Biomedical Engineering Reference
In-Depth Information
Different from Tetra-Cubes, once the generation of a component is begun, the
algorithm runs until it is completed. However, the algorithm needs a set of seed
simplices to be able to generate all the components of an isosurface. This is an
important point when comparing continuation and marching methods.
If we do not have guesses about seeds, every simplex should be visited. Thus,
the computational complexity of both methods is the same (
O
(
N
), where
N
is the
number of simplices). However, if we know in advance that the target boundary is
connected we do not need to search inside a connected component. Consequently,
the computational cost is reduced if continuation methods are applied.
We have emphasized in previous works that the isosurfaces methods are useful
not only for initialization but also for reparameterization of T-Surfaces [16, 15]. In
these studies, continuation methods were used for both those steps due to the fact
that topological restrictions were supposed for the target. If these constraints are
discarded, new considerations must be made to decide the more suitable isosurface
method. This is demonstrated in the next section.
4.
T-SURFACES AND ISOSURFACE METHODS
The reparameterization of T-Surfaces gives the link between the isosurface
generation methods, described above, and the T-Surfaces model. To explain this,
let us take the Object Characteristic Function defined in Eq. (7). If we apply tetra-
cubes or continuation methods to this field, we get a set of piecewise linear (PL)
surfaces that involve the structures of interest. It can be verified that each obtained
connected component
M
has the following properties: (1) the intersection
σ
1
∩
σ
2
of two distinct triangles
σ
1
,σ
2
∈
M
is empty, a common vertex or edge of both
triangles; (2) an edge
τ
∈
M
is common to at most two triangles of
M
; (3)
M
is locally finite, that is, any compact subset of
3
meets only finitely many cells
of
M
.
A polygonal surface with such a property is called a Piecewise Linear Manifold
(
PLManifold
) [13]. From the reparameterization process of Section 2.1, we can see
that a T-Surface is also a PL Manifold. Thus, the isosurface extraction methods
can be used in a straightforward fashion to initialize T-Surfaces. Besides, the
Object Characteristic Function (Eq. (7)) gives the initial Characteristic Function.
However, what kind of isosurface method should be used? Based on the discussion
about tetra-cubes and PL generation (Section 3), we can conclude that if we do
not have the topological and scale restrictions (Section 2.2), marching methods are
more appropriate to initialize the T-Surfaces. In this case, it is not worthwhile to
attempt to reconstruct the surface into neighboring simplices because all simplices
should be visited to find surface patches.
However, for the T-Surfaces reparameterization (steps (1)-(4) of Section 2.1),
the situation is different. Now, each connected component is evolved at a time.
