Biomedical Engineering Reference
In-Depth Information
5.2. Using Characteristic Function Estimates to Define the STOP
and GO Field
The choice of the function
g
depends on the particular segmenting problem
we handle. Classic applications of deformable models deal with contour-based
spaces. Note that contour spaces are a degenerate version of the characteristic
function. In this sense they can be used in a STOP and GO formulation without
altering the scheme. However, in more complex problems the use of alternate
spaces like the filter space, likelihood map, or confidence rate spaces allow the use
of deformable models at any step of the classic classification pipeline. Take, for
instance, a color or texture feature space. In these spaces it is nearly unthinkable
to use a contour-based approach; however, it is simple to create either of the
alternatives introduced in this chapter. Therefore, we define the STOP term based
on the characteristic function estimate as
∇
(1
−
L
). Moreover, since the former
gradient is negligible outside a band around the contours, we can merge the two
STOP factors as follows:
∂
∂t
=
ακ L
∇
(1
−
L
)
,n>
n
+
V
0
(1
−
L
)
·
·
n
+
β<
·
n.
(10)
5.3. STOP and GO Numeric Formulation
Evolution of an initial snake Γ
0
under (10) is implemented using the Level
Sets [10] formulation. That is, given any initial surface (
φ
0
) properly defining the
interior of Γ
0
, the snake evolution at time
t
coincides with the 0 level contour of
the solution to:
∂φ
∂t
=(
α L
div(
∇
φ
)+
V
0
(1
−
L
))
|∇
∇
(1
−
L
)
,
φ
|
+
<
∇
φ>.
|∇
φ
|
The explicit Euler scheme we use in the numeric implementation of the former
equation is given by
φ
t
+1
=
φ
t
+(
α L
u
xx
u
y
−
2
u
xy
u
x
u
y
+
u
yy
u
x
|∇
+
|
2
u
+
V
0
(1
−
L
)
|∇
∇
(1
−
L
)
,
φ
t
|
+
<
∇
φ
t
>
)∆
t,
(11)
where
φ
t
stands for the solution at time
t
and derivatives are computed using
centered finite differences. Notice that the speed of convergence hinges upon the
magnitude of the time step ∆
t
; the higher it is, the less iterations the algorithm
requires. Accuracy is determined by
V
0
.
6.
EXPERIMENTAL RESULTS
In this section we provide different experiments results. In the first subsection
the general behavior of the STOP and GO active models is shown. The next
