The name is absent

(a) Two material (b) Two mate- (c) Three material (d) Three material

binary classification rial +/- bilinear binary classification bilinear contour

contour

Figure 1.1 : 2D example of segmented volume visualization. 1.1(a) shows the input
segmented volume, and 1.1(b) renders a piecewise bi-linear function over the seg-
mented domain. Note that the bi-linear classification produces smoother boundary
but also maintains the correct partition. 1.1(c) and 1.1(d) are the three material
variants.

multiple material has received only limited attention. A simple representation of a
segmented volume map is to attach an auxiliary discrete volumetric grid whose voxels
each consist of a single material, which can be denoted with integers. The materials
comprising the volumetric grid can be stored as a 3D array of integers. While this
approach is fairly simple to implement, its drawbacks are obvious. The resulting
materials have blocky boundaries that are hard to shade in a natural manner and have
jagged silhouettes and inter-material boundaries [7] (see Figures 1.1(c) and 1.2(c)).

In Ju et al’s Dual Contouring, in which material indices are attached to a grid
to represent three or more materials [9]. In this approach, the boundaries between
materials are represented as polygonal surfaces created from point and normal data
stored on edges in the grid. Ju et al’s work assumes that auxiliary data (such as
edge intersections) is provided in addition to the volume. In our work, we make no
assumption about the data beyond the segmentation of the volume. Other researchers
have studied multiple material rendering in different contexts. Fujimori et al. and
Shammaa et al. focus on the related problem of extracting multi-material surfaces
from volume maps that do not have labeled voxels [6, 25]. Their extraction involves

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