Biomedical Engineering Reference
In-Depth Information
The forces defined by the Eqs. (3)-(4) are internal forces. The external force
is defined as an image data function, according to the interested features. Several
different approaches have been adopted according to the application [18, 25]. In
our case, it can be defined as follows:
2
.
Image :: Force ::
f
i
=
−
γ
i
∇
P
;
P
=
∇
I
(5)
The surface evolution is controlled by the following discrete dynamical
system:
=
v
i
+
h
i
α
i
t
t
,
+
F
i
+
f
i
t
v
(
t
+∆
t
)
i
(6)
where
h
i
is an evolution step.
During the T-Surfaces evolution, some grid nodes become interior to the sur-
face. Such nodes are called
burnt nodes
, and their identification is required by
update of the characteristic function [10]. To deal with self-intersections, the T-
Surfaces model incorporates an entropy condition:
once a node is burnt it stays
burnt
. A termination condition is set based on the number of deformation steps in
which a simplex has remained a transverse one.
Now, it is important to make considerations about the T-Surfaces model, which
will guarantee efficiency by initializing the T-Surfaces through isosurface methods.
This is the starting point for our previous works [20, 17].
2.2. Initialization of T-Surfaces
The threshold
T
, used in the normal force definition (Eq. (4)), plays an im-
portant role in the T-Surfaces model [11, 10]. If it was not properly chosen, the
T-Surfaces can be frozen in a region far from the target(s) [20, 17].
The choice of
T
is critical when two target objects are closer, as shown in
Figure 3. In this example, the marked grid nodes (spheres) are those whose image
intensity falls below threshold
T
. They belong to the target objects that are the
two cells enclosed by the white-colored T-Snake.
For the T-Snakes model to accurately segment the pictured objects, it has to
burn the marked grid nodes between the two objects. However, the normal force
(given by Eq. (4)) changes its signal while the T-Snakes gets closer. So, the force
parameters in Eqs. (3)-(4) should be properly chosen to guarantee advance over
the narrow region. However, parameter choice remains an open problem in snake
models [26]. One solution is increasing the grid resolution as it controls T-Surface
flexibility. However, this increases the computational cost of the method.
To address the tradeoff between model flexibility and computational cost, we
proposed in [16, 15] to get a rough approximation of the target surfaces through
isosurfaces generation methods and then applying the T-Surfaces model. The
topological capabilities of T-Surfaces enable evolving the extracted isosurfaces in
