Biomedical Engineering Reference
In-Depth Information
An example of decreasing function g(x)
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
x
Figure 10.2:
Plot of the monotonically decreasing function
g
(
x
)
=
1
/
(1
+
x
).
for the minimum length curve in the modified Euclidean metric:
min
1
0
g
(
|∇
I
(
C
(
q
))
|
)
|
C
(
q
)
|
dq
.
(10.6)
Caselles
et al
. [4] have shown that this steady state is achieved by determining
how each point in the active contour should move along the normal direction in
order to decrease the length. The Euler-Lagrange of (10.6) gives the right-hand
side of (10.7), i.e., the desired steady state:
C
t
=
g
(
|∇
I
|
)
κ N
−
(
∇
g
(
|∇
I
|
)
·
N
)
N.
(10.7)
Two forces are represented by (10.7). The first is the curvature term multi-
plied by the weighting function
g
(
·
) and moves the curve toward object bound-
aries constrained by the curvature flow that ensure regularity during propaga-
tion. In application to shape modeling, the weighting factor could be an edge
indication function that has larger values in homogeneous regions and very small
values on the edges. Since (10.7) is slow, Caselles
et al
. [4] added a constant in-
flation term to speed up the convergence. The constant flow is given by
C
t
=
N
showing each point on the contour moves in the direction of its normal and on
