Biomedical Engineering Reference
In-Depth Information
Fig. 4.2
Process of the iterative deformation of a surface using forces
ones described in Sect.
4.7
, is the first step. If a force is calculated for an edge or a
triangle, it has to be distributed to the corresponding vertices. The next step is the
movement of the vertices. The translation of each vertex is determined according to
the selected numerical implementation. If the volume or shape of the models changes
significantly, resampling processes are often needed. They adjust the mesh to ensure
the geometrical quality of the triangles and modify the mesh in a way that triangles
are small enough to fit through the desired structures. For example, the segmentation
of a small vein needs a constant refinement as the model grows along the tubular
structure. In a final step, a check is performed to see if the iteration process should
be terminated. This can be triggered for example after a certain number of steps or
once the vertices have reached equilibrium.
4.4.1 Implementation
There are several possibilities to implement the physical properties of a discrete de-
formable model. The forces may be modeled directly or as hookean spring forces,
as presented by Gilles and Magnenat-Thalmann [
55
] and Schmid [
10
]. In their for-
mulation, a translation from the point
x
i
to
x
j
is weighted by a factor
α
h
:
F
h
=
α
h
(
x
j
−
x
i
)
(4.7)
A common approach to describe the forces is by applying Newtonian physics, but
a popular alternative is to use Lagrangian dynamics [
11
]. Other approaches include
the Finite Difference Methods (FDM) [
2
] and Finite Element Methods (FEM) [
25
,
56
-
58
].
4.4.2 Mesh Types
Simple segmentation algorithms often create point clouds as a result. To initialize a
deformable model, an initial shape has to be provided, which can be derived from
point clouds. The conversion from a point cloud can be done using the marching
cube algorithm as proposed by Lorensen and Cline [
59
] in 1987. The resulting mesh
will be closed but also will contain a high number of faces, higher than needed in
