Biomedical Engineering Reference
In-Depth Information
2.1.2. GVF snake model
The classical active contour models have several limitations that weaken their
practicability in resolving image segmentation problems [46]. One limitation is
associated with initialization, that is, the initial contour must be close to the true
boundary or it will likely converge to a wrong result. To address this problem,
several novel methods have been proposed, including pressure forces [44] and
distance potentials [63]. The basic idea is to increase the capture range of the
external force fields and guide the contour toward the desired boundary. Another
limitation is the poor convergence of classical snakes to boundary concavities. The
GVF snake is an effective model that can be employed to solve this problem.
According to the Helmholtz theorem [64], rewriting Eq. (4) and replacing the
−∇
E
ext
with Θ, we can obtain
x
(
s
)
−
x
(
s
)+Θ=0
,
α
β
(7)
where Θ is the gradient vector flow, defined as Θ(
x, y
)=(
u
(
x, y
)
,v
(
x, y
)), and
it is usually generated by the energy functional
µ
u
x
+
u
y
+
v
x
+
v
y
+
|∇
f
|
2
·|
Θ
−∇
f
|
2
dxdy,
ε
=
(8)
where
is the gradient of the given image
f
(
x, y
), and it can also be substituted
with its Gaussian smoothing version as Eq. (3). Equation (8) is dominated by the
sum of squares of the partial derivatives of the GVF vector field when
|∇
f
|
|∇
f
|
is
small, yielding a slowly varying field. On the other hand, when
is large, the
second term dominates the integrand and is minimized by setting Θ=
∇
|∇
f
|
f
. This
results in the desired effect of keeping Θ nearly equal to the gradient of the edge
map when it is large, but forcing the field to be slowly varying in homogeneous
regions. The parameter
µ
is a weighting parameter for governing the tradeoff
between the two cases.
The physical nature of Eq. (8) is to create an energy field containing both the
degree of divergence and curl for a vector field. Using the calculus of variation
and the finite-difference method again, Θ can be calculated according to
∇
2
u
t
−
(
u
t
−
f
x
)(
f
x
+
f
y
)
,
u
t
+1
=
µ
(9)
∇
2
v
t
−
(
v
t
−
f
y
)(
f
x
+
f
y
)
.
v
t
+1
=
µ
Deliberately developed for overcoming the above-mentioned limitations, the GVF
snake can expand the capture range remarkably, so the initial contour need not
be as close to the true boundary as before. However, proper initialization is still
necessary, or else the snake may converge to a wrong result.
