Biomedical Engineering Reference
In-Depth Information
In the external energy
E
2
,
P
is a potential related with the features we seek.
For edge detection in a grayscale image a possible definition is [6]:
2
P
=−∇
I
,
(7.5)
where
I
is the image intensity.
The process of minimizing the functional given in (7.2) can be viewed from a
dynamic point of view by using the Lagrangian mechanics. This leads to dynamic
deformable models that unify the description of shape and motion. In these
models the deformable contour is viewed as a time-varying curve:
c
(
s
,
t
)
=
(
x
(
s
,
t
)
,
y
(
s
,
t
))
,
(7.6)
with a mass density
µ
and a damping density
γ
.
In this formulation, the Lagrange equations of motion for a snake with po-
tential energy given by expression (7.2) have the form [44, 50]:
µ
∂
2
c
∂
t
+
w
1
c
(
s
)
+
w
2
c
(
s
)
+∇
P
(
c
(
s
))
=
0
,
∂
t
2
+
γ
∂
c
(7.7)
where the first two terms represent the inertial and damping forces while the
third and fourth terms give the forces related to the internal energy (Eq. (7.2)).
The last term in Eq. (7.7) is the external force due to the external potential
P
in
expression (7.5). Equilibrium is achieved when the internal and external forces
balance and the contour comes to rest; which implies that:
2
c
/∂
t
2
∂
c
/∂
t
=
∂
=
0
.
(7.8)
In general, Eq. (7.7) does not have analytical solutions. Thus, numerical meth-
ods must be considered. Henceforth, in order to solve this equation, for an initial
closed contour, we have to discretize the snake in space and time by using
finite
differences
or
finite elements
methods, each of them with trade-offs between
performance and numerical efficiency [19,44]. We also have to use a
termination
condition
, based on Eq. (7.8), to stop the numerical interactions [44].
It is important to observe that the space
Ad
in expression (7.2) does not
include contours with more than one connected component. So the classical
snake model does not incorporate topological changes of the contour
c
during
its evolution given by Eq. (7.7). Besides, the contraction force generated by
the third and fourth terms in this equation is shape dependent and makes the
stabilization of the snake too dependent on the parameters
w
1
and
w
2
. While in
theory it is possible to compute a pair of proper weights of the internal energy
for each point, it is very difficult in practice [79].
