The name is absent

work on volume visualization and segmentation. We will also briefly describe our
approach, which is elaborated in the following chapters.

1.1 Visualizing Volume Data

Visualizing volume data requires interpreting the discretely-sampled volume as a con-
tinuous function. One approach is to interpret the space between samples as a tri-
Iinear interpolation function by treating the discrete samples as the interpolant at the
corners of the function domain. The entire domain of the volume is then covered by
piece-wise tri-linear functions. To display the function defined on the volume, we can
choose a threshold value and extract a level-set from the combined function. This
approach is a form of implicit modeling, which is a standard technique for modeling
shapes in computer graphics [1]. Given a function f(x,y,z) (our previously defined
volume function), the set of all points such that f(x,y, z) < 0 defines the one region
and the set of all points such that f(x,y,z,) > 0 defines a second, complementary
region (excluding the surface where the function is zero). For a multiple material
viewpoint, the implicit function f(x,y,z) partitions space into two materials; one
material where the function f(x, y, z) is positive and another where f(x, y, z) is nega-
tive. Numerous contouring method such as Marching Cubes [15], Dual Contouring [9]
and others [5, 12, 20] can then generate a polygonal surface that separates the positive
space from negative space.

Alternatively, the two spaces can be visualized using various volumetric approaches
implemented on the GPU [28, 4, 22]. The key ideas behind these approaches are that
the signed grid can be stored as a 3D texture and that a single texture load can be
used to evaluate f(x,y,z^y) via tri-linear interpolation at an arbitrary point. In prac-
tice, this tri-linear boundary surface provides better normals for shading and better
silhouettes than the discrete voxelized approach.

While the use of two signs to distinguish between two materials is simple and
elegant, the idea of using three or more signs to represent a partition of space into

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