Biomedical Engineering Reference
In-Depth Information
In order to reduce computation time, in [21] the IFT algorithm was applied
at a lower image resolution. Another advantage of this procedure is that at the
lower resolution, small background artifacts become less significant relative to
the object(s) of interest. This philosophy, very well known in the field of multi-
resolution image processing [24], was also incorporated in our work, as we shall
see later.
Recently, in [17], we showed that most of the mentioned problems for ini-
tialization of deformable models through isosurfaces are naturally addressed when
using the T-Surfaces model. The reparameterization process of this model can deal
naturally with self-intersections and also efficiently perform topological changes
(merges/splits) over the surface(s) during the evolution process. These capabilities
enable automatic corrections of topological defects of the initial isosurface. We
next describe the T-Surfaces model.
2.1. T-Surfaces
The T-Surfaces approach is composed of three components [10]: (1) a tetra-
hedral decomposition (CF-Triangulation) of the image domain
D
⊂
3
; (2) a
particle model of the deformable surface; (3) a
Characteristic Function
,
χ
, de-
fined on the grid nodes that distinguishes the interior (
Int
(
S
)) from the exterior
(
Ext
(
S
)) of a surface
S
:
⊂
3
→{
0
,
1
}
χ
:
D
(2)
∈
where
χ
(
p
)=1if
p
Int
(
S
) and
χ
(
p
)=0otherwise, where
p
is a node of the
grid.
Following the classical nomenclature, a vertex of a tetrahedron is called a
node
and the collection of nodes and triangle edges is called the grid Γ
s
. A tetrahedron
(also called a simplex)
σ
is a
transverse
one if the characteristic function
χ
in
Eq. (2) changes its value in
σ
, analogously, for an edge.
In the framework composed by both the simplicial decomposition and the
characteristic function, the reparameterization of a surface is done by [10]: (1)
computing the intersections points of the surface with the grid; (2) finding the set
of transverse tetrahedrons (
Combinatorial Manifold
); (3) choosing an intersection
point for each transverse edge; (4) connecting the selected points.
In this reparameterization process, the transverse simplices play a central role.
Given such a simplex, we choose in each transverse edge an intersection point to
generate the new surface patch. In general, we will obtain three or four transverse
edges in each transverse tetrahedron (Figure 5). The former gives a triangular
patch, and the latter defines two triangles. So at the end of the step (4), a triangular
mesh is obtained. Each triangle is called a
triangular element
[10].
Taking a 2D example, let us consider the characteristic functions (
χ
1
and
χ
2
)
relative to the two contours depicted in Figure 2. The functions are defined on
the vertices of a CF-triangulation of the plane.
The vertices marked are those
