approach would produce a classification that has ripple-like effect (see Figure 1.4(b)).
Our method guarantees classification for all points within a voxel (see Figure 1.3(d))
and provides greater flexibility in sub-voxel classification (see Figure 1.3(e)), hence
capable of representing smooth inter-material boundary (see Figure 1.4(c)) . Also
note that both Tiede et al. and Hadwiger et al. tackled the problem from a visual-
ization perspective, where they improved multi-material rendering for one particular
visualization technique. Our approach is to present a geometric representation for
multi-material volume that can be used for various visualization methods.
Our approach is to generalize the idea of two-sign tri-linear contouring to that
of multi-sign tri-linear contouring with the goal of creating a partition of space into
disjoint materials whose boundaries are piecewise tri-linear surfaces. The advantage
of this approach is that the resulting material boundaries would better approximate
given boundaries and reduce rendering artifacts inherent in voxel rendering. Our
fundamental approach to multi-material contouring is viable for any type of implicit
function and not restricted to the tri-linear case. Figure 1.5 shows how our approach
reduces the artifacts of the input segmented volume.
1.2 Volume Segmentation
In conjunction with our contribution on visualization, we also present a simple method
for rapid segmentation. Segmentation is the process of partitioning a set voxels into
subsets, each of which is called a segment. More formally, segmentation can be
viewed as the partition of voxels coordinates. Let V be the input volume. V defines a
function in 3-Space where V : N3 → Y and Y can be a space of any finite dimension
(usually Y = R). The volume data is discrete so there exists a finite set X G N3
such that a point p is in X if and only if V(p) is well-defined. Finally, segmentation
is the construction of sets S1, S2,..., Sn where Si C X, and each of the Si is called a
segment. Two other common requirements are that Ui=ι...n5'i = X and Si ∩ Sj = 0
for i ψ j. These two conditions express the disjoint-union property for segmentation.