contours. These operations will provide the basis
for the painting operations described in the next section.

2.2.1 Operations on Two Materials

One of the primary attractions of implicit modeling is the ease with which it can model
Boolean operations from constructive solid geometry [23, 18]. In the signed (two-

<<   22   23   24   25   26   27   28   >>

More intriguing information