Biomedical Engineering Reference
In-Depth Information
Figure 10.1: Motion under curvature flow: A simple closed curve will (become
smoother and) disappear in a circular shape no matter how twisted it is.
a conformal metric given by
ds
g
=
g
(
|∇
I
(
C
(
q
))
|
)
2
(
dx
2
+
dy
2
)
,
(10.3)
where
g
(
·
) represents a monotonically decreasing function such that
g
(
x
)
→
0
as
x
→∞
, and
g
(
x
)
→
1as
x
→
0. A typical function for
g
(
x
) can be
1
1
+
x
.
(10.4)
g
(
x
)
=
This is plotted in Fig. 10.2. Using this metric, a new length definition in Rieman-
nian space is given by
1
g
(
|∇
I
(
C
(
q
))
|
)
|
C
(
q
)
|
dq
.
L
:
=
(10.5)
0
Then it is no longer necessary that the minimum path between two points in
this metric be a straight line, which is the case in the standard Euclidean metric.
The minimum path is now affected by the weighting function
g
(
·
). Two distant
points in the standard Euclidean metric can be considered to be very close to
each other in this metric if there exists a route along which values of
g
(
·
) are
nearer to zero. The steady state of the active contour is achieved by searching
