Biomedical Engineering Reference
In-Depth Information
For boundary extraction and segmentation tasks, in general we use a simpli-
fied version of Eq. (7.7) in which we take
µ
=
0. Hence, the model has no inertial
forces, which avoids oscillations near the equilibrium point [31].
Snake models can be extended to 3D, generating deformable surface models.
The traditional mathematical description for these models is given next.
7.2.2 Deformable Surfaces
Let us consider the following balloon-like model for closed surfaces [19]:
3
v
:
+
×
[0
,
1]
×
[0
,
1]
→
v
(
t
,
r
,
s
)
=
(
v
1
(
t
,
r
,
s
)
,v
2
(
t
,
r
,
s
)
,v
3
(
t
,
r
,
s
))
,
,
2
2
4
4
4
∂v
∂
t
−
ω
10
∂
∂
s
2
−
ω
01
∂
v
v
∂
r
2
+
2
ω
11
∂
∂
s
2
+
ω
20
∂
v
∂
s
4
+
ω
02
∂
v
v
∂
r
4
=
F
(
v
)
−
kn
(
v
)
,
(7.9)
∂
r
2
Initial estimation
:
v
(0
,
r
,
s
)
=
v
0
(
r
,
s
)
,
where
n
(
v
) is the normal (unitary) field over the surface
v
,
F
is the image force
field (may be normalized), and
k
is a force scale factor. The parameters
ω
ij
control the smoothing and flexibility of the model.
By using the internal pressure force (
kn
(
v
)), the model behaves like a balloon,
which is inflated, passing over regions in which the external force is too weak.
Consequently, the model becomes less sensitive to initialization, which is an
advantage over more traditional active models [6, 18].
If finite differences is used to numerically solve Eq. (7.9), the continuous
surface
v
(
r
,
s
) is discretized, generating a polygonal mesh. During the mesh
evolution, self-intersections must be avoided.
This problem has been efficiently addressed in the context of
discrete de-
formable models
. Differently from the above formulation, in which the mesh
arises due to a discretization of the continuous model (defined by Eq. (7.9)),
discrete surface models start from a two-dimensional mesh. The mesh nodes
are updated by a system of forces that resembles a discrete dynamical system.
The T-surfaces model is such a system, which is fundamental for our work. It is
summarized next.
7.2.3 T-Surfaces
The T-surfaces approach is composed of three components [49]: (1) a tetrahedral
decomposition (CF-triangulation) of the image domain
D
⊂
3
; (2) a particle
