Biomedical Engineering Reference
In-Depth Information
We can do a step further, shown in Fig. 7.8( b), where we present a curve
which belongs to the transverse triangles. Observe that this curve approximates
the boundary we seek. This curve (or surface for 3D) can be obtained by isosur-
face extraction methods and can be used to efficiently initialize the T-surfaces
model, as we already pointed out before.
If we take a grid resolution coarser than
r
min
, the isosurface method might
split the objects. Also, in [22, 29] 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 component.
In [63] 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
the method to be used for isosurfaces generation. In [22, 63] we consider two
kinds of isosurface generation methods: the marching ones and continuation
ones.
In marching cubes, each surface-finding phase visits all cells of the volume,
normally by varying coordinate values in a triple “for” loop [45]. As each cell
that intersects the isosurface is found, the necessary polygon(s) to represent
the portion of the isosurface within the cell is generated. There is no attempt
to trace the surface into neighboring cells. Space subdivision schemes (such as
Octree and k-d-tree) have been used to avoid the computational cost of visiting
cells that the surface does not cut [17, 64].
Once the T-surfaces grid is a CF one, the tetra-cubes is especially interesting
for this discussion [10]. As in the marching cubes, its search is linear: Each
cell of the volume is visited and its simplices (tetrahedrons) are searched to
find surfaces patches. Following marching cubes implementations, tetra-cubes
uses auxiliary structures based on the fact that the topology of the intersections
between a plane and a tetrahedron can be reduced to three basic configurations
pictured in Fig. 7.1 (Section 7.2.3).
Unlike tetra-cubes, continuation algorithms attempt to trace the surface
into neighboring simplices [1]. Thus, given a transverse simplex, the algorithm
searches its neighbors to continue the surface reconstruction. The key idea is
to generate the combinatorial manifold (set of transverse simplices) that holds
the isosurface.
The following definition will be useful. Let us suppose two simplices
σ
0
,
σ
1
,
which have a common face and the vertices
v
∈
σ
0
and
v
∈
σ
1
both opposite
