Master's Thesis - Studierstube Augmented Reality Project - Graz ...

2.1 Technical Visualization
relatively easy.
The term ”iso”, from the greek word ισoζ which means ”similar”, is used in this
context to underline what the visualiztion is actually doing: It tries to produce a
triangle mesh by computing an iso-surface with a certain iso-value representing the
same property of the volume. A very popular method to extract a certain property in
a given volume is the so called marching cubes algorithm [Lorensen1987].
The underlying voxel grid is divided into small cubes where always four corners are
built up of four pixels in adjacent slices. The goal is to determine how this cube is
intersected by the desired object’s surface and thus the adjacent slices are intersected.
Choosing an iso-value will result rarely in one of the cubes corners directly; in most cases
the positions of the desired value will not be found without an interpolation along the
intersected edges of the cube. In the majority of cases a surface will intersect three or
more edges of these cubes consisting of vertices with values greater and smaller than the
iso-value. A connection of these intersections will lead to one or at most three triangles
of the desired iso-surface and subsequent processing of all cubes in the volumetric
dataset (”marching”) consequently unfolds the whole surface. Then the algorithm tries
to find the edges which are intersected. This reduces the intersection problem for eight
vertices to 2 8 = 256 cases which can be further generalized by rotations and symmetries
to 15 cases. Finding these intersected edges is rather easy. Every cube’s corner with a
value greater or equal the iso-value will be assigned to one and the others to zero. In
the first case the vertex is behind or on the surface and in the latter it is in front of it.
The creation of a lookup table for these cases enables an efficient build-up; the table
can be accessed via an adopted notation referring to each case by an index created from
a binary eight bit interpretation of the corner’s state. The exact surface intersection on
the edges can then be linear interpolated with the real image values where the vertices
are placed.
Finally the vertex normals for correct shading have to be investigated. These components
can be estimated along the tree coordinate axis by the gradient vectors at the
cube vertices and linearly interpolated depending on the light model.
Two ambiguous cases and certain incompatible triangulated cube-types in the above
described approach have been resolved by introducing new types of intersecting families
[Nielson1991; Montani1994; Chernyaev1995, and others]. Based on this idea many
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