Biomedical Engineering Reference
In-Depth Information
where
I
(
x, y
) is the image intensity at point (
x, y
),
G
σ
(
x,y
)
is the two-dimensional
Gaussian kernel with
σ
as standard deviation. Therefore, the external energy in
the traditional deformable model is defined as the minus value of the gradient of
the Gaussian smoothing image.
The goal of the active contour models is to find the local minima of
E
df
defined
in Eq. (1), based on the Euler-Lagrange principle. Equation (1) has a minimum
only if the Euler-Lagrange differential equation is satisfied as
∗
x
ss
(
s
)
−
∗
x
ssss
(
s
)
−
∗∇
E
ext
(
x
(
s
))=0
,
α
β
γ
(4)
where
x
ssss
(
s
) are the second and fourth derivatives of the curve with
respect to the parameter
s
.
In order to find the solution for Eq. (4), the deformablemodel is made dynamic
by defining
x
ss
(
s
) and
x
as the function of time
t
and
s
as follows:
x
t
(
s, t
)=
α
∗
x
ss
(
s, t
)
−
β
∗
x
ssss
(
s, t
)
−
γ
∗∇
E
ext
(
x
(
s, t
))
.
(5)
Then the partial derivative of
x
with respect to
t
is the same as the left side in Eq.
(4). When
becomes a stable value, its partial derivative with respect to
t
will
be 0 and we get the solution for Eq. (4). The solution to Eq. (5) can be achieved
by solving the discrete equations iteratively.
The success of the deformable model depends greatly on the design of the
external force. The external force of the traditional deformable model, defined as
the gradient magnitude of the original image, suffers from the following weakness:
x
1.
Initialization problem
: Because the external force is only around the
edges and the gradient magnitude of many other homogeneous places
will be zero, the capture range of this kind of deformable model is quite
limited. Therefore, the deformable model fails to capture the edges even
if initialized even just a small distance from the edges.
2.
Boundary concavities
: Because the forces in concave regions act in oppo-
site directions, they can become balanced in these areas and will therefore
fail to enable the deformable model to detect the edges.
3.
Topological changes
: There is noway for this deformablemodel to handle
topological changes in an image. Therefore, it is quite difficult to segment
multiply objects in the image without conducting special procedures.
1.2. Introduction to Essential Tools and Methods
This chapter is written for the purpose of introducing readers to the essen-
tial elements, tools, and methods of deformable models and to provide a basic
understanding as to when it is appropriate to utilize them in practical problems
