Biomedical Engineering Reference
In-Depth Information
to define this decoupling, a characteristic function defining the region of interest
is used. Given the region of interest
R
, the characteristic function is defined as
follows:
1
if (
x, y
)
∈
R,
I
(
x, y
)=
0
otherwise
.
Let us assume that the evolving curve is outside the region of interest. In
that case, the GO term corresponds to an evolution process that shrinks the model
outside of
R
:
V
GO
=(1
−
I
)
·
V
0
·
n.
(4)
The above equation creates a constant “inward” motion toward the region of in-
terest. On the other hand, in order to define the “outward” motion we just have to
create a STOP field around the region of interest. In this sense, the easiest way to
create the STOP term is to use the “outward” gradient of any function, namely
g
,
locally defining the contours of the object of interest:
V
STOP
=
I
·∇
g,n
n.
(5)
Therefore, combining both terms, the STOP and GO motions, we can define
the whole evolution of a deformable model initialized outside the region of interest.
Hence, the formulation is as follows:
∂
∂t
=
<
I
·∇
g,n>
n
+
V
0
·
(1
−
I
)
·
n
.
(6)
Stop
Go
Note that the equilibrium solution is obtained if we ensure that the condition
V
0
≤
0 holds along the boundary of
R
. This formulation is mainly
governed by
V
0
. The change of this parameter allows different behaviors of the
curve: on one hand, we can ignore small activations of the potential
g
; on the
other, we can overpass areas with low value.
Figure 2 shows the basic decomposition of the force field in two sets. Figure 2a
shows the mask
I
defining the region of interest. The GO term is represented in
Figure 2b and the STOP term in Figure 2c.
<
∇
g,n>
Joining both terms results in the
complete deformation process.
3.2. Improving thePerformance of the STOPandGOBasicFormulation
The formulationwe have just described leads the curve to the desired boundary
defined by the characteristic function. As a result of this process, we will have
the borders of the region of interest accurately located. However, one of the
attractive features of the deformable models is the possibility of controlling the
smoothness and continuity of the resulting model. Since this regularity behavior is
only needed in the final stages of the snake deformation, we can bound its scope to
