Biomedical Engineering Reference
In-Depth Information
Figure 7.2:
Two snakes colliding with the inside grid nodes and snaxels marked.
Each mesh node is called a
node element
and each pair of connected nodes
v
i
,v
j
is called a
model element
.
The node elements are linked by springs, whose natural length we set to
zero. Hence, a tensile force can be defined by:
S
ij
,
where
S
ij
=
c
·
r
ij
,
α
i
=
(7.11)
j
c
is a scale factor and
r
ij
=
v
i
−
v
j
is the length of the corresponding model
element. The model also has a normal force which can be weighted as follows
[49]:
F
i
=
k
(sign
i
)
n
i
,
(7.12)
where
n
i
is the normal vector at node
i
,
k
is a scale factor, and sign
i
=+
1if
I
(
v
i
)
>
T
and sign
i
=−
1 otherwise (
T
is a threshold of the image
I
). This force
is used to push the model toward image edges until it is opposed by external
image forces.
The forces defined by Eqs. (7.11) and (7.12) are internal forces. The exter-
nal force is defined as a function of the image data, according to the interested
features. Several different approaches have been adopted according to the ap-
plication [34, 77]. In our case, it can be defined as follows:
image
::
force
::
f
i
=−
γ
i
∇
P
,
2
P
= ∇
I
.
(7.13)
The evolution of the surface is controlled by the following dynamical system:
i
+
h
i
α
i
t
t
+
F
i
+
f
i
t
(
t
+
t
)
i
t
(7.14)
v
=
v
,
where
h
i
is an evolution step.
During the T-surfaces evolution, some grid nodes become interior to a sur-
face. Such nodes are called
burnt nodes
and its identification is required by the
update of the characteristic function [49]. To deal with self-intersections, the
T-surfaces model incorporates an entropy condition:
Once a node is burnt it
