Biomedical Engineering Reference
In-Depth Information
Now, we can define a simple function, called an
Object Characteristic Func-
tion
, similar to function (2):
χ
(
p
)=1
,
fI
(
p
)
<T,
(7)
χ
(
p
)=0
,
otherwise
,
where
p
is a node of the triangulation (marked grid nodes in Figure 4a).
We can take it a step further, shown in Figure 4b, where we present a curve
that belongs to the transverse triangles. Observe that this curve approximates the
boundary we seek. This curve (or surface for 3D) can be obtained by isosurface
extraction methods and can be used to efficiently initialize the T-Surfaces model,
as we pointed out earlier.
If we take a grid resolution coarser than
r
min
, the isosurface method might
split the objects. Also, in [16, 15] it is supposed that the object boundaries are
closed and connected. These topological restrictions imply that we do not need to
search inside a generated connected components.
In this chapter we discard the mentioned scale and topological constraints.
As a consequence, the target topology may be corrupted. So a careful approach
will be required to deal with topological defects. An important point is the choice
of method for isosurfaces generation, which will be discussed next.
3.
ISOSURFACE EXTRACTION METHODS
Isosurface extraction is one of the most often-used techniques for visualization
of volume data sets. Due to the data type (time-varying or stationary) and the data
set size, many works have been done to improve the basic methods in this area
[12]. In this chapter we consider two kinds of isosurface generation methods:
marching and continuation.
In Marching Cubes, each surface-finding phase visits all cells of the volume,
normally by varying coordinate values in a triple
for
loop [14]. As each cell
that intersects the isosurface is found, the necessary polygon(s) representing the
portion of the isosurface within the cell is generated. There is no attempt to trace
the surface into neighboring cells. Space subdivision schemes (like Octree and
k-d-tree) have been used to avoid the computational cost of visiting cells that the
surface does not cut [27, 12].
Once the T-Surfaces grid is a CF one, the Tetra-Cubes is specially interesting
for this discussion [28]. As in the marching cubes, its search is linear: each cell of
the volume is visited and its simplices (tetrahedrons) are searched to find surface's
patches. Following marching cubes implementations, Tetra-Cubes uses auxiliary
structures based on the idea that the topology of the intersections between a plane
and a tetrahedron can be reduced to the three basic configurations pictured on
Figure 5.
